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class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_sl.wikipedia.org&uselang=sl" class=""><span>Denarni prispevki</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Posebno:Registracija&returnto=Planckova+dol%C5%BEina" title="Predlagamo vam, da si ustvarite račun in se prijavite, vendar to ni obvezno." class=""><span>Ustvari račun</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Posebno:Prijava&returnto=Planckova+dol%C5%BEina" title="Prijava je zaželena, vendar ni obvezna [o]" accesskey="o" class=""><span>Prijava</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Več možnosti" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Osebna orodja" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Osebna orodja</span> </label> <div 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title="Prijava je zaželena, vendar ni obvezna [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Prijava</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Strani za neprijavljene urejevalce <a href="/wiki/Pomo%C4%8D:Uvod" aria-label="Več o urejanju"><span>več o tem</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Posebno:MojiPrispevki" title="Seznam urejanj s tega IP-naslova [y]" accesskey="y"><span>Prispevki</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Posebno:MojPogovor" title="Pogovor o urejanjih s tega IP-naslova [n]" accesskey="n"><span>Pogovorna stran</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Projekt"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Vsebina" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Vsebina</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">prestavi v stransko letvico</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrij</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Uvod</div> </a> </li> <li id="toc-Vrednost" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vrednost"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Vrednost</span> </div> </a> <ul id="toc-Vrednost-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zgodovina" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zgodovina"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Zgodovina</span> </div> </a> <ul id="toc-Zgodovina-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teoretični_in_fizikalni_pomen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teoretični_in_fizikalni_pomen"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Teoretični in fizikalni pomen</span> </div> </a> <ul id="toc-Teoretični_in_fizikalni_pomen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Planckova_dolžina_in_evklidska_geometrija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Planckova_dolžina_in_evklidska_geometrija"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Planckova dolžina in evklidska geometrija</span> </div> </a> <ul id="toc-Planckova_dolžina_in_evklidska_geometrija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Glej_tudi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Glej_tudi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Glej tudi</span> </div> </a> <ul id="toc-Glej_tudi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sklici" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sklici"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sklici</span> </div> </a> <ul id="toc-Sklici-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Viri" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Viri"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Viri</span> </div> </a> <ul id="toc-Viri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zunanje_povezave" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zunanje_povezave"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Zunanje povezave</span> </div> </a> <ul id="toc-Zunanje_povezave-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="Vsebina" 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="Vklopi kazalo vsebine" > <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">Vklopi kazalo vsebine</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">Planckova dolžina</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="P9jdi na članek v drugem jeziku. Na voljo v 51 jezikih." > <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-51" 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">51 jezikov</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Longaria_de_Planck" title="Longaria de Planck – aragonščina" lang="an" hreflang="an" data-title="Longaria de Planck" data-language-autonym="Aragonés" data-language-local-name="aragonščina" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D9%88%D9%84_%D8%A8%D9%84%D8%A7%D9%86%D9%83" title="طول بلانك – arabščina" lang="ar" hreflang="ar" data-title="طول بلانك" data-language-autonym="العربية" data-language-local-name="arabščina" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D0%BD%D0%BA%D0%B0%D1%9E%D1%81%D0%BA%D0%B0%D1%8F_%D0%B4%D0%B0%D1%9E%D0%B6%D1%8B%D0%BD%D1%8F" title="Планкаўская даўжыня – beloruščina" lang="be" hreflang="be" data-title="Планкаўская даўжыня" data-language-autonym="Беларуская" data-language-local-name="beloruščina" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D1%8A%D0%BB%D0%B6%D0%B8%D0%BD%D0%B0_%D0%BD%D0%B0_%D0%9F%D0%BB%D0%B0%D0%BD%D0%BA" title="Дължина на Планк – bolgarščina" lang="bg" hreflang="bg" data-title="Дължина на Планк" data-language-autonym="Български" data-language-local-name="bolgarščina" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%8D%E0%A6%B2%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%82%E0%A6%95%E0%A7%87%E0%A6%B0_%E0%A6%A6%E0%A7%88%E0%A6%B0%E0%A7%8D%E0%A6%98%E0%A7%8D%E0%A6%AF" title="প্ল্যাংকের দৈর্ঘ্য – bengalščina" lang="bn" hreflang="bn" data-title="প্ল্যাংকের দৈর্ঘ্য" data-language-autonym="বাংলা" data-language-local-name="bengalščina" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Planckova_du%C5%BEina" title="Planckova dužina – bosanščina" lang="bs" hreflang="bs" data-title="Planckova dužina" data-language-autonym="Bosanski" data-language-local-name="bosanščina" 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/Longitud_de_Planck" title="Longitud de Planck – katalonščina" lang="ca" hreflang="ca" data-title="Longitud de Planck" data-language-autonym="Català" data-language-local-name="katalonščina" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Planckova_d%C3%A9lka" title="Planckova délka – češčina" lang="cs" hreflang="cs" data-title="Planckova délka" data-language-autonym="Čeština" data-language-local-name="češčina" 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/Planck-l%C3%A6ngde" title="Planck-længde – danščina" lang="da" hreflang="da" data-title="Planck-længde" data-language-autonym="Dansk" data-language-local-name="danščina" 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/Planck-L%C3%A4nge" title="Planck-Länge – nemščina" lang="de" hreflang="de" data-title="Planck-Länge" data-language-autonym="Deutsch" data-language-local-name="nemščina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en badge-Q70893996 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Planck_length" title="Planck length – angleščina" lang="en" hreflang="en" data-title="Planck length" data-language-autonym="English" data-language-local-name="angleščina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Longo_de_Planck" title="Longo de Planck – esperanto" lang="eo" hreflang="eo" data-title="Longo de Planck" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Longitud_de_Planck" title="Longitud de Planck – španščina" lang="es" hreflang="es" data-title="Longitud de Planck" data-language-autonym="Español" data-language-local-name="španščina" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Plancki_pikkus" title="Plancki pikkus – estonščina" lang="et" hreflang="et" data-title="Plancki pikkus" data-language-autonym="Eesti" data-language-local-name="estonščina" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Planck_luzera" title="Planck luzera – baskovščina" lang="eu" hreflang="eu" data-title="Planck luzera" data-language-autonym="Euskara" data-language-local-name="baskovščina" 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%B7%D9%88%D9%84_%D9%BE%D9%84%D8%A7%D9%86%DA%A9" title="طول پلانک – perzijščina" lang="fa" hreflang="fa" data-title="طول پلانک" data-language-autonym="فارسی" data-language-local-name="perzijščina" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Planckin_pituus" title="Planckin pituus – finščina" lang="fi" hreflang="fi" data-title="Planckin