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id="toc-Newtonian_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Newtonian_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Newtonian mechanics</span> </div> </a> <ul id="toc-Newtonian_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Remarks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Remarks"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Remarks</span> </div> </a> <ul id="toc-Remarks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Special_relativity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Special relativity</span> </div> </a> <ul id="toc-Special_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-inertial_frames" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-inertial_frames"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Non-inertial frames</span> </div> </a> <button aria-controls="toc-Non-inertial_frames-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Non-inertial frames subsection</span> </button> <ul id="toc-Non-inertial_frames-sublist" class="vector-toc-list"> <li id="toc-General_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>General relativity</span> </div> </a> <ul id="toc-General_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inertial_frames_and_rotation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inertial_frames_and_rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Inertial frames and rotation</span> </div> </a> <ul id="toc-Inertial_frames_and_rotation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primed_frames" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primed_frames"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Primed frames</span> </div> </a> <ul id="toc-Primed_frames-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Separating_non-inertial_from_inertial_reference_frames" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Separating_non-inertial_from_inertial_reference_frames"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Separating non-inertial from inertial reference frames</span> </div> </a> <button aria-controls="toc-Separating_non-inertial_from_inertial_reference_frames-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Separating non-inertial from inertial reference frames subsection</span> </button> <ul id="toc-Separating_non-inertial_from_inertial_reference_frames-sublist" class="vector-toc-list"> <li id="toc-Theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Theory</span> </div> </a> <ul id="toc-Theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Simple_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Simple example</span> </div> </a> <ul id="toc-Simple_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additional_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Additional example</span> </div> </a> <ul id="toc-Additional_example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Inertial frame of reference</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 52 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-52" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">52 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Inersiestelsel" title="Inersiestelsel – Afrikaans" lang="af" hreflang="af" data-title="Inersiestelsel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Inertialsystem" title="Inertialsystem – Alemannic" lang="gsw" hreflang="gsw" data-title="Inertialsystem" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A5%D8%B7%D8%A7%D8%B1_%D9%85%D8%B1%D8%AC%D8%B9%D9%8A_%D9%82%D8%B5%D9%88%D8%B1%D9%8A" title="إطار مرجعي قصوري – Arabic" lang="ar" hreflang="ar" data-title="إطار مرجعي قصوري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Ftal%C9%99t_hesablama_sistemi" title="Ətalət hesablama sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Ətalət hesablama sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9C%E0%A6%A1%E0%A6%BC_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%B8%E0%A6%99%E0%A7%8D%E0%A6%97_%E0%A6%95%E0%A6%BE%E0%A6%A0%E0%A6%BE%E0%A6%AE%E0%A7%8B" title="জড় প্রসঙ্গ কাঠামো – Bangla" lang="bn" hreflang="bn" data-title="জড় প্রসঙ্গ কাঠামো" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%BD%D0%B5%D1%80%D1%86%D1%8B%D1%8F%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%81%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%B0%D0%B4%D0%BB%D1%96%D0%BA%D1%83" title="Інерцыяльная сістэма адліку – Belarusian" lang="be" hreflang="be" data-title="Інерцыяльная сістэма адліку" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%86%D0%BD%D1%8D%D1%80%D1%86%D1%8B%D0%B9%D0%BD%D0%B0%D1%8F_%D1%81%D1%8B%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%B0%D0%B4%D0%BB%D1%96%D0%BA%D1%83" title="Інэрцыйная сыстэма адліку – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Інэрцыйная сыстэма адліку" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%9C%E0%A4%A1%E0%A4%BC_%E0%A4%AA%E0%A4%B0%E0%A4%B8%E0%A4%82%E0%A4%97_%E0%A4%AA%E0%A4%B0%E0%A4%A8%E0%A4%BE%E0%A4%B2%E0%A4%BF" title="जड़ परसंग परनालि – Bhojpuri" lang="bh" hreflang="bh" data-title="जड़ परसंग परनालि" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Inercijalni_referentni_okvir" title="Inercijalni referentni okvir – Bosnian" lang="bs" hreflang="bs" data-title="Inercijalni referentni okvir" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_de_refer%C3%A8ncia_inercial" title="Sistema de referència inercial – Catalan" lang="ca" hreflang="ca" data-title="Sistema de referència inercial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D0%B0%D0%BB%D0%BB%C4%83_%D0%BF%D1%83%C3%A7%D0%BB%D0%B0%D0%B2_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Инерциаллă пуçлав тытăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Инерциаллă пуçлав тытăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Inerci%C3%A1ln%C3%AD_vzta%C5%BEn%C3%A1_soustava" title="Inerciální vztažná soustava – Czech" lang="cs" hreflang="cs" data-title="Inerciální vztažná soustava" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Inertialsystem" title="Inertialsystem – Danish" lang="da" hreflang="da" data-title="Inertialsystem" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Inertialsystem" title="Inertialsystem – German" lang="de" hreflang="de" data-title="Inertialsystem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Inertsiaals%C3%BCsteem" title="Inertsiaalsüsteem – Estonian" lang="et" hreflang="et" data-title="Inertsiaalsüsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%B4%CF%81%CE%B1%CE%BD%CE%B5%CE%B9%CE%B1%CE%BA%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CE%B1%CE%BD%CE%B1%CF%86%CE%BF%CF%81%CE%AC%CF%82" title="Αδρανειακό σύστημα αναφοράς – Greek" lang="el" hreflang="el" data-title="Αδρανειακό σύστημα αναφοράς" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_de_referencia_inercial" title="Sistema de referencia inercial – Spanish" lang="es" hreflang="es" data-title="Sistema de referencia inercial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Inercia_kadro_de_referenco" title="Inercia kadro de referenco – Esperanto" lang="eo" hreflang="eo" data-title="Inercia kadro de referenco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erreferentzia-sistema_inertzial" title="Erreferentzia-sistema inertzial – Basque" lang="eu" hreflang="eu" data-title="Erreferentzia-sistema inertzial" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%B1%D8%AC%D8%B9_%D9%84%D8%AE%D8%AA" title="دستگاه مرجع لخت – Persian" lang="fa" hreflang="fa" data-title="دستگاه مرجع لخت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9f%C3%A9rentiel_galil%C3%A9en" title="Référentiel galiléen – French" lang="fr" hreflang="fr" data-title="Référentiel galiléen" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_inercial" title="Sistema inercial – Galician" lang="gl" hreflang="gl" data-title="Sistema inercial" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B4%80%EC%84%B1_%EC%A2%8C%ED%91%9C%EA%B3%84" title="관성 좌표계 – Korean" lang="ko" hreflang="ko" data-title="관성 좌표계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%B6%D5%A5%D6%80%D6%81%D5%AB%D5%A1%D5%AC_%D5%B0%D5%A1%D5%B7%D5%BE%D5%A1%D6%80%D5%AF%D5%B4%D5%A1%D5%B6_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Իներցիալ հաշվարկման համակարգ – Armenian" lang="hy" hreflang="hy" data-title="Իներցիալ հաշվարկման համակարգ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9C%E0%A4%A1%E0%A4%BC%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A5%80%E0%A4%AF_%E0%A4%AB%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%AE" title="जड़त्वीय फ्रेम – Hindi" lang="hi" hreflang="hi" data-title="जड़त्वीय फ्रेम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Inercijski_referentni_okvir" title="Inercijski referentni okvir – Croatian" lang="hr" hreflang="hr" data-title="Inercijski referentni okvir" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kerangka_acuan_inersia" title="Kerangka acuan inersia – Indonesian" lang="id" hreflang="id" data-title="Kerangka acuan inersia" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_di_riferimento_inerziale" title="Sistema di riferimento inerziale – Italian" lang="it" hreflang="it" data-title="Sistema di riferimento inerziale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%9C%E1%83%94%E1%83%A0%E1%83%AA%E1%83%98%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%90%E1%83%97%E1%83%95%E1%83%9A%E1%83%98%E1%83%A1_%E1%83%A1%E1%83%98%E1%83%A1%E1%83%A2%E1%83%94%E1%83%9B%E1%83%90" title="ინერციული ათვლის სისტემა – Georgian" lang="ka" hreflang="ka" data-title="ინერციული ათვლის სისტემა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Inerciarendszer" title="Inerciarendszer – Hungarian" lang="hu" hreflang="hu" data-title="Inerciarendszer" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D0%B0%D0%BB_%D1%82%D0%BE%D0%BE%D0%BB%D0%BB%D1%8B%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Инерциал тооллын систем – Mongolian" lang="mn" hreflang="mn" data-title="Инерциал тооллын систем" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inertiaalstelsel" title="Inertiaalstelsel – Dutch" lang="nl" hreflang="nl" data-title="Inertiaalstelsel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%85%A3%E6%80%A7%E7%B3%BB" title="慣性系 – Japanese" lang="ja" hreflang="ja" data-title="慣性系" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Treghetssystem" title="Treghetssystem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Treghetssystem" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tregleikssystem" title="Tregleikssystem – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Tregleikssystem" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Inersial_sanoq_sistemasi" title="Inersial sanoq sistemasi – Uzbek" lang="uz" hreflang="uz" data-title="Inersial sanoq sistemasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_inercjalny" title="Układ inercjalny – Polish" lang="pl" hreflang="pl" data-title="Układ inercjalny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Referencial_inercial" title="Referencial inercial – Portuguese" lang="pt" hreflang="pt" data-title="Referencial inercial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_de_referin%C8%9B%C4%83_iner%C8%9Bial" title="Sistem de referință inerțial – Romanian" lang="ro" hreflang="ro" data-title="Sistem de referință inerțial" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BE%D1%82%D1%81%D1%87%D1%91%D1%82%D0%B0" title="Инерциальная система отсчёта – Russian" lang="ru" hreflang="ru" data-title="Инерциальная система отсчёта" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Inerci%C3%A1lna_vz%C5%A5a%C5%BEn%C3%A1_s%C3%BAstava" title="Inerciálna vzťažná sústava – Slovak" lang="sk" hreflang="sk" data-title="Inerciálna vzťažná sústava" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Inercialni_opazovalni_sistem" title="Inercialni opazovalni sistem – Slovenian" lang="sl" hreflang="sl" data-title="Inercialni opazovalni sistem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D1%80%D0%B5%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B5" title="Инерцијални систем референције – Serbian" lang="sr" hreflang="sr" data-title="Инерцијални систем референције" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Inercijski_referentni_okvir" title="Inercijski referentni okvir – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Inercijski referentni okvir" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Inertiaalikoordinaatisto" title="Inertiaalikoordinaatisto – Finnish" lang="fi" hreflang="fi" data-title="Inertiaalikoordinaatisto" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Inertialsystem" title="Inertialsystem – Swedish" lang="sv" hreflang="sv" data-title="Inertialsystem" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A3%E0%B8%AD%E0%B8%9A%E0%B8%AD%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%AD%E0%B8%B4%E0%B8%87%E0%B9%80%E0%B8%89%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%A2" title="กรอบอ้างอิงเฉื่อย – Thai" lang="th" hreflang="th" data-title="กรอบอ้างอิงเฉื่อย" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Eylemsiz_referans_%C3%A7er%C3%A7evesi" title="Eylemsiz referans çerçevesi – Turkish" lang="tr" hreflang="tr" data-title="Eylemsiz referans çerçevesi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D0%B5%D1%80%D1%86%D1%96%D0%B9%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%B2%D1%96%D0%B4%D0%BB%D1%96%D0%BA%D1%83" title="Інерційна система відліку – Ukrainian" lang="uk" hreflang="uk" data-title="Інерційна система відліку" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_quy_chi%E1%BA%BFu_qu%C3%A1n_t%C3%ADnh" title="Hệ quy chiếu quán tính – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ quy chiếu quán tính" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%85%A3%E6%80%A7%E5%8F%83%E8%80%83%E7%B3%BB" title="慣性參考系 – Cantonese" lang="yue" hreflang="yue" data-title="慣性參考系" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%83%AF%E6%80%A7%E5%8F%82%E8%80%83%E7%B3%BB" title="惯性参考系 – Chinese" lang="zh" hreflang="zh" data-title="惯性参考系" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q192735#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-left:0.9em;padding-right:0.9em;"><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"></span><div class="sidebar-caption" style="font-size:90%;padding:0.6em 0;font-style:italic;"><a href="/wiki/Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading" style="font-weight: bold; display:block;margin-bottom:1.0em;"> <div class="hlist"> <ul><li><a href="/wiki/History_of_classical_mechanics" title="History of classical mechanics">History</a></li> <li><a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul> </div></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li> <li><a href="/wiki/Analytical_dynamics" class="mw-redirect" title="Analytical dynamics">Dynamics</a></li> <li><a href="/wiki/Classical_field_theory" title="Classical field theory">Field theory</a></li> <li><a href="/wiki/Kinematics" title="Kinematics">Kinematics</a></li> <li><a href="/wiki/Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li> <li><a href="/wiki/Statics" title="Statics">Statics</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Acceleration" title="Acceleration">Acceleration</a></li> <li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li> <li><a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">potential</a></li></ul></li> <li><a href="/wiki/Force" title="Force">Force</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a class="mw-selflink selflink">Inertial frame of reference</a></li> <li><a href="/wiki/Impulse_(physics)" title="Impulse (physics)">Impulse</a></li> <li><span class="nowrap"><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><br /><a href="/wiki/Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Mechanical work</a></li> <li><br /><a href="/wiki/Moment_(physics)" title="Moment (physics)">Moment</a></li> <li><a href="/wiki/Momentum" title="Momentum">Momentum</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Speed" title="Speed">Speed</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><span class="nowrap"><a class="mw-selflink selflink">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li> <li><a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li> <li><a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Edward_Routh" title="Edward Routh">Routh</a></li> <li><a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville</a></li> <li><a href="/wiki/Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Bernard_Koopman" title="Bernard Koopman">Koopman</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a 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class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Classical_mechanics" title="Category:Classical mechanics">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output 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navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a> and <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, an <b><a href="/wiki/Inertia" title="Inertia">inertial</a> frame of reference</b> (also called <b>inertial reference frame</b>, <b>inertial frame</b>, <b>inertial space</b>, or <b>Galilean reference frame</b>) is a <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a> that is not undergoing any <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>. It is a frame in which an isolated <a href="/wiki/Physical_object" title="Physical object">physical object</a> — an object with zero <a href="/wiki/Net_force" title="Net force">net force</a> acting on it — is perceived to move with a constant velocity (it might be a zero velocity) or, equivalently, it is a frame of reference in which <a href="/wiki/Newton%27s_laws_of_motion#Newton's_first_law" title="Newton's laws of motion">Newton's first law of motion</a> holds. <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> All inertial frames are in a state of constant, <a href="/wiki/Rectilinear_motion" class="mw-redirect" title="Rectilinear motion">rectilinear motion</a> with respect to one another; in other words, an <a href="/wiki/Accelerometer" title="Accelerometer">accelerometer</a> moving with any of them would detect zero acceleration. </p><p>It has been observed that celestial objects which are far away from other objects and which are in uniform motion with respect to the <a href="/wiki/Cosmic_microwave_background#Features" title="Cosmic microwave background">cosmic microwave background radiation</a> maintain such uniform motion.