pituus" data-language-autonym="Suomi" data-language-local-name="finščina" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Longueur_de_Planck" title="Longueur de Planck – francoščina" lang="fr" hreflang="fr" data-title="Longueur de Planck" data-language-autonym="Français" data-language-local-name="francoščina" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Fad_Planck" title="Fad Planck – irščina" lang="ga" hreflang="ga" data-title="Fad Planck" data-language-autonym="Gaeilge" data-language-local-name="irščina" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A8%D7%9A_%D7%A4%D7%9C%D7%90%D7%A0%D7%A7" title="אורך פלאנק – hebrejščina" lang="he" hreflang="he" data-title="אורך פלאנק" data-language-autonym="עברית" data-language-local-name="hebrejščina" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Planckova_duljina" title="Planckova duljina – hrvaščina" lang="hr" hreflang="hr" data-title="Planckova duljina" data-language-autonym="Hrvatski" data-language-local-name="hrvaščina" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Planck-hossz" title="Planck-hossz – madžarščina" lang="hu" hreflang="hu" data-title="Planck-hossz" data-language-autonym="Magyar" data-language-local-name="madžarščina" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%AC%D5%A1%D5%B6%D5%AF%D5%AB_%D5%A5%D6%80%D5%AF%D5%A1%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Պլանկի երկարություն – armenščina" lang="hy" hreflang="hy" data-title="Պլանկի երկարություն" data-language-autonym="Հայերեն" data-language-local-name="armenščina" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Panjang_Planck" title="Panjang Planck – indonezijščina" lang="id" hreflang="id" data-title="Panjang Planck" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijščina" 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/Lunghezza_di_Planck" title="Lunghezza di Planck – italijanščina" lang="it" hreflang="it" data-title="Lunghezza di Planck" data-language-autonym="Italiano" data-language-local-name="italijanščina" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%97%E3%83%A9%E3%83%B3%E3%82%AF%E9%95%B7" title="プランク長 – japonščina" lang="ja" hreflang="ja" data-title="プランク長" data-language-autonym="日本語" data-language-local-name="japonščina" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%8C%EB%9E%91%ED%81%AC_%EA%B8%B8%EC%9D%B4" title="플랑크 길이 – korejščina" lang="ko" hreflang="ko" data-title="플랑크 길이" data-language-autonym="한국어" data-language-local-name="korejščina" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Longitudo_Planckiana" title="Longitudo Planckiana – latinščina" lang="la" hreflang="la" data-title="Longitudo Planckiana" data-language-autonym="Latina" data-language-local-name="latinščina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Planko_ilgis" title="Planko ilgis – litovščina" lang="lt" hreflang="lt" data-title="Planko ilgis" data-language-autonym="Lietuvių" data-language-local-name="litovščina" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D0%BD%D0%BA%D0%BE%D0%B2%D0%B0_%D0%B4%D0%BE%D0%BB%D0%B6%D0%B8%D0%BD%D0%B0" title="Планкова должина – makedonščina" lang="mk" hreflang="mk" data-title="Планкова должина" data-language-autonym="Македонски" data-language-local-name="makedonščina" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Panjang_Planck" title="Panjang Planck – malajščina" lang="ms" hreflang="ms" data-title="Panjang Planck" data-language-autonym="Bahasa Melayu" data-language-local-name="malajščina" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Plancklengte" title="Plancklengte – nizozemščina" lang="nl" hreflang="nl" data-title="Plancklengte" data-language-autonym="Nederlands" data-language-local-name="nizozemščina" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A8%B2%E0%A9%88%E0%A8%82%E0%A8%95_%E0%A8%B2%E0%A9%B0%E0%A8%AC%E0%A8%BE%E0%A8%88" title="ਪਲੈਂਕ ਲੰਬਾਈ – pandžabščina" lang="pa" hreflang="pa" data-title="ਪਲੈਂਕ ਲੰਬਾਈ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžabščina" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/D%C5%82ugo%C5%9B%C4%87_Plancka" title="Długość Plancka – poljščina" lang="pl" hreflang="pl" data-title="Długość Plancka" data-language-autonym="Polski" data-language-local-name="poljščina" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%BE%D9%84%D8%A7%D9%86%DA%A9_%D9%84%D9%85%D8%A8%D8%A7%D8%A6%DB%8C" title="پلانک لمبائی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="پلانک لمبائی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Comprimento_de_Planck" title="Comprimento de Planck – portugalščina" lang="pt" hreflang="pt" data-title="Comprimento de Planck" data-language-autonym="Português" data-language-local-name="portugalščina" 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/Lungime_Planck" title="Lungime Planck – romunščina" lang="ro" hreflang="ro" data-title="Lungime Planck" data-language-autonym="Română" data-language-local-name="romunščina" 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%9F%D0%BB%D0%B0%D0%BD%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D0%B4%D0%BB%D0%B8%D0%BD%D0%B0" title="Планковская длина – ruščina" lang="ru" hreflang="ru" data-title="Планковская длина" data-language-autonym="Русский" data-language-local-name="ruščina" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Lunchizza_di_Planck" title="Lunchizza di Planck – sicilijanščina" lang="scn" hreflang="scn" data-title="Lunchizza di Planck" data-language-autonym="Sicilianu" data-language-local-name="sicilijanščina" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Planckova_duljina" title="Planckova duljina – srbohrvaščina" lang="sh" hreflang="sh" data-title="Planckova duljina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbohrvaščina" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E0%B6%BD%E0%B7%8F%E0%B6%B1%E0%B7%8A%E0%B6%9A%E0%B7%8A_%E0%B6%AF%E0%B7%96%E0%B6%BB%E0%B6%9A%E0%B6%BA" title="ප්ලාන්ක් දූරකය – sinhalščina" lang="si" hreflang="si" data-title="ප්ලාන්ක් දූරකය" data-language-autonym="සිංහල" data-language-local-name="sinhalščina" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Planck_length" title="Planck length – Simple English" lang="en-simple" hreflang="en-simple" data-title="Planck length" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Planckova_d%C4%BA%C5%BEka" title="Planckova dĺžka – slovaščina" lang="sk" hreflang="sk" data-title="Planckova dĺžka" data-language-autonym="Slovenčina" data-language-local-name="slovaščina" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D0%BD%D0%BA%D0%BE%D0%B2%D0%B0_%D0%B4%D1%83%D0%B6%D0%B8%D0%BD%D0%B0" title="Планкова дужина – srbščina" lang="sr" hreflang="sr" data-title="Планкова дужина" data-language-autonym="Српски / srpski" data-language-local-name="srbščina" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Planckl%C3%A4ngd" title="Plancklängd – švedščina" lang="sv" 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class="noprint">Iz Wikipedije, proste enciklopedije</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sl" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r5913888">.mw-parser-output .infobox{border:1px solid #a2a9b1;border-spacing:3px;background-color:#f8f9fa;color:black;margin:0.5em 0 0.5em 1em;padding:0.2em;float:right;clear:right;font-size:88%;line-height:1.5em;width:22em}.mw-parser-output .infobox-header,.mw-parser-output .infobox-label,.mw-parser-output .infobox-above,.mw-parser-output .infobox-full-data,.mw-parser-output .infobox-data,.mw-parser-output .infobox-below,.mw-parser-output .infobox-subheader,.mw-parser-output .infobox-image,.mw-parser-output .infobox-navbar,.mw-parser-output .infobox th,.mw-parser-output .infobox td{vertical-align:top}.mw-parser-output .infobox-label,.mw-parser-output .infobox-data,.mw-parser-output .infobox th,.mw-parser-output .infobox td{text-align:left}.mw-parser-output .infobox .infobox-header,.mw-parser-output .infobox .infobox-subheader,.mw-parser-output .infobox .infobox-image,.mw-parser-output .infobox .infobox-full-data,.mw-parser-output .infobox .infobox-below{text-align:center}.mw-parser-output .infobox .infobox-above,.mw-parser-output .infobox .infobox-title,.mw-parser-output .infobox caption{font-size:125%;font-weight:bold;text-align:center}.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}</style><table class="infobox" style="background:;"><tbody><tr><th colspan="2" class="infobox-above">Planckova dolžina</th></tr><tr><th scope="row" class="infobox-label">Sistem enote</th><td class="infobox-data"><a href="/wiki/Planckov_sistem_enot" title="Planckov sistem enot">Planckove enote</a></td></tr><tr><th scope="row" class="infobox-label">Enota</th><td class="infobox-data"><a href="/wiki/Dol%C5%BEina" title="Dolžina">dolžina</a></td></tr><tr><th scope="row" class="infobox-label">Simbol</th><td class="infobox-data"><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 \ell _{\rm {P}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {P}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faac47638f1b6e0f4e5fdd0492be67ba0aacc152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.321ex; height:2.509ex;" alt="{\displaystyle \ell _{\rm {P}}\!