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Measurements in one inertial frame can be converted to measurements in another by a simple transformation, the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a> in Newtonian physics and the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> in special relativity. <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Analytical_mechanics" title="Analytical mechanics">analytical mechanics</a>, an inertial frame of reference can be defined as a frame of reference that describes time and space <a href="/wiki/Homogeneity_(physics)" title="Homogeneity (physics)">homogeneously</a>, <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropically</a>, and in a time-independent manner. <sup id="cite_ref-LandauMechanics_4-0" class="reference"><a href="#cite_note-LandauMechanics-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/General_relativity" title="General relativity">general relativity</a> </p> <ul><li>in any region small enough for the curvature of spacetime and <a href="/wiki/Tidal_forces" class="mw-redirect" title="Tidal forces">tidal forces</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> to be negligible, one can find a set of inertial frames that approximately describe that region.<sup id="cite_ref-Einstein0_6-0" class="reference"><a href="#cite_note-Einstein0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>the <a href="/wiki/Physics" title="Physics">physics of a system</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> can be described in terms of an inertial frame without causes external to the respective system, with the exception of an apparent effect due to so-called distant masses.<sup id="cite_ref-Ferraro_8-0" class="reference"><a href="#cite_note-Ferraro-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <p>In a <i><a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial reference frame</a></i>, viewed from a classical physics and special relativity perspective, the interactions between the fundamental constituents of the observable universe (the <a href="/wiki/Physics" title="Physics">physics of a system</a>) vary depending on the acceleration of that frame with respect to an inertial frame. Viewed from this perspective and due to the phenomenon of inertia the 'usual' physical forces between two bodies have to be supplemented by <a href="/wiki/Fictitious_force" title="Fictitious force">apparently sourceless inertial forces</a>.<sup id="cite_ref-Rothman_9-0" class="reference"><a href="#cite_note-Rothman-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Borowitz_10-0" class="reference"><a href="#cite_note-Borowitz-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Viewed from a general relativity theory perspective appearing inertial forces (the supplementary external causes) are attributed to <a href="/wiki/Geodesic_(general_relativity)" class="mw-redirect" title="Geodesic (general relativity)">geodesic motion in spacetime</a>. </p><p>In classical physics, for example, a ball dropped towards the ground does not move exactly straight down because the <a href="/wiki/Earth" title="Earth">Earth</a> is rotating. This means the frame of reference of an observer on Earth is not inertial. As a consequence the science of physics has to take into account the <a href="/wiki/Coriolis_effect" class="mw-redirect" title="Coriolis effect">Coriolis effect</a>—an <a href="/wiki/Apparent_force" class="mw-redirect" title="Apparent force">apparent force</a>— to predict the respective small horizontal motion. Another example of an apparent force appearing in rotating reference frames concerns the <a href="/wiki/Centrifugal_effect" class="mw-redirect" title="Centrifugal effect">centrifugal effect</a>, the centrifugal force. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="A_set_of_frames_where_the_laws_of_physics_are_simple">A set of frames where the laws of physics are simple</h2></div> <p>The motion of a body can only be described relative to something else—other bodies, observers, or a set of spacetime coordinates. These are called <a href="/wiki/Frame_of_reference" title="Frame of reference">frames of reference</a>. If the coordinates are chosen badly, the laws of motion may appear to be more complex than necessary. For example, suppose a free body that has no external forces acting on it is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, even though there are no forces on it. However, a frame of reference can always be chosen in which it remains stationary. Similarly, if space is not described uniformly or time independently, a coordinate system could describe the simple flight of a free body in space as a complicated zig-zag in its coordinate system. Indeed, an intuitive summary of inertial frames can be given: in an inertial reference frame, the laws of mechanics take their simplest form.<sup id="cite_ref-LandauMechanics_4-1" class="reference"><a href="#cite_note-LandauMechanics-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p> According to the first postulate of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform <a href="/wiki/Translation_(physics)" class="mw-redirect" title="Translation (physics)">translation</a>: <span class="anchor" id="principle"></span><sup id="cite_ref-Einstein_11-0" class="reference"><a href="#cite_note-Einstein-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style></p><blockquote class="templatequote"><p>Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.</p><div class="templatequotecite">— <cite>Albert Einstein: <i>The foundation of the general theory of relativity</i>, Section A, §1</cite></div></blockquote> <p>This simplicity manifests itself in that inertial frames have self-contained physics without the need for external causes, while physics in <b>non-inertial frames</b> have external causes.<sup id="cite_ref-Ferraro_8-1" class="reference"><a href="#cite_note-Ferraro-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The principle of simplicity can be used within Newtonian physics as well as in special relativity; see Nagel<sup id="cite_ref-Nagel_12-0" class="reference"><a href="#cite_note-Nagel-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and also Blagojević.<sup id="cite_ref-Blagojević_13-0" class="reference"><a href="#cite_note-Blagojević-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The laws of Newtonian mechanics do not always hold in their simplest form...If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a 'force' pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton's laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity:<br />   The laws of mechanics have the same form in all inertial frames.</p><div class="templatequotecite">— <cite>Milutin Blagojević: <i>Gravitation and Gauge Symmetries</i>, p. 4</cite></div></blockquote> <p>In practical terms, the equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment. Otherwise, the differences would set up an absolute standard reference frame.<sup id="cite_ref-Einstein2_14-0" class="reference"><a href="#cite_note-Einstein2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Feynman_15-0" class="reference"><a href="#cite_note-Feynman-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> of symmetry transformations, of which the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a> are a subgroup.<sup id="cite_ref-Wachter_16-0" class="reference"><a href="#cite_note-Wachter-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In Newtonian mechanics, which can be viewed as a limiting case of special relativity in which the speed of light is infinite, inertial frames of reference are related by the <a href="/wiki/Galilean_group" class="mw-redirect" title="Galilean group">Galilean group</a> of symmetries. </p> <div class="mw-heading mw-heading2"><h2 id="Newton's_inertial_frame_of_reference"><span id="Newton.27s_inertial_frame_of_reference"></span>Newton's inertial frame of reference</h2></div> <div class="mw-heading mw-heading3"><h3 id="Absolute_space">Absolute space</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Absolute_space_and_time" title="Absolute space and time">Absolute space and time</a></div> <p>Newton posited an absolute space considered well approximated by a frame of reference stationary relative to the <a href="/wiki/Fixed_stars" title="Fixed stars">fixed stars</a>. An inertial frame was then one in uniform translation relative to absolute space. However, some scientists (called "relativists" by Mach<sup id="cite_ref-Mach_17-0" class="reference"><a href="#cite_note-Mach-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>), even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced. </p><p>Indeed, the expression <i>inertial frame of reference</i> (<a href="/w/index.php?title=Template:Lang-de&action=edit&redlink=1" class="new" title="Template:Lang-de (page does not exist)">Template:Lang-de</a>) was coined by <a href="/wiki/Ludwig_Lange_(physicist)" title="Ludwig Lange (physicist)">Ludwig Lange</a> in 1885, to replace Newton's definitions of "absolute space and time" by a more <a href="/wiki/Operational_definition#Science" title="Operational definition">operational definition</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Barbour_19-0" class="reference"><a href="#cite_note-Barbour-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> As translated by Iro, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9a9KAAAAMAAJ&q=Inertialsystem+inauthor:%22von+Laue%22&dq=Inertialsystem+inauthor:%22von+Laue%22&lr=&as_brr=0&pgis=1">Lange proposed</a> the following definition:<sup id="cite_ref-Iro_20-0" class="reference"><a href="#cite_note-Iro-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.</p></blockquote> <p>A discussion of Lange's proposal can be found in Mach.<sup id="cite_ref-Mach_17-1" class="reference"><a href="#cite_note-Mach-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p> The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojević:<sup id="cite_ref-Blagojević2_21-0" class="reference"><a href="#cite_note-Blagojević2-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"></p><blockquote class="templatequote"> <ul><li>The existence of absolute space contradicts the internal logic of classical mechanics since, according to Galilean principle of relativity, none of the inertial frames can be singled out.</li> <li>Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.</li> <li>Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.</li></ul> <div class="templatequotecite">— <cite>Milutin Blagojević: <i>Gravitation and Gauge Symmetries</i>, p. 5</cite></div></blockquote> <p>The utility of operational definitions was carried much further in the special theory of relativity.<sup id="cite_ref-Woodhouse0_22-0" class="reference"><a href="#cite_note-Woodhouse0-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Some historical background including Lange's definition is provided by DiSalle, who says in summary:<sup id="cite_ref-DiSalle_23-0" class="reference"><a href="#cite_note-DiSalle-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. For the laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.</p><div class="templatequotecite">— <cite><a rel="nofollow" class="external text" href="http://plato.stanford.edu/archives/sum2002/entries/spacetime-iframes/#Oth">Robert DiSalle <i>Space and Time: Inertial Frames</i></a></cite></div></blockquote> <div class="mw-heading mw-heading3"><h3 id="Newtonian_mechanics">Newtonian mechanics</h3></div> <p>Classical theories that use the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a> postulate the equivalence of all inertial reference frames. Some theories may even postulate the existence of a <a href="/wiki/Privileged_frame" class="mw-redirect" title="Privileged frame">privileged frame</a> which provides <a href="/wiki/Absolute_space" class="mw-redirect" title="Absolute space">absolute space</a> and <a href="/wiki/Absolute_time" class="mw-redirect" title="Absolute time">absolute time</a>. The Galilean transformation transforms coordinates from one inertial reference frame, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644ae690160e658898a141e568a7fb0ee6040004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }"></span>, to another, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} ^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} ^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec764c39d8df68bbe99c1fc4aa74d55db8f28b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.74ex; height:2.509ex;" alt="{\displaystyle \mathbf {s} ^{\prime }}"></span>, by simple addition or subtraction of coordinates: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} ^{\prime }=\mathbf {r} -\mathbf {r} _{0}-\mathbf {v} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} ^{\prime }=\mathbf {r} -\mathbf {r} _{0}-\mathbf {v} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5732e938834b3f280a5190cae88ce0a55719e13d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.075ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} ^{\prime }=\mathbf {r} -\mathbf {r} _{0}-\mathbf {v} t}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{\prime }=t-t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{\prime }=t-t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a281e089522999211e80a4f9521a94ac744fe45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.197ex; height:2.843ex;" alt="{\displaystyle t^{\prime }=t-t_{0}}"></span></dd></dl> <p>where <b>r</b><sub>0</sub> and <i>t</i><sub>0</sub> represent shifts in the origin of space and time, and <b>v</b> is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time <i>t</i><sub>2</sub> − <i>t</i><sub>1</sub> between two events is the same for all reference frames and the <a href="/wiki/Distance" title="Distance">distance</a> between two simultaneous events (or, equivalently, the length of any object, |<b>r</b><sub>2</sub> − <b>r</b><sub>1</sub>|) is also the same. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Inertial_frames.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Inertial_frames.svg/250px-Inertial_frames.svg.png" decoding="async" width="250" height="277" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Inertial_frames.svg/375px-Inertial_frames.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Inertial_frames.svg/500px-Inertial_frames.svg.png 2x" data-file-width="704" data-file-height="779" /></a><figcaption>Figure 1: Two frames of reference moving with relative velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {\vec {v}}{}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\stackrel {\vec {v}}{}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc98c41db839891a8123500ff79707ea00ee367a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0.85ex; height:2.676ex;" alt="{\displaystyle {\stackrel {\vec {v}}{}}}"></span>. Frame <i>S' </i> has an arbitrary but fixed rotation with respect to frame <i>S</i>. They are both <i>inertial frames</i> provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.</figcaption></figure> <p>Within the realm of Newtonian mechanics, an <a href="/wiki/Inertia" title="Inertia">inertial</a> frame of reference, or inertial reference frame, is one in which <a href="/wiki/Newton%27s_laws_of_motion#Newton's_first_law" title="Newton's laws of motion">Newton's first law of motion</a> is valid.<sup id="cite_ref-Moeller_24-0" class="reference"><a href="#cite_note-Moeller-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> However, the <a href="#principle">principle of special relativity</a> generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law. </p><p>Newton viewed the first law as valid in any reference frame that is in uniform motion relative to the fixed stars;<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> that is, neither rotating nor accelerating relative to the stars.<sup id="cite_ref-Resnick_26-0" class="reference"><a href="#cite_note-Resnick-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Today the notion of "<a href="/wiki/Absolute_space" class="mw-redirect" title="Absolute space">absolute space</a>" is abandoned, and an inertial frame in the field of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> is defined as:<sup id="cite_ref-Takwale_27-0" class="reference"><a href="#cite_note-Takwale-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Woodhouse_28-0" class="reference"><a href="#cite_note-Woodhouse-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.