\,}"></span> </td></tr><tr><th scope="row" class="infobox-label">Poimenovano po</th><td class="infobox-data"><a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a></td></tr><tr><th colspan="2" class="infobox-header">Pretvorbe enote </th></tr><tr class="nowrap"><td><i>1 <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 \ell _{\rm {P}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {P}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faac47638f1b6e0f4e5fdd0492be67ba0aacc152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.321ex; height:2.509ex;" alt="{\displaystyle \ell _{\rm {P}}\!\,}"></span> v ...</i></td><td><i>... je enako ...</i></td></tr><tr style="display:none"><th colspan="2"> </th></tr><tr><th scope="row" class="infobox-label"><span class="nowrap">   </span><a href="/wiki/Mednarodni_sistem_enot" title="Mednarodni sistem enot">enote SI</a></th><td class="infobox-data"><span class="nowrap">   </span>1,616 255(18) <span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="2998650000000000000♠"></span>−35</span></sup> <a href="/wiki/Meter" title="Meter">m</a></td></tr><tr><th scope="row" class="infobox-label"><span class="nowrap">   </span><a href="/wiki/Naravne_enote" title="Naravne enote">naravne enote</a></th><td class="infobox-data"><span class="nowrap">   </span>11,706 <a href="/w/index.php?title=Stoneyjeve_enote&action=edit&redlink=1" class="new" title="Stoneyjeve enote (stran ne obstaja)"><span class="texhtml"><var>ℓ</var><sub>S</sub></span></a><br /> 3,054 2 <span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="2998750000000000000♠"></span>−25</span></sup> <a href="/wiki/Bohrov_polmer" title="Bohrov polmer"><i>a</i><sub>0</sub></a></td></tr></tbody></table> <p><b>Planckova dolžina</b> (oznake <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 \ell _{\rm {P}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {P}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faac47638f1b6e0f4e5fdd0492be67ba0aacc152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.321ex; height:2.509ex;" alt="{\displaystyle \ell _{\rm {P}}\!\,}"></span>, <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 l_{\rm {P}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{\rm {P}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a8868bfe00b891d95805114e5ad7e624fd9221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.045ex; height:2.509ex;" alt="{\displaystyle l_{\rm {P}}\!\,}"></span>, <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 L_{\rm {P}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\rm {P}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e1ce8107ad431b3a69eaa56252265f1775990a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle L_{\rm {P}}\!\,}"></span> in <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 l_{\rm {Pl}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{\rm {Pl}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5c44f92547551628666b608e09f42609e4d6ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.502ex; height:2.509ex;" alt="{\displaystyle l_{\rm {Pl}}\!\,}"></span>) je v <a href="/wiki/Fizika" title="Fizika">fiziki</a> <a href="/wiki/Naravne_enote" title="Naravne enote">naravna enota</a> za <a href="/wiki/Dol%C5%BEina" title="Dolžina">dolžino</a> in predstavlja <a href="/wiki/Razdalja" title="Razdalja">razdaljo</a>, ki jo prepotuje <a href="/wiki/Svetloba" title="Svetloba">svetloba</a> v <a href="/wiki/Planckov_%C4%8Das" title="Planckov čas">Planckovem času</a>. Je ena izmed osnovnih enot v <a href="/wiki/Planckov_sistem_enot" title="Planckov sistem enot">Planckovem sistemu enot</a> (druge so še Planckov čas, <a href="/wiki/Planckov_naboj" title="Planckov naboj">Planckov naboj</a> in <a href="/wiki/Planckova_temperatura" title="Planckova temperatura">Planckova temperatura</a>). Planckova dolžina se lahko izpelje iz treh <a href="/wiki/Osnovna_fizikalna_konstanta" class="mw-redirect" title="Osnovna fizikalna konstanta">osnovnih fizikalnih konstant</a>: <a href="/wiki/Hitrost_svetlobe" title="Hitrost svetlobe">hitrosti svetlobe</a> v <a href="/wiki/Vakuum" title="Vakuum">vakuumu</a>, <a href="/wiki/Planckova_konstanta" title="Planckova konstanta">Planckove konstante</a> in <a href="/wiki/Gravitacijska_konstanta" title="Gravitacijska konstanta">gravitacijske konstante</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Vrednost">Vrednost</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Planckova_dol%C5%BEina&veaction=edit&section=1" title="Uredi razdelek: Vrednost" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Planckova_dol%C5%BEina&action=edit&section=1" title="Urejanje izvorne kode razdelka: Vrednost"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Planckova dolžina je definirana kot: </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 \ell _{\rm {P}}={\sqrt {\frac {\hbar \kappa }{c^{3}}}}\approx 1,616\;255(18)\cdot 10^{-35}{\mbox{ m}}\!\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>κ</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> <mo>≈</mo> <mn>1</mn> <mo>,</mo> <mn>616</mn> <mspace width="thickmathspace" /> <mn>255</mn> <mo stretchy="false">(</mo> <mn>18</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>35</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> m</mtext> </mstyle> </mrow> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {P}}={\sqrt {\frac {\hbar \kappa }{c^{3}}}}\approx 1,616\;255(18)\cdot 10^{-35}{\mbox{ m}}\!\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ce229f9ad73bc7cd0808a9af78ff60042c2270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.596ex; height:6.343ex;" alt="{\displaystyle \ell _{\rm {P}}={\sqrt {\frac {\hbar \kappa }{c^{3}}}}\approx 1,616\;255(18)\cdot 10^{-35}{\mbox{ m}}\!\,,}"></span> <sup id="cite_ref-baez_1999_1-0" class="reference"><a href="#cite_note-baez_1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-nist_2019_2-0" class="reference"><a href="#cite_note-nist_2019-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <p>kjer je: </p> <ul><li><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 c\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829863009c9092f792db1e6e878d6badcc6eb71c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c\!\,}"></span> – <a href="/wiki/Hitrost_svetlobe" title="Hitrost svetlobe">hitrost svetlobe</a> v <a href="/wiki/Vakuum" title="Vakuum">vakuumu</a>,</li> <li><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 \kappa \!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa \!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec56180596f88f97104539b2291211bcfd7f634f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa \!\,}"></span> – <a href="/wiki/Gravitacijska_konstanta" title="Gravitacijska konstanta">gravitacijska konstanta</a>,</li> <li><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 \hbar =h/2\pi \!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =h/2\pi \!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c003cab79b611e3768ad46b48f10c7661f5f325b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.401ex; height:2.843ex;" alt="{\displaystyle \hbar =h/2\pi \!\,}"></span> – <a href="/wiki/Reducirana_Planckova_konstanta" class="mw-redirect" title="Reducirana Planckova konstanta">reducirana Planckova konstanta</a>.</li></ul> <p>Zadnji dve števki v oklepaju predstavljata ocenjeno <a href="/w/index.php?title=Standardna_napaka&action=edit&redlink=1" class="new" title="Standardna napaka (stran ne obstaja)">standardno napako</a>. Relativna standardna <a href="/w/index.php?title=Merilna_negotovost&action=edit&redlink=1" class="new" title="Merilna negotovost (stran ne obstaja)">merilna negotovost</a> vrednosti je <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 u_{\rm {r}}=1,1\cdot 10^{-5}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\rm {r}}=1,1\cdot 10^{-5}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6529eaea40ab394db1fbab8c8a98e6499a0c9a46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.001ex; height:3.009ex;" alt="{\displaystyle u_{\rm {r}}=1,1\cdot 10^{-5}\!\,}"></span><sup id="cite_ref-nist_2019_2-1" class="reference"><a href="#cite_note-nist_2019-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Planckova dolžina je približno 10<sup>-20</sup> krat manjša od velikosti <a href="/wiki/Proton" title="Proton">protona</a>.<sup id="cite_ref-physlink_3-0" class="reference"><a href="#cite_note-physlink-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-baez_1999_1-1" class="reference"><a href="#cite_note-baez_1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Njena vrednost se lahko definira s pomočjo polmera domnevnega <a href="/wiki/Planckov_delec" title="Planckov delec">Planckovega delca</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Zgodovina">Zgodovina</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Planckova_dol%C5%BEina&veaction=edit&section=2" title="Uredi razdelek: Zgodovina" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Planckova_dol%C5%BEina&action=edit&section=2" title="Urejanje izvorne kode razdelka: Zgodovina"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a> je leta <a href="/w/index.php?title=1899_v_znanosti&action=edit&redlink=1" class="new" title="1899 v znanosti (stran ne obstaja)">1899</a> predlagal nekatere osnovne naravne enote za dolžino, maso, čas in energijo.<sup id="cite_ref-planck_1899_4-0" class="reference"><a href="#cite_note-planck_1899-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gorelik_1992_5-0" class="reference"><a href="#cite_note-gorelik_1992-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Te enote je izpeljal s pomočjo <a href="/wiki/Razse%C5%BEnostna_analiza" title="Razsežnostna analiza">razsežnostne analize</a>, pri čemer je uporabil le gravitacijsko konstanto, hitrost svetlobe in »enoto akcije«, ki je kasneje postala znana kot »Planckova konstanta«. Te naravne enote, ki jih je naprej izpeljal, so postale znane kot »Planckova dolžina«, »Planckova masa«, »Planckov čas« in »Planckova energija«. </p> <div class="mw-heading mw-heading2"><h2 id="Teoretični_in_fizikalni_pomen"><span id="Teoreti.C4.8Dni_in_fizikalni_pomen"></span>Teoretični in fizikalni pomen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Planckova_dol%C5%BEina&veaction=edit&section=3" title="Uredi razdelek: Teoretični in fizikalni pomen" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Planckova_dol%C5%BEina&action=edit&section=3" title="Urejanje izvorne kode razdelka: Teoretični in fizikalni pomen"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fizikalni pomen Planckove dolžine še ni popolnoma jasen. Razsežnostna analiza kaže na to, da ima posebni pomen v <a href="/wiki/Kvantna_gravitacija" title="Kvantna gravitacija">kvantni gravitaciji</a>. Predvidevajo, da je na razdaljah z velikostjo 1 Planckove dolžine zgradba <a href="/wiki/Prostor-%C4%8Das" title="Prostor-čas">prostor-časa</a> drugačna od običajnega <a href="/w/index.php?title=%C5%A0tirirazse%C5%BEni_prostor&action=edit&redlink=1" class="new" title="Štirirazsežni prostor (stran ne obstaja)">4-razsežnega</a> sveta.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Imenujejo jo tudi <i>kvant dolžine</i>, ker naj bi imela velikost <a href="/w/index.php?title=Struna_(fizika)&action=edit&redlink=1" class="new" title="Struna (fizika) (stran ne obstaja)">strune</a> v <a href="/wiki/Teorija_superstrun" title="Teorija superstrun">teoriji superstrun</a>.<sup id="cite_ref-eos_7-0" class="reference"><a href="#cite_note-eos-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>V merilu Planckove dolžine tako kvantnogravitacijski pojavi verjetno postanejo vidni in je treba interakcije analizirati z veljavno teorijo kvantne gravitacije.<sup id="cite_ref-physicsforums_8-0" class="reference"><a href="#cite_note-physicsforums-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> V Planckovem območju površina sferične črne luknje naraste, ko črna luknja požre en bit informacije.<sup id="cite_ref-bekenstein_1973_9-0" class="reference"><a href="#cite_note-bekenstein_1973-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Če bi se v sfero s premerom <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 l_{\rm {P}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{\rm {P}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a8868bfe00b891d95805114e5ad7e624fd9221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.045ex; height:2.509ex;" alt="{\displaystyle l_{\rm {P}}\!\,}"></span> zaprlo telo, bi to zahtevalo toliko energije, da bi se telo sesedlo v miniaturno črno luknjo. V tem smislu večina fizikov meni, da je obravnavanje dolžin, manjših od Planckove dolžine, ali časov, manjših od Planckovega časa, brez pomena. Pri tem privzemajo, da če <a href="/wiki/Gola_singularnost" title="Gola singularnost">gole singularnosti</a> obstajajo, v njih zakoni splošne teorije relativnosti odpovedo in jih morajo zamenjati tisti iz kvantne teorije gravitacije.<sup id="cite_ref-aaronson_2005_10-0" class="reference"><a href="#cite_note-aaronson_2005-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Merjenje česarkoli v velikosti Planckove dolžine zahteva, da je zaradi Heisenbergovega <a href="/wiki/Na%C4%8Delo_nedolo%C4%8Denosti" title="Načelo nedoločenosti">načela nedoločenosti</a> <a href="/wiki/Gibalna_koli%C4%8Dina" title="Gibalna količina">gibalna količina</a> <a href="/wiki/Foton" title="Foton">fotona</a> zelo velika, tako, da bi zaradi toliko energije v tako majhnem prostoru nastala miniaturna črna luknja (<a href="/w/index.php?title=Planckov_geon&action=edit&redlink=1" class="new" title="Planckov geon (stran ne obstaja)">Planckov geon</a>) s premerom <a href="/wiki/Dogodkovno_obzorje" title="Dogodkovno obzorje">dogodkovnega obzorja</a> enakim Planckovi dolžini. </p><p>Glavni del vpliva kvantne gravitacije je odvisen od načela nedoločenosti <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 \delta r_{\rm {s}}\,\delta r\geq \ell _{\rm {P}}^{2}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>r</mi> <mo>≥</mo> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta r_{\rm {s}}\,\delta r\geq \ell _{\rm {P}}^{2}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39bc2611fece1fc8d98a22e3bbfd7ad01d3e4fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.881ex; height:3.176ex;" alt="{\displaystyle \delta r_{\rm {s}}\,\delta r\geq \ell _{\rm {P}}^{2}\!\,}"></span>, kjer je <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 r_{\rm {s}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\rm {s}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcb84004e678cabac839d5169aba6738cb4ffe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.929ex; height:2.009ex;" alt="{\displaystyle r_{\rm {s}}\!\,}"></span> <a href="/wiki/Schwarzschildov_polmer" title="Schwarzschildov polmer">gravitacijski polmer</a>, <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 r\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3764051de5c095ec1663f341d925fe85eb800f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r\!\,}"></span> <a href="/wiki/Polarni_koordinatni_sistem" title="Polarni koordinatni sistem">radialna koordinata</a> in <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 \ell _{\rm {P}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {P}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faac47638f1b6e0f4e5fdd0492be67ba0aacc152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.321ex; height:2.509ex;" alt="{\displaystyle \ell _{\rm {P}}\!\,}"></span> Planckova dolžina. To načelo nedoločenosti je druga oblika Heisenbergovega načela nedoločenosti med gibalno količino in koordinato v Planckovem merilu. To razmerje se res lahko napiše kot <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 \delta (2\kappa m/c^{2})\,\delta r\geq \kappa \hbar /c^{3}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>κ</mi> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>r</mi> <mo>≥</mo> <mi>κ</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (2\kappa m/c^{2})\,\delta r\geq \kappa \hbar /c^{3}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02bb6219fae645f862ac97d9102ba51208a7cad6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.075ex; height:3.176ex;" alt="{\displaystyle \delta (2\kappa m/c^{2})\,\delta r\geq \kappa \hbar /c^{3}\!\,}"></span>, kjer je <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 \kappa \!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa \!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec56180596f88f97104539b2291211bcfd7f634f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa \!\,}"></span> gravitacijska konstanta, <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 m\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1566f742a01abf7df6a1a331325f798b559b94d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m\!