</p></blockquote> <p>Hence, with respect to an inertial frame, an object or body <a href="/wiki/Acceleration" title="Acceleration">accelerates</a> only when a physical <a href="/wiki/Force" title="Force">force</a> is applied, and (following <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's first law of motion</a>), in the absence of a net force, a body at <a href="/wiki/Rest_(physics)" class="mw-redirect" title="Rest (physics)">rest</a> will remain at rest and a body in motion will continue to move uniformly—that is, in a straight line and at constant <a href="/wiki/Speed" title="Speed">speed</a>. Newtonian inertial frames transform among each other according to the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean group of symmetries</a>. </p><p>If this rule is interpreted as saying that <a href="/wiki/Straight-line_motion" class="mw-redirect" title="Straight-line motion">straight-line motion</a> is an indication of zero net force, the rule does not identify inertial reference frames because straight-line motion can be observed in a variety of frames. If the rule is interpreted as defining an inertial frame, then we have to be able to determine when zero net force is applied. The problem was summarized by Einstein:<sup id="cite_ref-Einstein5_29-0" class="reference"><a href="#cite_note-Einstein5-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration.</p><div class="templatequotecite">— <cite>Albert Einstein: <i><a href="/wiki/The_Meaning_of_Relativity" title="The Meaning of Relativity">The Meaning of Relativity</a></i>, p. 58</cite></div></blockquote> <p>There are several approaches to this issue. One approach is to argue that all real forces drop off with distance from their sources in a known manner, so we have only to be sure that a body is far enough away from all sources to ensure that no force is present.<sup id="cite_ref-Rosser_30-0" class="reference"><a href="#cite_note-Rosser-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> A possible issue with this approach is the historically long-lived view that the distant universe might affect matters (<a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a>). Another approach is to identify all real sources for real forces and account for them. A possible issue with this approach is that we might miss something, or account inappropriately for their influence, perhaps, again, due to Mach's principle and an incomplete understanding of the universe. A third approach is to look at the way the forces transform when we shift reference frames. Fictitious forces, those that arise due to the acceleration of a frame, disappear in inertial frames, and have complicated rules of transformation in general cases. On the basis of universality of physical law and the request for frames where the laws are most simply expressed, inertial frames are distinguished by the absence of such fictitious forces. </p><p> Newton enunciated a principle of relativity himself in one of his corollaries to the laws of motion:<sup id="cite_ref-Feynman2_31-0" class="reference"><a href="#cite_note-Feynman2-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Principia_32-0" class="reference"><a href="#cite_note-Principia-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"></p><blockquote class="templatequote"><p>The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.</p><div class="templatequotecite">— <cite>Isaac Newton: <i>Principia</i>, Corollary V, p. 88 in Andrew Motte translation</cite></div></blockquote> <p>This principle differs from the <a href="#principle">special principle</a> in two ways: first, it is restricted to mechanics, and second, it makes no mention of simplicity. It shares with the special principle the invariance of the form of the description among mutually translating reference frames.<sup id="cite_ref-note1_33-0" class="reference"><a href="#cite_note-note1-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> The role of fictitious forces in classifying reference frames is pursued further below. </p> <div class="mw-heading mw-heading3"><h3 id="Remarks">Remarks</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Special_theory_of_relativity" class="mw-redirect" title="Special theory of relativity">Special theory of relativity</a> and <a href="/wiki/General_theory_of_relativity" class="mw-redirect" title="General theory of relativity">General theory of relativity</a></div> <p>It is important to note some assumptions made above about the various inertial frames of reference. Newton, for instance, employed universal time, as explained by the following example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronize them so that they both display exactly the same time. The two clocks are now separated and one clock is on a fast moving train, traveling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept of time and simultaneity was later generalized by Einstein in his <a href="/wiki/Special_theory_of_relativity" class="mw-redirect" title="Special theory of relativity">special theory of relativity</a> (1905) where he developed transformations between inertial frames of reference based upon the universal nature of physical laws and their economy of expression (<a href="/wiki/Lorentz_transformations" class="mw-redirect" title="Lorentz transformations">Lorentz transformations</a>). </p><p>Frames of reference are especially important in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another frame. The speed of light is considered to be the only true constant between moving frames of reference. </p><p>The definition of inertial reference frame can also be extended beyond three-dimensional Euclidean space. Newton's assumed a Euclidean space, but <a href="/wiki/General_relativity" title="General relativity">general relativity</a> uses a more general geometry. As an example of why this is important, consider the <a href="/wiki/Geometry" title="Geometry">geometry</a> of an ellipsoid. In this geometry, a "free" particle is defined as one at rest or traveling at constant speed on a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> path. Two free particles may begin at the same point on the surface, traveling with the same constant speed in different directions. After a length of time, the two particles collide at the opposite side of the ellipsoid. Both "free" particles traveled with a constant speed, satisfying the definition that no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now common to describe<sup id="cite_ref-caveatondescription_34-0" class="reference"><a href="#cite_note-caveatondescription-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> that we exist in a four-dimensional geometry known as <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. In this picture, the curvature of this 4D space is responsible for the way in which two bodies with mass are drawn together even if no forces are acting. This curvature of spacetime replaces the force known as gravity in Newtonian mechanics and special relativity. </p> <div class="mw-heading mw-heading2"><h2 id="Special_relativity">Special relativity</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></div> <p><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein's</a> <a href="/wiki/Special_relativity" title="Special relativity">theory of special relativity</a>, like Newtonian mechanics, postulates the equivalence of all inertial reference frames. However, because special relativity postulates that the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in <a href="/wiki/Free_space" class="mw-redirect" title="Free space">free space</a> is <a href="/wiki/Invariant_(physics)" title="Invariant (physics)">invariant</a>, the transformation between inertial frames is the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a>, not the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a> which is used in Newtonian mechanics. The invariance of the speed of light leads to counter-intuitive phenomena, such as <a href="/wiki/Time_dilation" title="Time dilation">time dilation</a> and <a href="/wiki/Length_contraction" title="Length contraction">length contraction</a>, and the <a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">relativity of simultaneity</a>, which have been extensively verified experimentally.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> The Lorentz transformation reduces to the Galilean transformation as the speed of light approaches infinity or as the relative velocity between frames approaches zero.<sup id="cite_ref-Landau_36-0" class="reference"><a href="#cite_note-Landau-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Non-inertial_frames">Non-inertial frames</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a>, <a href="/wiki/Non-inertial_frame" class="mw-redirect" title="Non-inertial frame">Non-inertial frame</a>, and <a href="/wiki/Rotating_frame_of_reference" class="mw-redirect" title="Rotating frame of reference">Rotating frame of reference</a></div> <p>Here the relation between inertial and non-inertial observational frames of reference is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below. </p> <div class="mw-heading mw-heading3"><h3 id="General_relativity">General relativity</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/General_relativity" title="General relativity">General relativity</a> and <a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction to general relativity</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a> and <a href="/wiki/E%C3%B6tv%C3%B6s_experiment" title="Eötvös experiment">Eötvös experiment</a></div><p> General relativity is based upon the principle of equivalence:<sup id="cite_ref-Morin_37-0" class="reference"><a href="#cite_note-Morin-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Giancoli_38-0" class="reference"><a href="#cite_note-Giancoli-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"></p><blockquote class="templatequote"><p>There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating.</p><div class="templatequotecite">— <cite>Douglas C. Giancoli, <i>Physics for Scientists and Engineers with Modern Physics</i>, p. 155.</cite></div></blockquote> <p>This idea was introduced in Einstein's 1907 article "Principle of Relativity and Gravitation" and later developed in 1911.<sup id="cite_ref-General_theory_39-0" class="reference"><a href="#cite_note-General_theory-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Support for this principle is found in the <a href="/wiki/E%C3%B6tv%C3%B6s_experiment" title="Eötvös experiment">Eötvös experiment</a>, which determines whether the ratio of inertial to gravitational mass is the same for all bodies, regardless of size or composition. To date no difference has been found to a few parts in 10<sup>11</sup>.<sup id="cite_ref-NRC_40-0" class="reference"><a href="#cite_note-NRC-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> For some discussion of the subtleties of the Eötvös experiment, such as the local mass distribution around the experimental site (including a quip about the mass of Eötvös himself), see Franklin.<sup id="cite_ref-Franklin_41-0" class="reference"><a href="#cite_note-Franklin-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>Einstein's <a href="/wiki/General_relativity" title="General relativity">general theory</a> modifies the distinction between nominally "inertial" and "non-inertial" effects by replacing special relativity's "flat" <a href="/wiki/Minkowski_Space" class="mw-redirect" title="Minkowski Space">Minkowski Space</a> with a metric that produces non-zero curvature. In general relativity, the principle of inertia is replaced with the principle of <a href="/wiki/Geodesic_(general_relativity)" class="mw-redirect" title="Geodesic (general relativity)">geodesic motion</a>, whereby objects move in a way dictated by the curvature of spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so. This phenomenon of <a href="/wiki/Geodesic_deviation" title="Geodesic deviation">geodesic deviation</a> means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity. </p><p>However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> Consequently, modern special relativity is now sometimes described as only a "local theory".<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> "Local" can encompass, for example, the entire Milky Way galaxy: The astronomer <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Karl Schwarzschild</a> observed the motion of pairs of stars orbiting each other. He found that the two orbits of the stars of such a system lie in a plane, and the perihelion of the orbits of the two stars remains pointing in the same direction with respect to the solar system. Schwarzschild pointed out that that was invariably seen: the direction of the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> of all observed double star systems remains fixed with respect to the direction of the angular momentum of the Solar System. These observations allowed him to conclude that inertial frames inside the galaxy do not rotate with respect to one another, and that the space of the Milky Way is approximately Galilean or Minkowskian.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Inertial_frames_and_rotation">Inertial frames and rotation</h3></div> <p>In an inertial frame, <a href="/wiki/Newton%27s_first_law" class="mw-redirect" title="Newton's first law">Newton's first law</a>, the <i>law of inertia</i>, is satisfied: Any free motion has a constant magnitude and direction.<sup id="cite_ref-LandauMechanics_4-2" class="reference"><a href="#cite_note-LandauMechanics-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> for a <a href="/wiki/Point_particle" title="Point particle">particle</a> takes the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m\mathbf {a} \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89afe672e24bb812537066cb740cc2646cafb58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.349ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} \ ,}"></span></dd></dl> <p>with <b>F</b> the net force (a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a>), <i>m</i> the mass of a particle and <b>a</b> the <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> of the particle (also a vector) which would be measured by an observer at rest in the frame. The force <b>F</b> is the <a href="/wiki/Vector_sum" class="mw-redirect" title="Vector sum">vector sum</a> of all "real" forces on the particle, such as <a href="/wiki/Contact_force" title="Contact force">contact forces</a>, electromagnetic, gravitational, and nuclear forces. </p><p>In contrast, Newton's second law in a <a href="/wiki/Rotating_frame_of_reference" class="mw-redirect" title="Rotating frame of reference">rotating frame of reference</a> (a <b>non-inertial frame of reference</b>), rotating at angular rate <i>Ω</i> about an axis, takes the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} '=m\mathbf {a} \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} '=m\mathbf {a} \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1356aefccabd944bdee072beba8017fa5d46b44e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.033ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} '=m\mathbf {a} \ ,}"></span></dd></dl> <p>which looks the same as in an inertial frame, but now the force <b>F</b>′ is the resultant of not only <b>F</b>, but also additional terms (the paragraph following this equation presents the main points without detailed mathematics): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} '=\mathbf {F} -2m\mathbf {\Omega } \times \mathbf {v} _{B}-m\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {x} _{B})-m{\frac {d\mathbf {\Omega } }{dt}}\times \mathbf {x} _{B}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} '=\mathbf {F} -2m\mathbf {\Omega } \times \mathbf {v} _{B}-m\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {x} _{B})-m{\frac {d\mathbf {\Omega } }{dt}}\times \mathbf {x} _{B}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc3391b76469c57b7613df1ea14d21060239975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:55.8ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} '=\mathbf {F} -2m\mathbf {\Omega } \times \mathbf {v} _{B}-m\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {x} _{B})-m{\frac {d\mathbf {\Omega } }{dt}}\times \mathbf {x} _{B}\ ,}"></span></dd></dl> <p>where the angular rotation of the frame is expressed by the vector <b>Ω</b> pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation <i>Ω</i>, symbol × denotes the <a href="/wiki/Vector_cross_product" class="mw-redirect" title="Vector cross product">vector cross product</a>, vector <b>x</b><sub><i>B</i></sub> locates the body and vector <b>v</b><sub><i>B</i></sub> is the <a href="/wiki/Velocity" title="Velocity">velocity</a> of the body according to a rotating observer (different from the velocity seen by the inertial observer). </p><p>The extra terms in the force <b>F</b>′ are the "fictitious" forces for this frame, whose causes are external to the system in the frame. The first extra term is the <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a>, the second the <a href="/wiki/Centrifugal_force_(rotating_reference_frame)" class="mw-redirect" title="Centrifugal force (rotating reference frame)">centrifugal force</a>, and the third the <a href="/wiki/Euler_force" title="Euler force">Euler force</a>. These terms all have these properties: they vanish when <i>Ω</i> = 0; that is, they are zero for an inertial frame (which, of course, does not rotate); they take on a different magnitude and direction in every rotating frame, depending upon its particular value of <b>Ω</b>; they are ubiquitous in the rotating frame (affect every particle, regardless of circumstance); and they have no apparent source in identifiable physical sources, in particular, <a href="/wiki/Matter" title="Matter">matter</a>. Also, fictitious forces do not drop off with distance (unlike, for example, <a href="/wiki/Nuclear_force" title="Nuclear force">nuclear forces</a> or <a href="/wiki/Electrical_force" class="mw-redirect" title="Electrical force">electrical forces</a>). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis. </p><p>All observers agree on the real forces, <b>F</b>; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present. </p><p>In Newton's time the <a href="/wiki/Fixed_stars" title="Fixed stars">fixed stars</a> were invoked as a reference frame, supposedly at rest relative to <a href="/wiki/Absolute_space" class="mw-redirect" title="Absolute space">absolute space</a>. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of <a href="/wiki/Fictitious_forces" class="mw-redirect" title="Fictitious forces">fictitious forces</a>, for example, the <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a> and the <a href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal force</a>. Two experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking <a href="/wiki/Rotating_spheres" title="Rotating spheres">two spheres rotating</a> about their center of gravity, and the example of the curvature of the surface of water in a <a href="/wiki/Bucket_argument" title="Bucket argument">rotating bucket</a>. In both cases, application of <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> would not work for the rotating observer without invoking centrifugal and Coriolis forces to account for their observations (tension in the case of the spheres; parabolic water surface in the case of the rotating bucket). </p><p>As we now know, the fixed stars are not fixed. Those that reside in the <a href="/wiki/Milky_Way" title="Milky Way">Milky Way</a> turn with the galaxy, exhibiting <a href="/wiki/Proper_motion" title="Proper motion">proper motions</a>. Those that are outside our galaxy (such as nebulae once mistaken to be stars) participate in their own motion as well, partly due to <a href="/wiki/Expansion_of_the_universe" title="Expansion of the universe">expansion of the universe</a>, and partly due to <a href="/wiki/Peculiar_velocity" title="Peculiar velocity">peculiar velocities</a>.<sup id="cite_ref-Balbi_46-0" class="reference"><a href="#cite_note-Balbi-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> For instance, the <a href="/wiki/Andromeda_Galaxy" title="Andromeda Galaxy">Andromeda Galaxy</a> is on <a href="/wiki/Andromeda%E2%80%93Milky_Way_collision" title="Andromeda–Milky Way collision">collision course with the Milky Way</a> at a speed of 117 km/s.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial frame is based on the simplicity of the laws of physics in the frame. John Stachel wrote: once one gave up the existence of a privileged frame of reference (the ether frame) there was no reason why one should stop at the relativity of inertial frames. The <i>conventional answer</i> to such doubts was that the laws of nature took a simpler form in the inertial frames of reference because in these frames one did not have to introduce inertial forces when writing down Newton's law of motion. <sup id="cite_ref-Stachel_48-0" class="reference"><a href="#cite_note-Stachel-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p>In practice, although not a requirement, using a frame of reference based upon the fixed stars as though it were an inertial frame of reference introduces very little discrepancy. For example, the centrifugal acceleration of the Earth because of its rotation about the Sun is about thirty million times greater than that of the Sun about the galactic center.<sup id="cite_ref-Graneau_49-0" class="reference"><a href="#cite_note-Graneau-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p>To illustrate further, consider the question: "Does our Universe rotate?" To answer, we might attempt to explain the shape of the <a href="/wiki/Milky_Way" title="Milky Way">Milky Way</a> galaxy using the laws of physics,<sup id="cite_ref-Genz_50-0" class="reference"><a href="#cite_note-Genz-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> although other observations might be more definitive; that is, provide larger <a href="/wiki/Observational_error" title="Observational error">discrepancies</a> or less <a href="/wiki/Measurement_uncertainty" title="Measurement uncertainty">measurement uncertainty</a>, like the anisotropy of the <a href="/wiki/Microwave_background_radiation" class="mw-redirect" title="Microwave background radiation">microwave background radiation</a> or <a href="/wiki/Big_Bang_nucleosynthesis" title="Big Bang nucleosynthesis">Big Bang nucleosynthesis</a>.<sup id="cite_ref-Thompson_51-0" class="reference"><a href="#cite_note-Thompson-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Szydlowski_52-0" class="reference"><a href="#cite_note-Szydlowski-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> The flatness of the Milky Way depends on its rate of rotation in an inertial frame of reference. If we attribute its apparent rate of rotation entirely to rotation in an inertial frame, a different "flatness" is predicted than if we suppose part of this rotation actually is due to rotation of the universe and should not be included in the rotation of the galaxy itself. Based upon the laws of physics, a model is set up in which one parameter is the rate of rotation of the Universe. If the laws of physics agree more accurately with observations in a model with rotation than without it, we are inclined to select the best-fit value for rotation, subject to all other pertinent experimental observations. If no value of the rotation parameter is successful and theory is not within observational error, a modification of physical law is considered, for example, <a href="/wiki/Dark_matter" title="Dark matter">dark matter</a> is invoked to explain the <a href="/wiki/Galactic_rotation_curve" class="mw-redirect" title="Galactic rotation curve">galactic rotation curve</a>. So far, observations show any rotation of the universe is very slow, no faster than once every <span class="nowrap"><span data-sort-value="7013600000000000000♠"></span>6<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>13</sup></span> years (10<sup>−13</sup> rad/yr),<sup id="cite_ref-Birch_53-0" class="reference"><a href="#cite_note-Birch-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> and debate persists over whether there is <i>any</i> rotation. However, if rotation were found, interpretation of observations in a frame tied to the universe would have to be corrected for the fictitious forces inherent in such rotation in classical physics and special relativity, or interpreted as the curvature of spacetime and the motion of matter along the geodesics in general relativity.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p><p>When <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum</a> effects are important, there are additional conceptual complications that arise in <a href="/wiki/Quantum_reference_frame" title="Quantum reference frame">quantum reference frames</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Primed_frames">Primed frames</h3></div> <p>An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. <i>x′</i>, <i>y′</i>, <i>a′</i>. </p><p>The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as <b>R</b>. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called <b>r</b>, and the vector from the accelerated origin to the point is called <b>r′</b>. From the geometry of the situation, we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =\mathbf {R} +\mathbf {r} '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {R} +\mathbf {r} '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b33cdfda02deffda79d1d07a687eb7ce4495e80f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.478ex; height:2.676ex;" alt="{\displaystyle \mathbf {r} =\mathbf {R} +\mathbf {r} '.}"></span></dd></dl> <p>Taking the first and second derivatives of this with respect to time, we obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =\mathbf {V} +\mathbf {v} ',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\mathbf {V} +\mathbf {v} ',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683fd58bef82b0094dbb0fc98955494634a698b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.112ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} =\mathbf {V} +\mathbf {v} ',}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =\mathbf {A} +\mathbf {a} '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {A} +\mathbf {a} '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dfe8686e939b77361f9e6da4d932516ba1c721f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.889ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} =\mathbf {A} +\mathbf {a} '.}"></span></dd></dl> <p>where <b>V</b> and <b>A</b> are the velocity and acceleration of the accelerated system with respect to the inertial system and <b>v</b> and <b>a</b> are the velocity and acceleration of the point of interest with respect to the inertial frame. </p><p>These equations allow transformations between the two coordinate systems; for example, we can now write <a href="/wiki/Newton%27s_laws_of_motion#Newton.27s_second_law" title="Newton's laws of motion">Newton's second law</a> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m\mathbf {a} =m\mathbf {A} +m\mathbf {a} '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} =m\mathbf {A} +m\mathbf {a} '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f221f7df920bddb817aef12aea200466756451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.791ex; height:2.676ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} =m\mathbf {A} +m\mathbf {a} '.}"></span></dd></dl> <p>When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in <a href="/wiki/Centrifugal_force_(rotating_reference_frame)" class="mw-redirect" title="Centrifugal force (rotating reference frame)">centrifugal</a> direction, or in a direction orthogonal to an object's motion, the <a href="/wiki/Coriolis_effect" class="mw-redirect" title="Coriolis effect">Coriolis effect</a>). </p><p>A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation (see <a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a> for a derivation): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =\mathbf {a} '+{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '+2{\boldsymbol {\omega }}\times \mathbf {v} '+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')+\mathbf {A} _{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {a} '+{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '+2{\boldsymbol {\omega }}\times \mathbf {v} '+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')+\mathbf {A} _{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6161139365cfbb93fe3ec1c13b89e8326a10fea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.142ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} =\mathbf {a} '+{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '+2{\boldsymbol {\omega }}\times \mathbf {v} '+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')+\mathbf {A} _{0},}"></span></dd></dl> <p>or, to solve for the acceleration in the accelerated frame, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} '=\mathbf {a} -{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '-2{\boldsymbol {\omega }}\times \mathbf {v} '-{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')-\mathbf {A} _{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} '=\mathbf {a} -{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '-2{\boldsymbol {\omega }}\times \mathbf {v} '-{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')-\mathbf {A} _{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91716e58d3c0e1235c33d193b89bd5e85fa8bcb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.142ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} '=\mathbf {a} -{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '-2{\boldsymbol {\omega }}\times \mathbf {v} '-{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')-\mathbf {A} _{0}.}"></span></dd></dl> <p>Multiplying through by the mass <i>m</i> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} '=\mathbf {F} _{\mathrm {physical} }+\mathbf {F} '_{\mathrm {Euler} }+\mathbf {F} '_{\mathrm {Coriolis} }+\mathbf {F} '_{\mathrm {centripetal} }-m\mathbf {A} _{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>−</mo> <mi>m</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} '=\mathbf {F} _{\mathrm {physical} }+\mathbf {F} '_{\mathrm {Euler} }+\mathbf {F} '_{\mathrm {Coriolis} }+\mathbf {F} '_{\mathrm {centripetal} }-m\mathbf {A} _{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67681cc0daccb4c46a69ed357b4d857360848c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:53.098ex; height:3.343ex;" alt="{\displaystyle \mathbf {F} '=\mathbf {F} _{\mathrm {physical} }+\mathbf {F} '_{\mathrm {Euler} }+\mathbf {F} '_{\mathrm {Coriolis} }+\mathbf {F} '_{\mathrm {centripetal} }-m\mathbf {A} _{0},}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} '_{\mathrm {Euler} }=-m{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} '_{\mathrm {Euler} }=-m{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814c3319423b9c9e93b456b783328a42a63bb5fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.023ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} '_{\mathrm {Euler} }=-m{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '}"></span> (<a href="/wiki/Euler_force" title="Euler force">Euler force</a>),</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} '_{\mathrm {Coriolis} }=-2m{\boldsymbol {\omega }}\times \mathbf {v} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} '_{\mathrm {Coriolis} }=-2m{\boldsymbol {\omega }}\times \mathbf {v} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578db398cc7acfb76e4532beef337d7fc199a643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.124ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} '_{\mathrm {Coriolis} }=-2m{\boldsymbol {\omega }}\times \mathbf {v} '}"></span> (<a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a>),</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} '_{\mathrm {centrifugal} }=-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')=m(\omega ^{2}\mathbf {r} '-({\boldsymbol {\omega }}\cdot \mathbf {r} '){\boldsymbol {\omega }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} '_{\mathrm {centrifugal} }=-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')=m(\omega ^{2}\mathbf {r} '-({\boldsymbol {\omega }}\cdot \mathbf {r} '){\boldsymbol {\omega }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f021888918850838d80c20be7e2154ed87157ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.907ex; height:3.509ex;" alt="{\displaystyle \mathbf {F} '_{\mathrm {centrifugal} }=-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')=m(\omega ^{2}\mathbf {r} '-({\boldsymbol {\omega }}\cdot \mathbf {r} '){\boldsymbol {\omega }})}"></span> (<a href="/wiki/Centrifugal_force_(rotating_reference_frame)" class="mw-redirect" title="Centrifugal force (rotating reference frame)">centrifugal force</a>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Separating_non-inertial_from_inertial_reference_frames">Separating non-inertial from inertial reference frames</h2></div> <div class="mw-heading mw-heading3"><h3 id="Theory">Theory</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Non-inertial_frame" class="mw-redirect" title="Non-inertial frame">Non-inertial frame</a>, <a href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</a>, and <a href="/wiki/Bucket_argument" title="Bucket argument">Bucket argument</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Rotating_spheres.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Rotating_spheres.svg/180px-Rotating_spheres.svg.png" decoding="async" width="180" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Rotating_spheres.svg/270px-Rotating_spheres.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Rotating_spheres.svg/360px-Rotating_spheres.svg.png 2x" data-file-width="465" data-file-height="398" /></a><figcaption>Figure 2: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rotating-sphere_forces.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Rotating-sphere_forces.svg/220px-Rotating-sphere_forces.svg.png" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Rotating-sphere_forces.svg/330px-Rotating-sphere_forces.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Rotating-sphere_forces.svg/440px-Rotating-sphere_forces.svg.png 2x" data-file-width="512" data-file-height="355" /></a><figcaption>Figure 3: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.</figcaption></figure><p> Inertial and non-inertial reference frames can be distinguished by the absence or presence of <a href="/wiki/Fictitious_force" title="Fictitious force">fictitious forces</a>, as explained shortly.<sup id="cite_ref-Rothman_9-1" class="reference"><a href="#cite_note-Rothman-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Borowitz_10-1" class="reference"><a href="#cite_note-Borowitz-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"></p><blockquote class="templatequote"><p>The effect of this being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations….</p><div class="templatequotecite">— <cite>Sidney Borowitz and Lawrence A Bornstein in <i>A Contemporary View of Elementary Physics</i>, p. 138</cite></div></blockquote> <p>The presence of fictitious forces indicates the physical laws are not the simplest laws available so, in terms of the <a href="#principle">special principle of relativity</a>, a frame where fictitious forces are present is not an inertial frame:<sup id="cite_ref-Arnold2_55-0" class="reference"><a href="#cite_note-Arnold2-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.</p><div class="templatequotecite">— <cite>V. I. Arnol'd: <i>Mathematical Methods of Classical Mechanics</i> Second Edition, p. 129</cite></div></blockquote> <p>Bodies in <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial reference frames</a> are subject to so-called <i>fictitious</i> forces (pseudo-forces); that is, <a href="/wiki/Force" title="Force">forces</a> that result from the acceleration of the <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frame</a> itself and not from any physical force acting on the body. Examples of fictitious forces are the <a href="/wiki/Centrifugal_force_(fictitious)" class="mw-redirect" title="Centrifugal force (fictitious)">centrifugal force</a> and the <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a> in <a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">rotating reference frames</a>. </p><p>How then, are "fictitious" forces to be separated from "real" forces? It is hard to apply the Newtonian definition of an inertial frame without this separation. For example, consider a stationary object in an inertial frame. Being at rest, no net force is applied. But in a frame rotating about a fixed axis, the object appears to move in a circle, and is subject to centripetal force (which is made up of the Coriolis force and the centrifugal force). How can we decide that the rotating frame is a non-inertial frame? There are two approaches to this resolution: one approach is to look for the origin of the fictitious forces (the Coriolis force and the centrifugal force). We will find there are no sources for these forces, no associated <a href="/wiki/Force_carrier" title="Force carrier">force carriers</a>, no originating bodies.<sup id="cite_ref-note2_56-0" class="reference"><a href="#cite_note-note2-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> A second approach is to look at a variety of frames of reference. For any inertial frame, the Coriolis force and the centrifugal force disappear, so application of the principle of special relativity would identify these frames where the forces disappear as sharing the same and the simplest physical laws, and hence rule that the rotating frame is not an inertial frame. </p><p>Newton examined this problem himself using rotating spheres, as shown in Figure 2 and Figure 3. He pointed out that if the spheres are not rotating, the tension in the tying string is measured as zero in every frame of reference.<sup id="cite_ref-tension_57-0" class="reference"><a href="#cite_note-tension-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> If the spheres only appear to rotate (that is, we are watching stationary spheres from a rotating frame), the zero tension in the string is accounted for by observing that the centripetal force is supplied by the centrifugal and Coriolis forces in combination, so no tension is needed. If the spheres really are rotating, the tension observed is exactly the centripetal force required by the circular motion. Thus, measurement of the tension in the string identifies the inertial frame: it is the one where the tension in the string provides exactly the centripetal force demanded by the motion as it is observed in that frame, and not a different value. That is, the inertial frame is the one where the fictitious forces vanish. </p><p>So much for fictitious forces due to rotation. However, for <a href="/wiki/Linear_acceleration" class="mw-redirect" title="Linear acceleration">linear acceleration</a>, Newton expressed the idea of undetectability of straight-line accelerations held in common:<sup id="cite_ref-Principia_32-1" class="reference"><a href="#cite_note-Principia-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces. </p><div class="templatequotecite">— <cite>Isaac Newton: <i>Principia</i> Corollary VI, p. 89, in Andrew Motte translation</cite></div></blockquote> <p>This principle generalizes the notion of an inertial frame. For example, an observer confined in a free-falling lift will assert that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift. So, strictly speaking, inertial frame is a relative concept. With this in mind, we can define inertial frames collectively as a set of frames which are stationary or moving at constant velocity with respect to each other, so that a single inertial frame is defined as an element of this set. </p><p>For these ideas to apply, everything observed in the frame has to be subject to a base-line, common acceleration shared by the frame itself. That situation would apply, for example, to the elevator example, where all objects are subject to the same gravitational acceleration, and the elevator itself accelerates at the same rate. </p> <div class="mw-heading mw-heading3"><h3 id="Applications">Applications</h3></div> <p><a href="/wiki/Inertial_navigation_system" title="Inertial navigation system">Inertial navigation systems</a> used a cluster of <a href="/wiki/Gyroscope" title="Gyroscope">gyroscopes</a> and accelerometers to determine accelerations relative to inertial space. After a gyroscope is spun up in a particular orientation in inertial space, the law of conservation of angular momentum requires that it retain that orientation as long as no external forces are applied to it.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 59">: 59 </span></sup> Three orthogonal gyroscopes establish an inertial reference frame, and the accelerators measure acceleration relative to that frame. The accelerations, along with a clock, can then be used to calculate the change in position. Thus, inertial navigation is a form of <a href="/wiki/Dead_reckoning" title="Dead reckoning">dead reckoning</a> that requires no external input, and therefore cannot be jammed by any external or internal signal source.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Gyrocompass" title="Gyrocompass">gyrocompass</a>, employed for navigation of seagoing vessels, finds the geometric north. It does so, not by sensing the Earth's magnetic field, but by using inertial space as its reference. The outer casing of the gyrocompass device is held in such a way that it remains aligned with the local plumb line. When the gyroscope wheel inside the gyrocompass device is spun up, the way the gyroscope wheel is suspended causes the gyroscope wheel to gradually align its spinning axis with the Earth's axis. Alignment with the Earth's axis is the only direction for which the gyroscope's spinning axis can be stationary with respect to the Earth and not be required to change direction with respect to inertial space. After being spun up, a gyrocompass can reach the direction of alignment with the Earth's axis in as little as a quarter of an hour.<sup id="cite_ref-l_60-0" class="reference"><a href="#cite_note-l-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Inertial_frame_of_reference" title="Special:EditPage/Inertial frame of reference">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">July 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Simple_example">Simple example</h3></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Two_reference_frames.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Two_reference_frames.PNG/320px-Two_reference_frames.PNG" decoding="async" width="320" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Two_reference_frames.PNG/480px-Two_reference_frames.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Two_reference_frames.PNG/640px-Two_reference_frames.PNG 2x" data-file-width="725" data-file-height="435" /></a><figcaption>Figure 1: Two cars moving at different but constant velocities observed from stationary inertial frame <i>S</i> attached to the road and moving inertial frame <i>S′</i> attached to the first car.</figcaption></figure> <p>Consider a situation common in everyday life. Two cars travel along a road, both moving at constant velocities. See Figure 1. At some particular moment, they are separated by 200 metres. The car in front is travelling at 22 metres per second and the car behind is travelling at 30 metres per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose. </p><p>First, we could observe the two cars from the side of the road. We define our "frame of reference" <i>S</i> as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance <span class="nowrap"><i>d</i> = 200 m</span> apart. Since neither of the cars is accelerating, we can determine their positions by the following formulas, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edc245640b5794e3f754bee7b713761282b60c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:2.843ex;" alt="{\displaystyle x_{1}(t)}"></span> is the position in meters of car one after time <i>t</i> in seconds and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8821f9acaecc995a1b0bdfea31bb01605db7142c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:2.843ex;" alt="{\displaystyle x_{2}(t)}"></span> is the position of car two after time <i>t</i>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}(t)=d+v_{1}t=200+22t,\quad x_{2}(t)=v_{2}t=30t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>=</mo> <mn>200</mn> <mo>+</mo> <mn>22</mn> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> <mo>=</mo> <mn>30</mn> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}(t)=d+v_{1}t=200+22t,\quad x_{2}(t)=v_{2}t=30t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e181865b9dbba562b0cc0ce38ebfda80f7458d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.219ex; height:2.843ex;" alt="{\displaystyle x_{1}(t)=d+v_{1}t=200+22t,\quad x_{2}(t)=v_{2}t=30t.}"></span></dd></dl> <p>Notice that these formulas predict at <i>t</i> = 0 s the first car is 200 m down the road and the second car is right beside us, as expected. We want to find the time at which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}=x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e7b902fd6d8f38f50e78211fd453a4e2651d66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.009ex;" alt="{\displaystyle x_{1}=x_{2}}"></span>. Therefore, we set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}=x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e7b902fd6d8f38f50e78211fd453a4e2651d66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.009ex;" alt="{\displaystyle x_{1}=x_{2}}"></span> and solve for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, that is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 200+22t=30t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>200</mn> <mo>+</mo> <mn>22</mn> <mi>t</mi> <mo>=</mo> <mn>30</mn> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 200+22t=30t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b91832f51b139f5abac96d320b41b599c4774c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.402ex; height:2.509ex;" alt="{\displaystyle 200+22t=30t,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8t=200,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>t</mi> <mo>=</mo> <mn>200</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8t=200,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163c0c1516800b6f949777d2b2c0d52984df5834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.235ex; height:2.509ex;" alt="{\displaystyle 8t=200,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=25\ \mathrm {seconds} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>25</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">s</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=25\ \mathrm {seconds} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fc96d5b245875de10d47e80eae5a1f5b628a63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.135ex; height:2.176ex;" alt="{\displaystyle t=25\ \mathrm {seconds} .}"></span></dd></dl> <p>Alternatively, we could choose a frame of reference <i>S′</i> situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of <span class="texhtml"><i>v</i><sub>2</sub> − <i>v</i><sub>1</sub> = 8 m/s</span>. In order to catch up to the first car, it will take a time of <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i></span><span class="sr-only">/</span><span class="den"><i>v</i><sub>2</sub> − <i>v</i><sub>1</sub></span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">200</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span> s</span>, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at <span class="nowrap">8 m/s</span>. </p><p>It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three). </p> <div class="mw-heading mw-heading3"><h3 id="Additional_example">Additional example</h3></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Wikipage_pic.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Wikipage_pic.PNG/250px-Wikipage_pic.PNG" decoding="async" width="250" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/98/Wikipage_pic.PNG 1.5x" data-file-width="287" data-file-height="161" /></a><figcaption>Figure 2: Simple-minded frame-of-reference example</figcaption></figure> <p>For a simple example involving only the orientation of two observers, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving towards the right. However, for the person facing west, the car was moving toward the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system. </p><p>For a more complex example involving observers in relative motion, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the <span class="texhtml"><i>x</i></span>-axis and the direction in front of him as the positive <span class="texhtml"><i>y</i></span>-axis. To him, the car moves along the <span class="texhtml"><i>x</i></span> axis with some <a href="/wiki/Velocity" title="Velocity">velocity</a> <span class="texhtml"><i>v</i></span> in the positive <span class="texhtml"><i>x</i></span>-direction. Alfred's frame of reference is considered an inertial frame of reference because he is not accelerating (ignoring effects such as Earth's rotation and gravity). </p><p>Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive <span class="texhtml"><i>x</i></span>-axis, and the direction in front of her as the positive <span class="texhtml"><i>y</i></span>-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving – for instance, as she drives past Alfred, she observes him moving with velocity <span class="texhtml"><i>v</i></span> in the negative <span class="texhtml"><i>y</i></span>-direction. If she is driving north, then north is the positive <span class="texhtml"><i>y</i></span>-direction; if she turns east, east becomes the positive <span class="texhtml"><i>y</i></span>-direction. </p><p>Finally, as an example of non-inertial observers, assume Candace is accelerating her car. As she passes by him, Alfred measures her <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> and finds it to be <span class="texhtml"><i>a</i></span> in the negative <span class="texhtml"><i>x</i></span>-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity <span class="texhtml"><i>v</i></span> is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same as Alfred in her frame of reference, <span class="texhtml"><i>a</i></span> in the negative <span class="texhtml"><i>y</i></span>-direction. However, if she is accelerating at rate <span class="texhtml"><i>A</i></span> in the negative <span class="texhtml"><i>y</i></span>-direction (in other words, slowing down), she will find Candace's acceleration to be <span class="texhtml"><i>a′</i> = <i>a</i> − <i>A</i></span> in the negative <span class="texhtml"><i>y</i></span>-direction—a smaller value than Alfred has measured. Similarly, if she is accelerating at rate <i>A</i> in the positive <span class="texhtml"><i>y</i></span>-direction (speeding up), she will observe Candace's acceleration as <span class="texhtml"><i>a′</i> = <i>a</i> + <i>A</i></span> in the negative <span class="texhtml"><i>y</i></span>-direction—a larger value than Alfred's measurement. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Absolute_rotation" title="Absolute rotation">Absolute rotation</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean invariance</a></li> <li><a href="/wiki/General_covariance" title="General covariance">General covariance</a></li> <li><a href="/wiki/Local_reference_frame" title="Local reference frame">Local reference frame</a></li> <li><a href="/wiki/Lorentz_covariance" title="Lorentz covariance">Lorentz covariance</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion#Newton's_first_law" title="Newton's laws of motion">Newton's first law</a></li> <li><a href="/wiki/Quantum_reference_frame" title="Quantum reference frame">Quantum reference frame</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFields2020" class="citation web cs1">Fields, Douglas E. (Spring 2020). <a rel="nofollow" class="external text" href="https://physics.unm.edu/Courses/Fields/Phys2310/Lectures/lecture25.pdf#page=8">"Lecture25: Galilean and Special Relativity"</a> <span class="cs1-format">(PDF)</span>. <i>PHYC 2310: Calculus Based Physics III</i>. <a href="/wiki/University_of_New_Mexico" title="University of New Mexico">University of New Mexico</a>. p. 8<span class="reference-accessdate">. Retrieved <span class="nowrap">7 November</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PHYC+2310%3A+Calculus+Based+Physics+III&rft.atitle=Lecture25%3A+Galilean+and+Special+Relativity&rft.ssn=spring&rft.pages=8&rft.date=2020&rft.aulast=Fields&rft.aufirst=Douglas+E.