\,}"></span> masa telesa, <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 c\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829863009c9092f792db1e6e878d6badcc6eb71c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c\!\,}"></span> hitrost svetlobe in <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 \hbar \!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56c657646cdc3361d84c54b55a9529a7b4fe461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar \!\,}"></span> reducirana Planckova konstanta. Če se na obeh straneh reducirajo enake konstante, izhaja Heisenbergovo načelo nedoločenosti <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 \delta (mc)\,\delta r\geq \hbar /2\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>r</mi> <mo>≥</mo> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (mc)\,\delta r\geq \hbar /2\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e305431083e054d5fb6f59c2108845f943841439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.119ex; height:2.843ex;" alt="{\displaystyle \delta (mc)\,\delta r\geq \hbar /2\!\,}"></span>. Načelo nedoločenosti <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 \delta r_{\rm {s}}\,\delta r\geq \ell _{\rm {P}}^{2}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>r</mi> <mo>≥</mo> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta r_{\rm {s}}\,\delta r\geq \ell _{\rm {P}}^{2}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39bc2611fece1fc8d98a22e3bbfd7ad01d3e4fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.881ex; height:3.176ex;" alt="{\displaystyle \delta r_{\rm {s}}\,\delta r\geq \ell _{\rm {P}}^{2}\!\,}"></span> napoveduje obstoj <a href="/w/index.php?title=Virtualna_%C4%8Drna_luknja&action=edit&redlink=1" class="new" title="Virtualna črna luknja (stran ne obstaja)">virtualne črne luknje</a> in <a href="/wiki/%C4%8Crvina" title="Črvina">črvine</a> (<a href="/w/index.php?title=Kvantna_pena&action=edit&redlink=1" class="new" title="Kvantna pena (stran ne obstaja)">kvantno peno</a>) v Planckovem merilu.<sup id="cite_ref-misner_1973_11-0" class="reference"><a href="#cite_note-misner_1973-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-klimets_2017_12-0" class="reference"><a href="#cite_note-klimets_2017-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Iz vsakega poskusa raziskovanja možnega obstoja krajših razdalj s pomočjo trkov višjih energij neizbežno sledi nastanek takšnih črnih lukenj. V trkih z višjo energijo bi namesto delitve snovi v še manjše dele preprosto nastajale večje črne luknje.<sup id="cite_ref-carr_2005_13-0" class="reference"><a href="#cite_note-carr_2005-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Zmanjšanje nedoločenosti radialne koordinate <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 \delta r\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>r</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta r\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18a7979534b49f240b499d107688a9ebbd604ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.097ex; height:2.343ex;" alt="{\displaystyle \delta r\!\,}"></span> povzroči povečanje nedoločenosti gravitacijskega polmera <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 \delta r_{\rm {s}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta r_{\rm {s}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7605140d1a7cbd83b082566dfc053f6054f631c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.977ex; height:2.676ex;" alt="{\displaystyle \delta r_{\rm {s}}\!\,}"></span> in obratno. </p><p>Nastanek virtualnih črnih lukenj je v Planckovem merilu najučinkovitejše v <a href="/wiki/Trirazse%C5%BEni_prostor" title="Trirazsežni prostor">trirazsežnem prostoru</a>, kjer ima <a href="/w/index.php?title=Polna_energija&action=edit&redlink=1" class="new" title="Polna energija (stran ne obstaja)">polna energija</a> Planckovega geona najnižji maksimum. Le v enorazsežnem prostoru takšne črne luknje ne morejo nastati, v vseh drugih pa lahko.<sup id="cite_ref-klimets_2017_12-1" class="reference"><a href="#cite_note-klimets_2017-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Planckovo dolžino včasih napačno istovetijo z <a href="/wiki/Kvantizacija_(fizika)" class="mw-redirect" title="Kvantizacija (fizika)">najmanjšo dolžino</a> prostor-časa, tega pa običajna fizika ne sprejema, saj bi to zahtevalo kršitev ali spremembo <a href="/w/index.php?title=Lorentzeva_simetrija&action=edit&redlink=1" class="new" title="Lorentzeva simetrija (stran ne obstaja)">Lorentzeve simetrije</a>.<sup id="cite_ref-physicsforums_8-1" class="reference"><a href="#cite_note-physicsforums-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Vendar nekatere teorije <a href="/w/index.php?title=Zan%C4%8Dna_kvantna_gravitacija&action=edit&redlink=1" class="new" title="Zančna kvantna gravitacija (stran ne obstaja)">zančne kvantne gravitacije</a> poskušajo uvesti najmanjšo dolžino v merilu Planckove dolžine, čeprav ne nujno Planckove dolžine same,<sup id="cite_ref-physicsforums_8-2" class="reference"><a href="#cite_note-physicsforums-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> ali uvesti Planckovo dolžino kot opazovalčevo invarianto, znano kot <a href="/w/index.php?title=Dvojna_posebna_teorija_relativnosti&action=edit&redlink=1" class="new" title="Dvojna posebna teorija relativnosti (stran ne obstaja)">dvojna posebna teorija relativnosti</a>. </p><p>Strune v teoriji strun so modelirane v velikosti reda Planckove dolžine.<sup id="cite_ref-physicsforums_8-3" class="reference"><a href="#cite_note-physicsforums-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-burgess_2007_14-0" class="reference"><a href="#cite_note-burgess_2007-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> V teorijah <a href="/w/index.php?title=Velika_dodatna_razse%C5%BEnost&action=edit&redlink=1" class="new" title="Velika dodatna razsežnost (stran ne obstaja)">velikih dodatnih razsežnostih</a> Planckova dolžina nima osnovnega fizikalnega pomena, kvantnogravitacijski pojavi pa se pojavljajo v drugih merilih. </p> <div class="mw-heading mw-heading2"><h2 id="Planckova_dolžina_in_evklidska_geometrija"><span id="Planckova_dol.C5.BEina_in_evklidska_geometrija"></span>Planckova dolžina in evklidska geometrija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Planckova_dol%C5%BEina&veaction=edit&section=4" title="Uredi razdelek: Planckova dolžina in evklidska geometrija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Planckova_dol%C5%BEina&action=edit&section=4" title="Urejanje izvorne kode razdelka: Planckova dolžina in evklidska geometrija"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Planckova dolžina je dolžina pri kateri kvantna ničelna nihanja jakosti <a href="/wiki/Gravitacijsko_polje" title="Gravitacijsko polje">gravitacijskega polja</a> popolnoma popačijo <a href="/wiki/Evklidska_geometrija" title="Evklidska geometrija">evklidsko geometrijo</a>.<sup id="cite_ref-migdal_1989_15-0" class="reference"><a href="#cite_note-migdal_1989-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Gravitacijsko polje izvaja ničelna nihanja in tudi geometrija povezana z njim niha. Razmerje med obsegom in polmerom se spreminja blizu evklidske vrednosti. Manjše je merilo, večji so odkloni od evklidske geometrije. Pri oceni reda valovne dolžine ničelnih gravitacijskih nihanj, pri kateri geometrija postane popolnoma različna od evklidske, je stopnja odklona <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 \zeta \!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta \!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e37e03ac25b904e0c3539fa9ae36f1b093711a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta \!\,}"></span> geometrije od evklidske v gravitacijskem polju, določena z razmerjem med <a href="/wiki/Gravitacijski_potencial" class="mw-redirect" title="Gravitacijski potencial">gravitacijskim potencialom</a> <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 \varphi \!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81933e4a7365a58ef1035a2114c0dd29e6a28b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi \!\,}"></span> in kvadratom hitrosti svetlobe <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 c\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829863009c9092f792db1e6e878d6badcc6eb71c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c\!\,}"></span>: <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 \zeta =\varphi /c^{2}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta =\varphi /c^{2}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8262934493686c453c470b6f3495dd61fc059f3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle \zeta =\varphi /c^{2}\!\,}"></span>. Ko je <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 \zeta \ll 1\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>≪</mo> <mn>1</mn> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta \ll 1\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e741bbe90128490c3cfc3d040d64bc13fab2db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.872ex; height:2.509ex;" alt="{\displaystyle \zeta \ll 1\!\,}"></span>, je geometrija enaka evklidski, pri <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 \zeta \sim 1\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>∼</mo> <mn>1</mn> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta \sim 1\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c55cd6d12a198ed3aea0fbf26a4ff4e33640d83f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.356ex; height:2.509ex;" alt="{\displaystyle \zeta \sim 1\!\,}"></span> pa vse podobnosti med geometrijama izginejo. Energija nihanja pri merilu <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 l\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d909866e84b9b9e47be2e69f1ea2ff180e780d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l\!\,}"></span> je enaka <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 E=\hbar \nu \sim \hbar c/l\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>ν</mi> <mo>∼</mo> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\hbar \nu \sim \hbar c/l\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f601215df40e908d0f316d348858a30c8c916b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.68ex; height:2.843ex;" alt="{\displaystyle E=\hbar \nu \sim \hbar c/l\!\,}"></span> (kjer je <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 c/l\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c/l\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36bdd75fdd479e6eb43c0028c3935a30688804e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.863ex; height:2.843ex;" alt="{\displaystyle c/l\!\,}"></span> red frekvence nihanja). Gravitacijski potencial, ki ga povzroča masa <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 m\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1566f742a01abf7df6a1a331325f798b559b94d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m\!\,}"></span> pri tej dolžini, je enak <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 \varphi =\kappa m/l\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>κ</mi> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\kappa m/l\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a974b496b6de7a81fb1167f5faebc39249ad3e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.854ex; height:2.843ex;" alt="{\displaystyle \varphi =\kappa m/l\!\,}"></span>, kjer je <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 \kappa \!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa \!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec56180596f88f97104539b2291211bcfd7f634f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa \!\,}"></span> gravitacijska konstanta. namesto mase <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 m\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1566f742a01abf7df6a1a331325f798b559b94d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m\!\,}"></span> jo je treba zamenjati z maso, ki ji po <a href="/wiki/Ekvivalentnost_mase_in_energije" class="mw-redirect" title="Ekvivalentnost mase in energije">Einsteinovi formuli</a> ustreza energija <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 E\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e207352add08c46cddd498fc639186ef9ca731f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E\!\,}"></span> (kjer je <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 m=E/c^{2}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=E/c^{2}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39447688f8e40c545cfc55001b3cda138f5e422b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.138ex; height:3.176ex;" alt="{\displaystyle m=E/c^{2}\!\,}"></span>). Tako je <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 \varphi =\kappa E/l\,c^{2}=\kappa \hbar /l^{2}c\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>κ</mi> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> <mspace width="thinmathspace" /> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>κ</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\kappa E/l\,c^{2}=\kappa \hbar /l^{2}c\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac681fc43f54e4abf45c90710ea03fa22bd6578e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.698ex; height:3.176ex;" alt="{\displaystyle \varphi =\kappa E/l\,c^{2}=\kappa \hbar /l^{2}c\!\,}"></span>. Če se ta izraz deli s <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 c^{2}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0432245b3b880e59b527dc4790955b2234798d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.061ex; height:2.676ex;" alt="{\displaystyle c^{2}\!\,}"></span>, izhaja vrednost odklona <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 \zeta =\kappa \hbar /c^{3}l^{2}=\ell _{\rm {P}}^{2}/l^{2}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mi>κ</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta =\kappa \hbar /c^{3}l^{2}=\ell _{\rm {P}}^{2}/l^{2}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33b1a4682a6b5bbd8d77c1100d6a98d800c8374c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.14ex; height:3.343ex;" alt="{\displaystyle \zeta =\kappa \hbar /c^{3}l^{2}=\ell _{\rm {P}}^{2}/l^{2}\!\,}"></span>. Vrednost <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 \zeta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63f7f0a961df774bd3db35e02af655591ad23c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.356ex; height:2.509ex;" alt="{\displaystyle \zeta =1}"></span> ustreza dolžini pri kateri je evklidska geometrija popolnoma popačena. Ta dolžina je enaka Planckovi <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="{\textstyle \ell _{\rm {P}}={\sqrt {\kappa \hbar /c^{3}}}\approx 10^{-35}\mathrm {m} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>κ</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </msqrt> </mrow> <mo>≈</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>35</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \ell _{\rm {P}}={\sqrt {\kappa \hbar /c^{3}}}\approx 10^{-35}\mathrm {m} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b684e6e6194a9f72aa6d6e3d17f63d8b195249" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.126ex; height:3.343ex;" alt="{\textstyle \ell _{\rm {P}}={\sqrt {\kappa \hbar /c^{3}}}\approx 10^{-35}\mathrm {m} }"></span>.<sup id="cite_ref-migdal_1985_16-0" class="reference"><a href="#cite_note-migdal_1985-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Za območje prostor-časa z razsežnostmi <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 l\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d909866e84b9b9e47be2e69f1ea2ff180e780d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l\!\,}"></span> je nedoločenost <a href="/w/index.php?title=Christoffelovi_simboli&action=edit&redlink=1" class="new" title="Christoffelovi simboli (stran ne obstaja)">Christoffelovih simbolov</a> <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 \delta \Gamma \!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi mathvariant="normal">Γ</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \Gamma \!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328df877b7cc5e27196ca1e7dd8cfd322ebf277a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.501ex; height:2.343ex;" alt="{\displaystyle \delta \Gamma \!\,}"></span> enaka redu <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 \ell _{\rm {P}}^{2}/l^{3}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {P}}^{2}/l^{3}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/430ef90732c72693f54122f0daee57805cd14a76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.231ex; height:3.343ex;" alt="{\displaystyle \ell _{\rm {P}}^{2}/l^{3}\!\,}"></span>, nedoločenost <a href="/w/index.php?title=Metri%C4%8Dni_tenzor&action=edit&redlink=1" class="new" title="Metrični tenzor (stran ne obstaja)">metričnega tenzorja</a> <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 \delta g\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>g</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta g\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1351f80b8e65aab2234bcb44b1882779ca2f8dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.165ex; height:2.676ex;" alt="{\displaystyle \delta g\!