&rft_id=https%3A%2F%2Fphysics.unm.edu%2FCourses%2FFields%2FPhys2310%2FLectures%2Flecture25.pdf%23page%3D8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChabaySherwood2015" class="citation book cs1">Chabay, Ruth; Sherwood, Bruce (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Gz4HBgAAQBAJ&pg=PA34"><i>Matter & Interactions Vol 1: Modern Mechanics</i></a> (Fourth ed.). Wiley. pp. 34–35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781118875865" title="Special:BookSources/9781118875865"><bdi>9781118875865</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matter+%26+Interactions+Vol+1%3A+Modern+Mechanics&rft.pages=34-35&rft.edition=Fourth&rft.pub=Wiley&rft.date=2015&rft.isbn=9781118875865&rft.aulast=Chabay&rft.aufirst=Ruth&rft.au=Sherwood%2C+Bruce&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGz4HBgAAQBAJ%26pg%3DPA34&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPuebe2009" class="citation book cs1">Puebe, Jean-Laurent (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pMHVJE8nL2YC&pg=PT62"><i>Fluid Mechanics</i></a>. p. 62. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84821-065-3" title="Special:BookSources/978-1-84821-065-3"><bdi>978-1-84821-065-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.pages=62&rft.date=2009&rft.isbn=978-1-84821-065-3&rft.aulast=Puebe&rft.aufirst=Jean-Laurent&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpMHVJE8nL2YC%26pg%3DPT62&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-LandauMechanics-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-LandauMechanics_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-LandauMechanics_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-LandauMechanics_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1960" class="citation book cs1">Landau, L. D.; Lifshitz, E. M. (1960). <a rel="nofollow" class="external text" href="https://ia903206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%201%20-%20Landau%2C%20Lifshitz%20-%20Mechanics%20%283rd%20ed%2C%201976%29.pdf#page=31"><i>Mechanics</i></a> <span class="cs1-format">(PDF)</span>. Pergamon Press. pp. 4–6.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics&rft.pages=4-6&rft.pub=Pergamon+Press&rft.date=1960&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Lifshitz%2C+E.+M.&rft_id=https%3A%2F%2Fia903206.us.archive.org%2F4%2Fitems%2Flandau-and-lifshitz-physics-textbooks-series%2FVol%25201%2520-%2520Landau%252C%2520Lifshitz%2520-%2520Mechanics%2520%25283rd%2520ed%252C%25201976%2529.pdf%23page%3D31&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheng2013" class="citation book cs1">Cheng, Ta-Pei (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=thXT19cY9jsC"><i>Einstein's Physics: Atoms, Quanta, and Relativity – Derived, Explained, and Appraised</i></a> (illustrated ed.). OUP Oxford. p. 219. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-966991-2" title="Special:BookSources/978-0-19-966991-2"><bdi>978-0-19-966991-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein%27s+Physics%3A+Atoms%2C+Quanta%2C+and+Relativity+%E2%80%93+Derived%2C+Explained%2C+and+Appraised&rft.pages=219&rft.edition=illustrated&rft.pub=OUP+Oxford&rft.date=2013&rft.isbn=978-0-19-966991-2&rft.aulast=Cheng&rft.aufirst=Ta-Pei&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DthXT19cY9jsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=thXT19cY9jsC&pg=PA219">Extract of page 219</a></span> </li> <li id="cite_note-Einstein0-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Einstein0_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlbert_Einstein2001" class="citation book cs1">Albert Einstein (2001) [Reprint of edition of 1920 translated by RQ Lawson]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YLsSxQqEww0C&pg=PA71"><i>Relativity: The Special and General Theory</i></a> (3rd ed.). Courier Dover Publications. p. 71. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-41714-X" title="Special:BookSources/0-486-41714-X"><bdi>0-486-41714-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%3A+The+Special+and+General+Theory&rft.pages=71&rft.edition=3rd&rft.pub=Courier+Dover+Publications&rft.date=2001&rft.isbn=0-486-41714-X&rft.au=Albert+Einstein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYLsSxQqEww0C%26pg%3DPA71&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Britannica. <a rel="nofollow" class="external text" href="https://www.britannica.com/science/physics-science">Physics science definition.</a></span> </li> <li id="cite_note-Ferraro-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ferraro_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ferraro_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFerraro2007" class="citation cs2">Ferraro, Rafael (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wa3CskhHaIgC&pg=PA209"><i>Einstein's Space-Time: An Introduction to Special and General Relativity</i></a>, Springer Science & Business Media, pp. 209–210, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007esti.book.....F">2007esti.book.....F</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387699462" title="Special:BookSources/9780387699462"><bdi>9780387699462</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein%27s+Space-Time%3A+An+Introduction+to+Special+and+General+Relativity&rft.pages=209-210&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft_id=info%3Abibcode%2F2007esti.book.....F&rft.isbn=9780387699462&rft.aulast=Ferraro&rft.aufirst=Rafael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dwa3CskhHaIgC%26pg%3DPA209&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Rothman-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rothman_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rothman_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilton_A._Rothman1989" class="citation book cs1">Milton A. Rothman (1989). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/discoveringnatur0000roth"><i>Discovering the Natural Laws: The Experimental Basis of Physics</i></a></span>. Courier Dover Publications. p. <a rel="nofollow" class="external text" href="https://archive.org/details/discoveringnatur0000roth/page/n37/mode/2up">23-24</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-26178-6" title="Special:BookSources/0-486-26178-6"><bdi>0-486-26178-6</bdi></a>. <q>reference laws of physics.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discovering+the+Natural+Laws%3A+The+Experimental+Basis+of+Physics&rft.pages=23-24&rft.pub=Courier+Dover+Publications&rft.date=1989&rft.isbn=0-486-26178-6&rft.au=Milton+A.+Rothman&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiscoveringnatur0000roth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Borowitz-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Borowitz_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Borowitz_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSidney_BorowitzLawrence_A._Bornstein1968" class="citation book cs1">Sidney Borowitz; Lawrence A. Bornstein (1968). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/contemporaryview00boro"><i>A Contemporary View of Elementary Physics</i></a></span>. McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/contemporaryview00boro/page/138">138</a>. <a href="/wiki/ASIN_(identifier)" class="mw-redirect" title="ASIN (identifier)">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.com/dp/B000GQB02A">B000GQB02A</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Contemporary+View+of+Elementary+Physics&rft.pages=138&rft.pub=McGraw-Hill&rft.date=1968&rft_id=https%3A%2F%2Fwww.amazon.com%2Fdp%2FB000GQB02A%23id-name%3DASIN&rft.au=Sidney+Borowitz&rft.au=Lawrence+A.+Bornstein&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcontemporaryview00boro&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Einstein-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Einstein_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinsteinLorentzMinkowskiWeyl1952" class="citation book cs1"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, A.</a>; <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz, H. A.</a>; <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski, H.</a>; <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl, H.</a> (1952). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yECokhzsJYIC&q=postulate+%22Principle+of+Relativity%22&pg=PA111"><i>The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity</i></a>. Courier Dover Publications. p. 111. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60081-5" title="Special:BookSources/0-486-60081-5"><bdi>0-486-60081-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Principle+of+Relativity%3A+a+collection+of+original+memoirs+on+the+special+and+general+theory+of+relativity&rft.pages=111&rft.pub=Courier+Dover+Publications&rft.date=1952&rft.isbn=0-486-60081-5&rft.aulast=Einstein&rft.aufirst=A.&rft.au=Lorentz%2C+H.+A.&rft.au=Minkowski%2C+H.&rft.au=Weyl%2C+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyECokhzsJYIC%26q%3Dpostulate%2B%2522Principle%2Bof%2BRelativity%2522%26pg%3DPA111&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Nagel-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nagel_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErnest_Nagel1979" class="citation book cs1">Ernest Nagel (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u6EycHgRfkQC&q=inertial+%22Foucault%27s+pendulum%22&pg=PA212"><i>The Structure of Science</i></a>. Hackett Publishing. p. 212. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-915144-71-9" title="Special:BookSources/0-915144-71-9"><bdi>0-915144-71-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Structure+of+Science&rft.pages=212&rft.pub=Hackett+Publishing&rft.date=1979&rft.isbn=0-915144-71-9&rft.au=Ernest+Nagel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Du6EycHgRfkQC%26q%3Dinertial%2B%2522Foucault%2527s%2Bpendulum%2522%26pg%3DPA212&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Blagojević-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Blagojević_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilutin_Blagojević2002" class="citation book cs1">Milutin Blagojević (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=N8JDSi_eNbwC&q=inertial+frame+%22absolute+space%22&pg=PA5"><i>Gravitation and Gauge Symmetries</i></a>. CRC Press. p. 4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7503-0767-6" title="Special:BookSources/0-7503-0767-6"><bdi>0-7503-0767-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation+and+Gauge+Symmetries&rft.pages=4&rft.pub=CRC+Press&rft.date=2002&rft.isbn=0-7503-0767-6&rft.au=Milutin+Blagojevi%C4%87&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DN8JDSi_eNbwC%26q%3Dinertial%2Bframe%2B%2522absolute%2Bspace%2522%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Einstein2-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Einstein2_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlbert_Einstein1920" class="citation book cs1">Albert Einstein (1920). <a rel="nofollow" class="external text" href="https://archive.org/details/relativityspeci00lawsgoog"><i>Relativity: The Special and General Theory</i></a>. H. Holt and Company. p. <a rel="nofollow" class="external text" href="https://archive.org/details/relativityspeci00lawsgoog/page/n38">17</a>. <q>The Principle of Relativity.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%3A+The+Special+and+General+Theory&rft.pages=17&rft.pub=H.+Holt+and+Company&rft.date=1920&rft.au=Albert+Einstein&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frelativityspeci00lawsgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Feynman-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Feynman_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Phillips_Feynman1998" class="citation book cs1">Richard Phillips Feynman (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ipY8onVQWhcC&q=%22The+Principle+of+Relativity%22&pg=PA49"><i>Six not-so-easy pieces: Einstein's relativity, symmetry, and space-time</i></a>. Basic Books. p. 73. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-32842-9" title="Special:BookSources/0-201-32842-9"><bdi>0-201-32842-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Six+not-so-easy+pieces%3A+Einstein%27s+relativity%2C+symmetry%2C+and+space-time&rft.pages=73&rft.pub=Basic+Books&rft.date=1998&rft.isbn=0-201-32842-9&rft.au=Richard+Phillips+Feynman&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DipY8onVQWhcC%26q%3D%2522The%2BPrinciple%2Bof%2BRelativity%2522%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Wachter-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wachter_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArmin_WachterHenning_Hoeber2006" class="citation book cs1">Armin Wachter; Henning Hoeber (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=j3IQpdkinxMC&q=%2210-parameter+proper+orthochronous+Poincare+group%22&pg=PA98"><i>Compendium of Theoretical Physics</i></a>. Birkhäuser. p. 98. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-25799-3" title="Special:BookSources/0-387-25799-3"><bdi>0-387-25799-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Compendium+of+Theoretical+Physics&rft.pages=98&rft.pub=Birkh%C3%A4user&rft.date=2006&rft.isbn=0-387-25799-3&rft.au=Armin+Wachter&rft.au=Henning+Hoeber&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dj3IQpdkinxMC%26q%3D%252210-parameter%2Bproper%2Borthochronous%2BPoincare%2Bgroup%2522%26pg%3DPA98&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Mach-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Mach_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Mach_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErnst_Mach1915" class="citation book cs1">Ernst Mach (1915). <a rel="nofollow" class="external text" href="https://archive.org/details/sciencemechanic01jourgoog"><i>The Science of Mechanics</i></a>. The Open Court Publishing Co. p. <a rel="nofollow" class="external text" href="https://archive.org/details/sciencemechanic01jourgoog/page/n59">38</a>. <q>rotating sphere Mach cord OR string OR rod.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Science+of+Mechanics&rft.pages=38&rft.pub=The+Open+Court+Publishing+Co.&rft.date=1915&rft.au=Ernst+Mach&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsciencemechanic01jourgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLange,_Ludwig1885" class="citation journal cs1">Lange, Ludwig (1885). "Über die wissenschaftliche Fassung des Galileischen Beharrungsgesetzes". <i>Philosophische Studien</i>. <b>2</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophische+Studien&rft.atitle=%C3%9Cber+die+wissenschaftliche+Fassung+des+Galileischen+Beharrungsgesetzes&rft.volume=2&rft.date=1885&rft.au=Lange%2C+Ludwig&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Barbour-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Barbour_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJulian_B._Barbour2001" class="citation book cs1">Julian B. Barbour (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WQidkYkleXcC&q=Ludwig+Lange+%22operational+definition%22&pg=PA645"><i>The Discovery of Dynamics</i></a> (Reprint of 1989 <i>Absolute or Relative Motion?</i> ed.). Oxford University Press. pp. 645–646. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-513202-5" title="Special:BookSources/0-19-513202-5"><bdi>0-19-513202-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Discovery+of+Dynamics&rft.pages=645-646&rft.edition=Reprint+of+1989+%27%27Absolute+or+Relative+Motion%3F%27%27&rft.pub=Oxford+University+Press&rft.date=2001&rft.isbn=0-19-513202-5&rft.au=Julian+B.+Barbour&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWQidkYkleXcC%26q%3DLudwig%2BLange%2B%2522operational%2Bdefinition%2522%26pg%3DPA645&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Iro-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Iro_20-0">^</a></b></span> <span class="reference-text">L. Lange (1885) as quoted by Max von Laue in his book (1921) <i>Die Relativitätstheorie</i>, p. 34, and translated by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarald_Iro2002" class="citation book cs1">Harald Iro (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-L5ckgdxA5YC&q=inertial+noninertial&pg=PA179"><i>A Modern Approach to Classical Mechanics</i></a>. World Scientific. p. 169. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/981-238-213-5" title="Special:BookSources/981-238-213-5"><bdi>981-238-213-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Modern+Approach+to+Classical+Mechanics&rft.pages=169&rft.pub=World+Scientific&rft.date=2002&rft.isbn=981-238-213-5&rft.au=Harald+Iro&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-L5ckgdxA5YC%26q%3Dinertial%2Bnoninertial%26pg%3DPA179&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Blagojević2-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Blagojević2_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilutin_Blagojević2002" class="citation book cs1">Milutin Blagojević (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=N8JDSi_eNbwC&q=inertial+frame+%22absolute+space%22&pg=PA5"><i>Gravitation and Gauge Symmetries</i></a>. CRC Press. p. 5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7503-0767-6" title="Special:BookSources/0-7503-0767-6"><bdi>0-7503-0767-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation+and+Gauge+Symmetries&rft.pages=5&rft.pub=CRC+Press&rft.date=2002&rft.isbn=0-7503-0767-6&rft.au=Milutin+Blagojevi%C4%87&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DN8JDSi_eNbwC%26q%3Dinertial%2Bframe%2B%2522absolute%2Bspace%2522%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Woodhouse0-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Woodhouse0_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNMJ_Woodhouse2003" class="citation book cs1">NMJ Woodhouse (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tM9hic_wo3sC&q=Woodhouse+%22operational+definition%22&pg=PA126"><i>Special relativity</i></a>. London: Springer. p. 58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-85233-426-6" title="Special:BookSources/1-85233-426-6"><bdi>1-85233-426-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+relativity&rft.place=London&rft.pages=58&rft.pub=Springer&rft.date=2003&rft.isbn=1-85233-426-6&rft.au=NMJ+Woodhouse&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtM9hic_wo3sC%26q%3DWoodhouse%2B%2522operational%2Bdefinition%2522%26pg%3DPA126&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-DiSalle-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-DiSalle_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_DiSalle2002" class="citation book cs1">Robert DiSalle (Summer 2002). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/archives/sum2002/entries/spacetime-iframes/#Oth">"Space and Time: Inertial Frames"</a>. In Edward N. Zalta (ed.). <i>The Stanford Encyclopedia of Philosophy</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Space+and+Time%3A+Inertial+Frames&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.date=2002&rft.au=Robert+DiSalle&rft_id=http%3A%2F%2Fplato.stanford.edu%2Farchives%2Fsum2002%2Fentries%2Fspacetime-iframes%2F%23Oth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Moeller-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Moeller_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC_Møller1976" class="citation book cs1">C Møller (1976). <i>The Theory of Relativity</i> (Second ed.). Oxford UK: Oxford University Press. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-560539-X" title="Special:BookSources/0-19-560539-X"><bdi>0-19-560539-X</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/220221617">220221617</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Relativity&rft.place=Oxford+UK&rft.pages=1&rft.edition=Second&rft.pub=Oxford+University+Press&rft.date=1976&rft_id=info%3Aoclcnum%2F220221617&rft.isbn=0-19-560539-X&rft.au=C+M%C3%B8ller&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">The question of "moving uniformly relative to what?" was answered by Newton as "relative to <a href="/wiki/Absolute_space" class="mw-redirect" title="Absolute space">absolute space</a>". As a practical matter, "absolute space" was considered to be the <a href="/wiki/Fixed_stars" title="Fixed stars">fixed stars</a>. For a discussion of the role of fixed stars, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenning_Genz2001" class="citation book cs1">Henning Genz (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Cn_Q9wbDOM0C&q=frame+Newton+%22fixed+stars%22&pg=PA150"><i>Nothingness: The Science of Empty Space</i></a>. Da Capo Press. p. 150. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7382-0610-5" title="Special:BookSources/0-7382-0610-5"><bdi>0-7382-0610-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nothingness%3A+The+Science+of+Empty+Space&rft.pages=150&rft.pub=Da+Capo+Press&rft.date=2001&rft.isbn=0-7382-0610-5&rft.au=Henning+Genz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCn_Q9wbDOM0C%26q%3Dframe%2BNewton%2B%2522fixed%2Bstars%2522%26pg%3DPA150&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Resnick-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Resnick_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_ResnickDavid_HallidayKenneth_S._Krane2001" class="citation book cs1">Robert Resnick; David Halliday; Kenneth S. Krane (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsofph02hall"><i>Physics</i></a></span> (5th ed.). Wiley. Volume 1, Chapter 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-32057-9" title="Special:BookSources/0-471-32057-9"><bdi>0-471-32057-9</bdi></a>. <q>physics resnick.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics&rft.pages=Volume+1%2C+Chapter+3&rft.edition=5th&rft.pub=Wiley&rft.date=2001&rft.isbn=0-471-32057-9&rft.au=Robert+Resnick&rft.au=David+Halliday&rft.au=Kenneth+S.+Krane&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffundamentalsofph02hall&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Takwale-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-Takwale_27-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRG_Takwale1980" class="citation book cs1">RG Takwale (1980). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=r5P29cN6s6QC&q=fixed+stars+%22inertial+frame%22&pg=PA70"><i>Introduction to classical mechanics</i></a>. New Delhi: Tata McGraw-Hill. p. 70. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-096617-6" title="Special:BookSources/0-07-096617-6"><bdi>0-07-096617-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+classical+mechanics&rft.place=New+Delhi&rft.pages=70&rft.pub=Tata+McGraw-Hill&rft.date=1980&rft.isbn=0-07-096617-6&rft.au=RG+Takwale&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dr5P29cN6s6QC%26q%3Dfixed%2Bstars%2B%2522inertial%2Bframe%2522%26pg%3DPA70&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Woodhouse-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-Woodhouse_28-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNMJ_Woodhouse2003" class="citation book cs1">NMJ Woodhouse (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ggPXQAeeRLgC"><i>Special relativity</i></a>. London/Berlin: Springer. p. 6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-85233-426-6" title="Special:BookSources/1-85233-426-6"><bdi>1-85233-426-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+relativity&rft.place=London%2FBerlin&rft.pages=6&rft.pub=Springer&rft.date=2003&rft.isbn=1-85233-426-6&rft.au=NMJ+Woodhouse&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DggPXQAeeRLgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Einstein5-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-Einstein5_29-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFA_Einstein1950" class="citation book cs1">A Einstein (1950). <a rel="nofollow" class="external text" href="https://books.google.com/books?num=10&btnG=Google+Search"><i>The Meaning of Relativity</i></a>. Princeton University Press. p. 58.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Meaning+of+Relativity&rft.pages=58&rft.pub=Princeton+University+Press&rft.date=1950&rft.au=A+Einstein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fnum%3D10%26btnG%3DGoogle%2BSearch&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Rosser-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rosser_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_Geraint_Vaughan_Rosser1991" class="citation book cs1">William Geraint Vaughan Rosser (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zpjBEBbIjAIC&q=reference+%22laws+of+physics%22&pg=PA94"><i>Introductory Special Relativity</i></a>. CRC Press. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-85066-838-7" title="Special:BookSources/0-85066-838-7"><bdi>0-85066-838-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+Special+Relativity&rft.pages=3&rft.pub=CRC+Press&rft.date=1991&rft.isbn=0-85066-838-7&rft.au=William+Geraint+Vaughan+Rosser&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzpjBEBbIjAIC%26q%3Dreference%2B%2522laws%2Bof%2Bphysics%2522%26pg%3DPA94&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Feynman2-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Feynman2_31-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Phillips_Feynman1998" class="citation book cs1">Richard Phillips Feynman (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ipY8onVQWhcC&q=%22The+Principle+of+Relativity%22&pg=PA49"><i>Six not-so-easy pieces: Einstein's relativity, symmetry, and space-time</i></a>. Basic Books. p. 50. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-32842-9" title="Special:BookSources/0-201-32842-9"><bdi>0-201-32842-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Six+not-so-easy+pieces%3A+Einstein%27s+relativity%2C+symmetry%2C+and+space-time&rft.pages=50&rft.pub=Basic+Books&rft.date=1998&rft.isbn=0-201-32842-9&rft.au=Richard+Phillips+Feynman&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DipY8onVQWhcC%26q%3D%2522The%2BPrinciple%2Bof%2BRelativity%2522%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Principia-32"><span class="mw-cite-backlink">^ <a href="#cite_ref-Principia_32-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Principia_32-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See the <i>Principia</i> on line at <a rel="nofollow" class="external text" href="https://archive.org/stream/newtonspmathema00newtrich#page/n7/mode/2up">Andrew Motte Translation</a></span> </li> <li id="cite_note-note1-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-note1_33-0">^</a></b></span> <span class="reference-text">However, in the Newtonian system the Galilean transformation connects these frames and in the special theory of relativity the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> connects them. The two transformations agree for speeds of translation much less than the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>.</span> </li> <li id="cite_note-caveatondescription-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-caveatondescription_34-0">^</a></b></span> <span class="reference-text">That is, both descriptions are equivalent and can be used as needed. This equivalence does not hold outside of general relativity, e.g., in <a href="/wiki/Entropic_gravity" title="Entropic gravity">entropic gravity</a>.</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSkinner2014" class="citation book cs1">Skinner, Ray (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pnlpAwAAQBAJ"><i>Relativity for Scientists and Engineers</i></a> (reprinted ed.). Courier Corporation. p. 27. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-79367-2" title="Special:BookSources/978-0-486-79367-2"><bdi>978-0-486-79367-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity+for+Scientists+and+Engineers&rft.pages=27&rft.edition=reprinted&rft.pub=Courier+Corporation&rft.date=2014&rft.isbn=978-0-486-79367-2&rft.aulast=Skinner&rft.aufirst=Ray&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpnlpAwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pnlpAwAAQBAJ&pg=PA27">Extract of page 27</a></span> </li> <li id="cite_note-Landau-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-Landau_36-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLD_LandauLM_Lifshitz1975" class="citation book cs1">LD Landau; LM Lifshitz (1975). <i>The Classical Theory of Fields</i> (4th Revised English ed.). Pergamon Press. pp. 273–274. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-2768-9" title="Special:BookSources/978-0-7506-2768-9"><bdi>978-0-7506-2768-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Classical+Theory+of+Fields&rft.pages=273-274&rft.edition=4th+Revised+English&rft.pub=Pergamon+Press&rft.date=1975&rft.isbn=978-0-7506-2768-9&rft.au=LD+Landau&rft.au=LM+Lifshitz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Morin-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Morin_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Morin2008" class="citation book cs1">David Morin (2008). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontocl00mori"><i>Introduction to Classical Mechanics</i></a></span>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontocl00mori/page/649">649</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87622-3" title="Special:BookSources/978-0-521-87622-3"><bdi>978-0-521-87622-3</bdi></a>. <q>acceleration azimuthal Morin.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Classical+Mechanics&rft.pages=649&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0-521-87622-3&rft.au=David+Morin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontocl00mori&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Giancoli-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-Giancoli_38-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDouglas_C._Giancoli2007" class="citation book cs1">Douglas C. Giancoli (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xz-UEdtRmzkC&q=%22principle+of+equivalence%22&pg=PA155"><i>Physics for Scientists and Engineers with Modern Physics</i></a>. Pearson Prentice Hall. p. 155. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-149508-1" title="Special:BookSources/978-0-13-149508-1"><bdi>978-0-13-149508-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers+with+Modern+Physics&rft.pages=155&rft.pub=Pearson+Prentice+Hall&rft.date=2007&rft.isbn=978-0-13-149508-1&rft.au=Douglas+C.+Giancoli&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dxz-UEdtRmzkC%26q%3D%2522principle%2Bof%2Bequivalence%2522%26pg%3DPA155&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-General_theory-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-General_theory_39-0">^</a></b></span> <span class="reference-text">A. Einstein, "<a rel="nofollow" class="external text" href="http://www.relativitycalculator.com/pdfs/On_the_influence_of_Gravitation_on_the_Propagation_of_Light_English2.pdf">On the influence of gravitation on the propagation of light</a>", <i>Annalen der Physik</i>, vol. 35, (1911) : 898–908</span> </li> <li id="cite_note-NRC-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-NRC_40-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNational_Research_Council_(US)1986" class="citation book cs1">National Research Council (US) (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Hk1wj61PlocC&q=equivalence+gravitation&pg=PA15"><i>Physics Through the Nineteen Nineties: Overview</i></a>. National Academies Press. p. 15. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-309-03579-1" title="Special:BookSources/0-309-03579-1"><bdi>0-309-03579-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+Through+the+Nineteen+Nineties%3A+Overview&rft.pages=15&rft.pub=National+Academies+Press&rft.date=1986&rft.isbn=0-309-03579-1&rft.au=National+Research+Council+%28US%29&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHk1wj61PlocC%26q%3Dequivalence%2Bgravitation%26pg%3DPA15&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Franklin-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-Franklin_41-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAllan_Franklin2007" class="citation book cs1">Allan Franklin (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_RN-v31rXuIC&q=%22Eotvos+experiment%22&pg=PA66"><i>No Easy Answers: Science and the Pursuit of Knowledge</i></a>. University of Pittsburgh Press. p. 66. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8229-5968-7" title="Special:BookSources/978-0-8229-5968-7"><bdi>978-0-8229-5968-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=No+Easy+Answers%3A+Science+and+the+Pursuit+of+Knowledge&rft.pages=66&rft.pub=University+of+Pittsburgh+Press&rft.date=2007&rft.isbn=978-0-8229-5968-7&rft.au=Allan+Franklin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_RN-v31rXuIC%26q%3D%2522Eotvos%2Bexperiment%2522%26pg%3DPA66&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreen2000" class="citation book cs1">Green, Herbert S. (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CUJiQjSVCu8C"><i>Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process</i></a>. Springer. p. 154. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/354066517X" title="Special:BookSources/354066517X"><bdi>354066517X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Theory+and+Quantum+Physics%3A+Physical+Foundations+for+Understanding+the+Conscious+Process&rft.pages=154&rft.pub=Springer&rft.date=2000&rft.isbn=354066517X&rft.aulast=Green&rft.aufirst=Herbert+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCUJiQjSVCu8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CUJiQjSVCu8C&pg=PA154">Extract of page 154</a></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBandyopadhyay2000" class="citation book cs1">Bandyopadhyay, Nikhilendu (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qMOyfi_i0j8C"><i>Theory of Special Relativity</i></a>. Academic Publishers. p. 116. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/8186358528" title="Special:BookSources/8186358528"><bdi>8186358528</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Special+Relativity&rft.pages=116&rft.pub=Academic+Publishers&rft.date=2000&rft.isbn=8186358528&rft.aulast=Bandyopadhyay&rft.aufirst=Nikhilendu&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqMOyfi_i0j8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qMOyfi_i0j8C&pg=PA116">Extract of page 116</a></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiddleLyth2000" class="citation book cs1">Liddle, Andrew R.; Lyth, David H. (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XmWauPZSovMC"><i>Cosmological Inflation and Large-Scale Structure</i></a>. Cambridge University Press. p. 329. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-57598-2" title="Special:BookSources/0-521-57598-2"><bdi>0-521-57598-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cosmological+Inflation+and+Large-Scale+Structure&rft.pages=329&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=0-521-57598-2&rft.aulast=Liddle&rft.aufirst=Andrew+R.&rft.au=Lyth%2C+David+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXmWauPZSovMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XmWauPZSovMC&pg=PA329">Extract of page 329</a></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.mpiwg-berlin.mpg.de/Preprints/P271.PDF">In the Shadow of the Relativity Revolution</a> Section 3: The Work of Karl Schwarzschild (2.2 MB PDF-file)</span> </li> <li id="cite_note-Balbi-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-Balbi_46-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmedeo_Balbi2008" class="citation book cs1">Amedeo Balbi (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vEJM7s909CYC&q=CMB+%22rotation+of+the+universe%22&pg=PA58"><i>The Music of the Big Bang</i></a>. Springer. p. 59. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-78726-6" title="Special:BookSources/978-3-540-78726-6"><bdi>978-3-540-78726-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Music+of+the+Big+Bang&rft.pages=59&rft.pub=Springer&rft.date=2008&rft.isbn=978-3-540-78726-6&rft.au=Amedeo+Balbi&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvEJM7s909CYC%26q%3DCMB%2B%2522rotation%2Bof%2Bthe%2Buniverse%2522%26pg%3DPA58&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbraham_LoebMark_J._ReidAndreas_BrunthalerHeino_Falcke2005" class="citation journal cs1">Abraham Loeb; Mark J. Reid; Andreas Brunthaler; Heino Falcke (2005). <a rel="nofollow" class="external text" href="http://www.mpifr-bonn.mpg.de/staff/abrunthaler/pub/loeb.pdf">"Constraints on the proper motion of the Andromeda Galaxy based on the survival of its satellite M33"</a> <span class="cs1-format">(PDF)</span>. <i>The Astrophysical Journal</i>. <b>633</b> (2): 894–898. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/0506609">astro-ph/0506609</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005ApJ...633..894L">2005ApJ...633..894L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F491644">10.1086/491644</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17099715">17099715</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astrophysical+Journal&rft.atitle=Constraints+on+the+proper+motion+of+the+Andromeda+Galaxy+based+on+the+survival+of+its+satellite+M33&rft.volume=633&rft.issue=2&rft.pages=894-898&rft.date=2005&rft_id=info%3Aarxiv%2Fastro-ph%2F0506609&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17099715%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1086%2F491644&rft_id=info%3Abibcode%2F2005ApJ...633..894L&rft.au=Abraham+Loeb&rft.au=Mark+J.+Reid&rft.au=Andreas+Brunthaler&rft.au=Heino+Falcke&rft_id=http%3A%2F%2Fwww.mpifr-bonn.mpg.de%2Fstaff%2Fabrunthaler%2Fpub%2Floeb.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Stachel-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stachel_48-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_J._Stachel2002" class="citation book cs1">John J. Stachel (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OAsQ_hFjhrAC&q=%22laws+of+nature+took+a+simpler+form%22&pg=PA235"><i>Einstein from "B" to "Z"</i></a>. Springer. pp. 235–236. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-4143-2" title="Special:BookSources/0-8176-4143-2"><bdi>0-8176-4143-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein+from+%22B%22+to+%22Z%22&rft.pages=235-236&rft.pub=Springer&rft.date=2002&rft.isbn=0-8176-4143-2&rft.au=John+J.+Stachel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOAsQ_hFjhrAC%26q%3D%2522laws%2Bof%2Bnature%2Btook%2Ba%2Bsimpler%2Bform%2522%26pg%3DPA235&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Graneau-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-Graneau_49-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_GraneauNeal_Graneau2006" class="citation book cs1">Peter Graneau; Neal Graneau (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xpIJZxDkWAUC&q=universe+%22fixed+stars%22+date:2004-2010&pg=PA144"><i>In the Grip of the Distant Universe</i></a>. World Scientific. p. 147. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/981-256-754-2" title="Special:BookSources/981-256-754-2"><bdi>981-256-754-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=In+the+Grip+of+the+Distant+Universe&rft.pages=147&rft.pub=World+Scientific&rft.date=2006&rft.isbn=981-256-754-2&rft.au=Peter+Graneau&rft.au=Neal+Graneau&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxpIJZxDkWAUC%26q%3Duniverse%2B%2522fixed%2Bstars%2522%2Bdate%3A2004-2010%26pg%3DPA144&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Genz-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-Genz_50-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenning_Genz2001" class="citation book cs1">Henning Genz (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Cn_Q9wbDOM0C&q=%22rotation+of+the+universe%22&pg=PA274"><i>Nothingness</i></a>. Da Capo Press. p. 275. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7382-0610-5" title="Special:BookSources/0-7382-0610-5"><bdi>0-7382-0610-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nothingness&rft.pages=275&rft.pub=Da+Capo+Press&rft.date=2001&rft.isbn=0-7382-0610-5&rft.au=Henning+Genz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCn_Q9wbDOM0C%26q%3D%2522rotation%2Bof%2Bthe%2Buniverse%2522%26pg%3DPA274&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Thompson-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-Thompson_51-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ_Garcio-Bellido2005" class="citation book cs1">J Garcio-Bellido (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3TrsMTmbr-sC&q=CMB+%22rotation+of+the+universe%22&pg=PA32">"The Paradigm of Inflation"</a>. In J. M. T. Thompson (ed.). <i>Advances in Astronomy</i>. Imperial College Press. p. 32, §9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-86094-577-5" title="Special:BookSources/1-86094-577-5"><bdi>1-86094-577-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Paradigm+of+Inflation&rft.btitle=Advances+in+Astronomy&rft.pages=32%2C+%C2%A79&rft.pub=Imperial+College+Press&rft.date=2005&rft.isbn=1-86094-577-5&rft.au=J+Garcio-Bellido&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3TrsMTmbr-sC%26q%3DCMB%2B%2522rotation%2Bof%2Bthe%2Buniverse%2522%26pg%3DPA32&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Szydlowski-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-Szydlowski_52-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWlodzimierz_GodlowskiMarek_Szydlowski2003" class="citation journal cs1">Wlodzimierz Godlowski; Marek Szydlowski (2003). "Dark energy and global rotation of the Universe". <i>General Relativity and Gravitation</i>. <b>35</b> (12): 2171–2187. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/0303248">astro-ph/0303248</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003GReGr..35.2171G">2003GReGr..35.2171G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1027301723533">10.1023/A:1027301723533</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118988129">118988129</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=General+Relativity+and+Gravitation&rft.atitle=Dark+energy+and+global+rotation+of+the+Universe&rft.volume=35&rft.issue=12&rft.pages=2171-2187&rft.date=2003&rft_id=info%3Aarxiv%2Fastro-ph%2F0303248&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118988129%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FA%3A1027301723533&rft_id=info%3Abibcode%2F2003GReGr..35.2171G&rft.au=Wlodzimierz+Godlowski&rft.au=Marek+Szydlowski&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Birch-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-Birch_53-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirch1982" class="citation journal cs1">Birch, P. (29 July 1982). <a rel="nofollow" class="external text" href="http://www.nature.com/nature/journal/v298/n5873/abs/298451a0.html">"Is the Universe rotating?"</a>. <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>298</b> (5873): 451–454. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982Natur.298..451B">1982Natur.298..451B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F298451a0">10.1038/298451a0</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4343095">4343095</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Is+the+Universe+rotating%3F&rft.volume=298&rft.issue=5873&rft.pages=451-454&rft.date=1982-07-29&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4343095%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F298451a0&rft_id=info%3Abibcode%2F1982Natur.298..451B&rft.aulast=Birch&rft.aufirst=P.&rft_id=http%3A%2F%2Fwww.nature.com%2Fnature%2Fjournal%2Fv298%2Fn5873%2Fabs%2F298451a0.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilson2004" class="citation cs2">Gilson, James G. (1 September 2004), <i>Mach's Principle II</i>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0409010">physics/0409010</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004physics...9010G">2004physics...9010G</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mach%27s+Principle+II&rft.date=2004-09-01&rft_id=info%3Aarxiv%2Fphysics%2F0409010&rft_id=info%3Abibcode%2F2004physics...9010G&rft.aulast=Gilson&rft.aufirst=James+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-Arnold2-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-Arnold2_55-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFV._I._Arnol'd1989" class="citation book cs1">V. I. Arnol'd (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?num=10&btnG=Google+Search"><i>Mathematical Methods of Classical Mechanics</i></a>. Springer. p. 129. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96890-2" title="Special:BookSources/978-0-387-96890-2"><bdi>978-0-387-96890-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+of+Classical+Mechanics&rft.pages=129&rft.pub=Springer&rft.date=1989&rft.isbn=978-0-387-96890-2&rft.au=V.+I.+Arnol%27d&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fnum%3D10%26btnG%3DGoogle%2BSearch&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-note2-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-note2_56-0">^</a></b></span> <span class="reference-text">For example, there is no body providing a gravitational or electrical attraction.</span> </li> <li id="cite_note-tension-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-tension_57-0">^</a></b></span> <span class="reference-text">That is, the universality of the laws of physics requires the same tension to be seen by everybody. For example, it cannot happen that the string breaks under extreme tension in one frame of reference and remains intact in another frame of reference, just because we choose to look at the string from a different frame.</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChatfield1997" class="citation book cs1">Chatfield, Averil B. (1997). <i>Fundamentals of High Accuracy Inertial Navigation, Volume 174</i>. AIAA. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781600864278" title="Special:BookSources/9781600864278"><bdi>9781600864278</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+High+Accuracy+Inertial+Navigation%2C+Volume+174&rft.pub=AIAA&rft.date=1997&rft.isbn=9781600864278&rft.aulast=Chatfield&rft.aufirst=Averil+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKenniePetrie1993" class="citation book cs1">Kennie, T.J.M.; Petrie, G., eds. (1993). <i>Engineering Surveying Technology</i> (pbk. ed.). Hoboken: Taylor & Francis. p. 95. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780203860748" title="Special:BookSources/9780203860748"><bdi>9780203860748</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Surveying+Technology&rft.place=Hoboken&rft.pages=95&rft.edition=pbk.&rft.pub=Taylor+%26+Francis&rft.date=1993&rft.isbn=9780203860748&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> <li id="cite_note-l-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-l_60-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation magazine cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=YlEEAAAAMBAJ&pg=PA82">"The gyroscope pilots ships & planes"</a>. <i>Life</i>. 15 March 1943. pp. 80–83.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Life&rft.atitle=The+gyroscope+pilots+ships+%26+planes&rft.pages=80-83&rft.date=1943-03-15&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYlEEAAAAMBAJ%26pg%3DPA82&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <ul><li><a href="/wiki/Edwin_F._Taylor" title="Edwin F. Taylor">Edwin F. Taylor</a> and <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a>, <i>Spacetime Physics</i>, 2nd ed. (Freeman, NY, 1992)</li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, <i>Relativity, the special and the general theories</i>, 15th ed. (1954)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré1900" class="citation journal cs1"><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré, Henri</a> (1900). "La théorie de Lorentz et le Principe de Réaction". <i>Archives Neerlandaises</i>. <b>V</b>: 253–78.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archives+Neerlandaises&rft.atitle=La+th%C3%A9orie+de+Lorentz+et+le+Principe+de+R%C3%A9action&rft.volume=V&rft.pages=253-78&rft.date=1900&rft.aulast=Poincar%C3%A9&rft.aufirst=Henri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, <i>On the Electrodynamics of Moving Bodies</i>, included in <i>The Principle of Relativity</i>, page 38. Dover 1923</li></ul> <dl><dt>Rotation of the Universe</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJulian_B._BarbourHerbert_Pfister1998" class="citation book cs1">Julian B. Barbour; Herbert Pfister (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fKgQ9YpAcwMC&q=Birch++%22rotation+of+the+universe%22+-religion+-astrology+date:1990-2000&pg=PA445"><i>Mach's Principle: From Newton's Bucket to Quantum Gravity</i></a>. Birkhäuser. p. 445. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3823-7" title="Special:BookSources/0-8176-3823-7"><bdi>0-8176-3823-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mach%27s+Principle%3A+From+Newton%27s+Bucket+to+Quantum+Gravity&rft.pages=445&rft.pub=Birkh%C3%A4user&rft.date=1998&rft.isbn=0-8176-3823-7&rft.au=Julian+B.+Barbour&rft.au=Herbert+Pfister&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfKgQ9YpAcwMC%26q%3DBirch%2B%2B%2522rotation%2Bof%2Bthe%2Buniverse%2522%2B-religion%2B-astrology%2Bdate%3A1990-2000%26pg%3DPA445&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPJ_Nahin1999" class="citation book cs1">PJ Nahin (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=39KQY1FnSfkC&pg=PA369"><i>Time Machines</i></a>. Springer. p. 369; Footnote 12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98571-9" title="Special:BookSources/0-387-98571-9"><bdi>0-387-98571-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Machines&rft.pages=369%3B+Footnote+12&rft.pub=Springer&rft.date=1999&rft.isbn=0-387-98571-9&rft.au=PJ+Nahin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D39KQY1FnSfkC%26pg%3DPA369&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.nipne.ro/rjp/2008_53_1-2/0405_0416.pdf">B Ciobanu, I Radinchi</a> <i>Modeling the electric and magnetic fields in a rotating universe</i> Rom. Journ. Phys., Vol. 53, Nos. 1–2, P. 405–415, Bucharest, 2008</li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0206080v1">Yuri N. Obukhov, Thoralf Chrobok, Mike Scherfner</a> <i>Shear-free rotating inflation</i> Phys. Rev. D 66, 043518 (2002) [5 pages]</li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/0008106v1">Yuri N. Obukhov</a> <i>On physical foundations and observational effects of cosmic rotation</i> (2000)</li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/9703082v1">Li-Xin Li</a> <i>Effect of the Global Rotation of the Universe on the Formation of Galaxies</i> General Relativity and Gravitation, <b>30</b> (1998) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1018867011142">10.1023/A:1018867011142</a></li> <li><a rel="nofollow" class="external text" href="http://www.nature.com/nature/journal/v298/n5873/abs/298451a0.html">P Birch</a> <i>Is the Universe rotating?</i> Nature 298, 451 – 454 (29 July 1982)</li> <li><a rel="nofollow" class="external text" href="http://www.springerlink.com/content/t13ul36l27222351/fulltext.pdf?page=1">Kurt Gödel</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged February 2020">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup> <i>An example of a new type of cosmological solutions of Einstein’s field equations of gravitation</i> Rev. Mod. Phys., Vol. 21, p. 447, 1949.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/spacetime-iframes/">Stanford Encyclopedia of Philosophy entry</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=49JwbrXcPjc"><span class="plainlinks">Animation clip</span></a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=NblR01hHK6U">"Is Gravity An Illusion?"</a>. <i><a href="/wiki/PBS_Space_Time" class="mw-redirect" title="PBS Space Time">PBS Space Time</a></i>. 3 June 2015. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211113/NblR01hHK6U">Archived</a> from the original on 13 November 2021 – via <a href="/wiki/YouTube" title="YouTube">YouTube</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PBS+Space+Time&rft.atitle=Is+Gravity+An+Illusion%3F&rft.date=2015-06-03&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DNblR01hHK6U&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInertial+frame+of+reference" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox 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