\,}"></span> pa je reda <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 \ell _{\rm {P}}^{2}/l^{2}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {P}}^{2}/l^{2}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81df99ad423edf2cefc9a4943c6478862d15616a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.231ex; height:3.343ex;" alt="{\displaystyle \ell _{\rm {P}}^{2}/l^{2}\!\,}"></span>.<sup id="cite_ref-regge_1958_17-0" class="reference"><a href="#cite_note-regge_1958-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Če je <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 l\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d909866e84b9b9e47be2e69f1ea2ff180e780d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l\!\,}"></span> makroskopska dolžina, so kvantne vezi izjemno majhne in se jih lahko zanemari celo v atomskih merilih. Če je vrednost <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 l\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d909866e84b9b9e47be2e69f1ea2ff180e780d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l\!\,}"></span> primerljiva z <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 \ell _{\rm {P}}\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {P}}\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faac47638f1b6e0f4e5fdd0492be67ba0aacc152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.321ex; height:2.509ex;" alt="{\displaystyle \ell _{\rm {P}}\!\,}"></span>, potem ohranjanje prejšnjega (običajnega) koncepta prostora postaja vedno težje, vpliv mikroukrivljenosti pa postaja vedno očitnejše. To domnevno lahko pomeni, da prostor-čas v Planckovem merilu postane kvantna pena.<sup id="cite_ref-wheeler_1955_18-0" class="reference"><a href="#cite_note-wheeler_1955-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Glej_tudi">Glej tudi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Planckova_dol%C5%BEina&veaction=edit&section=5" title="Uredi razdelek: Glej tudi" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Planckova_dol%C5%BEina&action=edit&section=5" title="Urejanje izvorne kode razdelka: Glej tudi"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small" style="-webkit-column-count:3; -moz-column-count:3; column-count:3;"> <ul><li><a href="/w/index.php?title=Fok-Lorentzeva_simetrija&action=edit&redlink=1" class="new" title="Fok-Lorentzeva simetrija (stran ne obstaja)">Fok-Lorentzeva simetrija</a></li> <li><a href="/w/index.php?title=Red_velikosti_(dol%C5%BEina)&action=edit&redlink=1" class="new" title="Red velikosti (dolžina) (stran ne obstaja)">red velikosti (dolžina)</a></li> <li><a href="/wiki/Planckova_energija" title="Planckova energija">Planckova energija</a></li> <li><a href="/wiki/Planckova_masa" title="Planckova masa">Planckova masa</a></li> <li><a href="/wiki/Planckova_doba" title="Planckova doba">Planckova doba</a></li> <li><a href="/wiki/Planckova_temperatura" title="Planckova temperatura">Planckova temperatura</a></li> <li><a href="/wiki/Planckov_%C4%8Das" title="Planckov čas">Planckov čas</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Sklici">Sklici</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Planckova_dol%C5%BEina&veaction=edit&section=6" title="Uredi razdelek: Sklici" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Planckova_dol%C5%BEina&action=edit&section=6" title="Urejanje izvorne kode razdelka: Sklici"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-width" style="column-width: 25em; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-baez_1999-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-baez_1999_1-0">1,0</a></sup> <sup><a href="#cite_ref-baez_1999_1-1">1,1</a></sup></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFBaez1999"> Baez (1999)</a>.</span> </li> <li id="cite_note-nist_2019-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-nist_2019_2-0">2,0</a></sup> <sup><a href="#cite_ref-nist_2019_2-1">2,1</a></sup></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r5980307">.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 a,.mw-parser-output .citation .cs1-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 a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-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 a,.mw-parser-output .citation .cs1-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}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.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}</style><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="http://physics.nist.gov/cgi-bin/cuu/Value?plkl">»Planck length«</a>. <i><a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a></i> (v angleščini). Maj 2019<span class="reference-accessdate">. Pridobljeno 2. julija 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=neznano&rft.jtitle=NIST&rft.atitle=Planck+length&rft.date=2019-05&rft_id=http%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fplkl&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></span> </li> <li id="cite_note-physlink-3"><span class="mw-cite-backlink"><a href="#cite_ref-physlink_3-0">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="http://www.physlink.com/education/askexperts/ae281.cfm">»What is Planck length? What is Planck time?«</a>. <i>Physlink.com</i> (v angleščini).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=neznano&rft.jtitle=Physlink.com&rft.atitle=What+is+Planck+length%3F+What+is+Planck+time%3F&rft_id=http%3A%2F%2Fwww.physlink.com%2Feducation%2Faskexperts%2Fae281.cfm&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></span> </li> <li id="cite_note-planck_1899-4"><span class="mw-cite-backlink"><a href="#cite_ref-planck_1899_4-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFPlanck1899"> Planck (1899)</a>.</span> </li> <li id="cite_note-gorelik_1992-5"><span class="mw-cite-backlink"><a href="#cite_ref-gorelik_1992_5-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFGorelik1992"> Gorelik (1992)</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/lengths.html">Dolžina v fiziki</a></span> </li> <li id="cite_note-eos-7"><span class="mw-cite-backlink"><a href="#cite_ref-eos_7-0">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090426050457/http://www.daviddarling.info/encyclopedia/P/Planck_length.html">»Encyclopedia of science«</a> (v angleščini). Arhivirano iz <a rel="nofollow" class="external text" href="http://www.daviddarling.info/encyclopedia/P/Planck_length.html">prvotnega spletišča</a> dne 26. aprila 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=neznano&rft.btitle=Encyclopedia+of+science&rft_id=http%3A%2F%2Fwww.daviddarling.info%2Fencyclopedia%2FP%2FPlanck_length.html&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></span> </li> <li id="cite_note-physicsforums-8"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-physicsforums_8-0">8,0</a></sup> <sup><a href="#cite_ref-physicsforums_8-1">8,1</a></sup> <sup><a href="#cite_ref-physicsforums_8-2">8,2</a></sup> <sup><a href="#cite_ref-physicsforums_8-3">8,3</a></sup></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFKlotz2015"> Klotz (2015)</a>.</span> </li> <li id="cite_note-bekenstein_1973-9"><span class="mw-cite-backlink"><a href="#cite_ref-bekenstein_1973_9-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFBekenstein1973"> Bekenstein (1973)</a>.</span> </li> <li id="cite_note-aaronson_2005-10"><span class="mw-cite-backlink"><a href="#cite_ref-aaronson_2005_10-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFAaronson2005"> Aaronson (2005)</a>.</span> </li> <li id="cite_note-misner_1973-11"><span class="mw-cite-backlink"><a href="#cite_ref-misner_1973_11-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFMisnerThorneWheeler1973"> Misner; Thorne; Wheeler (1973)</a>, str. 1190–1194, 1198–1201.</span> </li> <li id="cite_note-klimets_2017-12"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-klimets_2017_12-0">12,0</a></sup> <sup><a href="#cite_ref-klimets_2017_12-1">12,1</a></sup></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFKlimets2017"> Klimets (2017)</a>, str. 25–28.</span> </li> <li id="cite_note-carr_2005-13"><span class="mw-cite-backlink"><a href="#cite_ref-carr_2005_13-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFCarr2005"> Carr (2005)</a>.</span> </li> <li id="cite_note-burgess_2007-14"><span class="mw-cite-backlink"><a href="#cite_ref-burgess_2007_14-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFBurgessQuevedo2007"> Burgess; Quevedo (2007)</a>.</span> </li> <li id="cite_note-migdal_1989-15"><span class="mw-cite-backlink"><a href="#cite_ref-migdal_1989_15-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFMigdal1989"> Migdal (1989)</a>, str. 116–117.</span> </li> <li id="cite_note-migdal_1985-16"><span class="mw-cite-backlink"><a href="#cite_ref-migdal_1985_16-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFMigdal1985"> Migdal (1985)</a>.</span> </li> <li id="cite_note-regge_1958-17"><span class="mw-cite-backlink"><a href="#cite_ref-regge_1958_17-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFRegge1958"> Regge (1958)</a>.</span> </li> <li id="cite_note-wheeler_1955-18"><span class="mw-cite-backlink"><a href="#cite_ref-wheeler_1955_18-0">↑</a></span> <span class="reference-text"><a class="mw-selflink-fragment" href="#CITEREFWheeler1955"> Wheeler (1955)</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Viri">Viri</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Planckova_dol%C5%BEina&veaction=edit&section=7" title="Uredi razdelek: Viri" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Planckova_dol%C5%BEina&action=edit&section=7" title="Urejanje izvorne kode razdelka: Viri"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Osnovni viri</dt></dl> <style data-mw-deduplicate="TemplateStyles:r5453066">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-100{font-size:100%}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns dl,.mw-parser-output .refbegin-columns ol,.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li,.mw-parser-output .refbegin-columns dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="refbegin refbegin-columns references-column-count references-column-count-2" style="column-count: 2;"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFBekenstein1973" class="citation cs2"><a href="/w/index.php?title=Jacob_David_Bekenstein&action=edit&redlink=1" class="new" title="Jacob David Bekenstein (stran ne obstaja)">Bekenstein, Jacob David</a> (1973), »Black Holes and Entropy«, <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>, <b>7</b> (8): 2333–2346, <a href="/wiki/Bibcode_(identifikator)" class="mw-redirect" title="Bibcode (identifikator)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973PhRvD...7.2333B">1973PhRvD...7.2333B</a>, <a href="/wiki/Doi_(identifikator)" class="mw-redirect" title="Doi (identifikator)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.7.2333">10.1103/PhysRevD.7.2333</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=%C4%8Dlanek&rft.jtitle=Physical+Review+D&rft.atitle=Black+Holes+and+Entropy&rft.volume=7&rft.issue=8&rft.pages=2333-2346&rft.date=1973&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.7.2333&rft_id=info%3Abibcode%2F1973PhRvD...7.2333B&rft.aulast=Bekenstein&rft.aufirst=Jacob+David&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFPlanck1899" class="citation cs2"><a href="/wiki/Max_Planck" title="Max Planck">Planck, Max</a> (1899), »Naturlische Masseinheiten«, <i>Der Koniglich Preussischen Akademie Der Wissenschaften</i>: 479</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=%C4%8Dlanek&rft.jtitle=Der+Koniglich+Preussischen+Akademie+Der+Wissenschaften&rft.atitle=Naturlische+Masseinheiten&rft.pages=479&rft.date=1899&rft.aulast=Planck&rft.aufirst=Max&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></li></ul> </div> <dl><dt>Drugi viri</dt></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5453066"><div class="refbegin refbegin-columns references-column-count references-column-count-2" style="column-count: 2;"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFAaronson2005" class="citation cs2"><a href="/w/index.php?title=Scott_Aaronson&action=edit&redlink=1" class="new" title="Scott Aaronson (stran ne obstaja)">Aaronson, Scott</a> (2005), <a rel="nofollow" class="external text" href="https://www.scottaaronson.com/papers/npcomplete.pdf"><i>NP-complete Problems and Physical Reality</i></a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">, pridobljeno 2. julija 2019</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=knjiga&rft.btitle=NP-complete+Problems+and+Physical+Reality&rft.date=2005&rft.aulast=Aaronson&rft.aufirst=Scott&rft_id=https%3A%2F%2Fwww.scottaaronson.com%2Fpapers%2Fnpcomplete.pdf&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFBaez1999" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/John_Carlos_Baez" title="John Carlos Baez">Baez, John Carlos</a> (1999), <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/planck/node2.html"><i>The Planck Length</i></a> (v angleščini)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=knjiga&rft.btitle=The+Planck+Length&rft.date=1999&rft.aulast=Baez&rft.aufirst=John+Carlos&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fplanck%2Fnode2.html&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFBurgessQuevedo2007" class="citation news cs1"><a href="/w/index.php?title=Cliff_Burgess&action=edit&redlink=1" class="new" title="Cliff Burgess (stran ne obstaja)">Burgess, Cliff</a>; <a href="/w/index.php?title=Fernando_Quevedo&action=edit&redlink=1" class="new" title="Fernando Quevedo (stran ne obstaja)">Quevedo, Fernando</a> (<span dir="ltr">november 2007</span>). »The Great Cosmic Roller-Coaster Ride«. <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i> (print). 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Freeman and Company, Princeton University Press, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:ViriKnjig/0-7167-0344-0" title="Posebno:ViriKnjig/0-7167-0344-0"><bdi>0-7167-0344-0</bdi></a>, arhivirano iz <a rel="nofollow" class="external text" href="https://www.pdf-archive.com/2016/03/21/gravitation-misner-thorne-wheeler/">prvotnega spletišča</a> dne 1. julija 2019<span class="reference-accessdate">, pridobljeno 10. julija 2019</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=knjiga&rft.btitle=Gravitation&rft.pub=W.+H.+Freeman+and+Company%2C+Princeton+University+Press&rft.date=1973-09&rft.isbn=0-7167-0344-0&rft.aulast=Misner&rft.aufirst=Charles+William&rft.au=Thorne%2C+Kip+Stephen&rft.au=Wheeler%2C+John+Archibald&rft_id=https%3A%2F%2Fwww.pdf-archive.com%2F2016%2F03%2F21%2Fgravitation-misner-thorne-wheeler%2F&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Predloga:Citation" class="mw-redirect" title="Predloga:Citation">citation</a>}}</code>: Vzdrževanje CS1: samodejni prevod datuma (<a href="/wiki/Kategorija:Vzdr%C5%BEevanje_CS1:_samodejni_prevod_datuma" title="Kategorija:Vzdrževanje CS1: samodejni prevod datuma">povezava</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFRegge1958" class="citation cs2"><a href="/wiki/Tullio_Regge" title="Tullio Regge">Regge, Tulio</a> (1958), »Gravitational fields and quantum mechanics«, <i>Nuovo Cim.</i>, <b>7</b> (215)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=%C4%8Dlanek&rft.jtitle=Nuovo+Cim.&rft.atitle=Gravitational+fields+and+quantum+mechanics&rft.volume=7&rft.issue=215&rft.date=1958&rft.aulast=Regge&rft.aufirst=Tulio&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFWheeler1955" class="citation cs2"><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</a> (Januar 1955), »Geons«, <i><a href="/wiki/Physical_Review" title="Physical Review">Physical Review</a></i>, <b>97</b> (2): 511–536, <a href="/wiki/Bibcode_(identifikator)" class="mw-redirect" title="Bibcode (identifikator)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1955PhRv...97..511W">1955PhRv...97..511W</a>, <a href="/wiki/Doi_(identifikator)" class="mw-redirect" title="Doi (identifikator)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.97.511">10.1103/PhysRev.97.511</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=%C4%8Dlanek&rft.jtitle=Physical+Review&rft.atitle=Geons&rft.volume=97&rft.issue=2&rft.pages=511-536&rft.date=1955-01&rft_id=info%3Adoi%2F10.1103%2FPhysRev.97.511&rft_id=info%3Abibcode%2F1955PhRv...97..511W&rft.aulast=Wheeler&rft.aufirst=John+Archibald&rfr_id=info%3Asid%2Fsl.wikipedia.org%3APlanckova+dol%C5%BEina" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Zunanje_povezave">Zunanje povezave</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Planckova_dol%C5%BEina&veaction=edit&section=8" title="Uredi razdelek: Zunanje povezave" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Planckova_dol%C5%BEina&action=edit&section=8" title="Urejanje izvorne kode razdelka: Zunanje povezave"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/planck/node2.html">Opis Planckove dolžine</a> <span class="languageicon">(angleško)</span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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