De Sitter space - Wikipedia

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id="toc-Flat_slicing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Flat_slicing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Flat slicing</span> </div> </a> <ul id="toc-Flat_slicing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Open_slicing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Open_slicing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Open slicing</span> </div> </a> <ul id="toc-Open_slicing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed_slicing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_slicing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Closed slicing</span> </div> </a> <ul id="toc-Closed_slicing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-dS_slicing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#dS_slicing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>dS slicing</span> </div> </a> <ul id="toc-dS_slicing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a 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aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 15 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-15" 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">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_de_Sitter" title="Espai de Sitter – Catalan" lang="ca" hreflang="ca" data-title="Espai de Sitter" data-language-autonym="Català" data-language-local-name="Catalan" 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/De_Sitter%C5%AFv_prostoro%C4%8Das" title="De Sitterův prostoročas – Czech" lang="cs" hreflang="cs" data-title="De Sitterův prostoročas" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/De-Sitter-Raum" title="De-Sitter-Raum – German" lang="de" hreflang="de" data-title="De-Sitter-Raum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/M%C3%A9trica_de_De_Sitter" title="Métrica de De Sitter – Spanish" lang="es" hreflang="es" data-title="Métrica de De Sitter" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_de_de_Sitter" title="Espace de de Sitter – French" lang="fr" hreflang="fr" data-title="Espace de de Sitter" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8D%94%EC%8B%9C%ED%84%B0%EB%A5%B4_%EA%B3%B5%EA%B0%84" title="더시터르 공간 – Korean" lang="ko" hreflang="ko" data-title="더시터르 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%A5_%D5%8D%D5%AB%D5%BF%D5%A5%D6%80%D5%AB_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Դե Սիտերի տարածություն – Armenian" lang="hy" hreflang="hy" data-title="Դե Սիտերի տարածություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_di_de_Sitter" title="Spazio di de Sitter – Italian" lang="it" hreflang="it" data-title="Spazio di de Sitter" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/De_Sitter-metriek" title="De Sitter-metriek – Dutch" lang="nl" hreflang="nl" data-title="De Sitter-metriek" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%89%E3%83%BB%E3%82%B8%E3%83%83%E3%82%BF%E3%83%BC%E7%A9%BA%E9%96%93" title="ド・ジッター空間 – Japanese" lang="ja" hreflang="ja" data-title="ド・ジッター空間" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_de_Sittera" title="Przestrzeń de Sittera – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń de Sittera" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_de_De_Sitter" title="Espaço de De Sitter – Portuguese" lang="pt" hreflang="pt" data-title="Espaço de De Sitter" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_%D0%B4%D0%B5_%D0%A1%D0%B8%D1%82%D1%82%D0%B5%D1%80%D0%B0" title="Модель де Ситтера – Russian" lang="ru" hreflang="ru" data-title="Модель де Ситтера" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_%D0%B4%D0%B5_%D0%A1%D1%96%D1%82%D1%82%D0%B5%D1%80%D0%B0" title="Модель де Сіттера – Ukrainian" lang="uk" hreflang="uk" data-title="Модель де Сіттера" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BE%B7%E8%A5%BF%E7%89%B9%E7%A9%BA%E9%96%93" title="德西特空間 – Chinese" lang="zh" hreflang="zh" data-title="德西特空間" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Maximally symmetric Lorentzian manifold with a positive cosmological constant</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid 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ambox-content ambox-multiple_issues compact-ambox" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>This article has multiple issues.</b> Please help <b><a href="/wiki/Special:EditPage/De_Sitter_space" title="Special:EditPage/De Sitter space">improve it</a></b> or discuss these issues on the <b><a href="/wiki/Talk:De_Sitter_space" title="Talk:De Sitter space">talk 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class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>relies excessively on <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">references</a> to <a href="/wiki/Wikipedia:No_original_research#Primary,_secondary_and_tertiary_sources" title="Wikipedia:No original research">primary sources</a></b>.<span class="hide-when-compact"> Please improve this article by adding <a href="/wiki/Wikipedia:No_original_research#Primary,_secondary_and_tertiary_sources" title="Wikipedia:No original research">secondary or tertiary sources</a>. <br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22De+Sitter+space%22">"De Sitter space"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22De+Sitter+space%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22De+Sitter+space%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22De+Sitter+space%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22De+Sitter+space%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22De+Sitter+space%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">July 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> </div> </div><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>, <i>n</i>-dimensional <b>de Sitter space</b> (often denoted dS<sub><i>n</i></sub>) is a maximally symmetric <a href="/wiki/Lorentzian_manifold" class="mw-redirect" title="Lorentzian manifold">Lorentzian manifold</a> with constant positive <a href="/wiki/Scalar_curvature" title="Scalar curvature">scalar curvature</a>. It is the Lorentzian<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag needs further explanation. (February 2024)">further explanation needed</span></a></i>]</sup> analogue of an <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-sphere</a> (with its canonical <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a>). </p><p>The main application of de Sitter space is its use in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, where it serves as one of the simplest mathematical models of the universe consistent with the observed <a href="/wiki/Accelerating_expansion_of_the_universe" title="Accelerating expansion of the universe">accelerating expansion of the universe</a>. More specifically, de Sitter space is the maximally symmetric <a href="/wiki/Vacuum_solution" class="mw-redirect" title="Vacuum solution">vacuum solution</a> of <a href="/wiki/Einstein%27s_field_equations" class="mw-redirect" title="Einstein's field equations">Einstein's field equations</a> with a positive <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</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 \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> (corresponding to a positive vacuum energy density and negative pressure). </p><p>De Sitter space and <a href="/wiki/Anti-de_Sitter_space" title="Anti-de Sitter space">anti-de Sitter space</a> are named after <a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">Willem de Sitter</a> (1872–1934),<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> professor of astronomy at <a href="/wiki/Leiden_University" title="Leiden University">Leiden University</a> and director of the <a href="/wiki/Leiden_Observatory" title="Leiden Observatory">Leiden Observatory</a>. Willem de Sitter and <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> worked closely together in <a href="/wiki/Leiden" title="Leiden">Leiden</a> in the 1920s on the spacetime structure of our universe. De Sitter space was also discovered, independently, and about the same time, by <a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A de Sitter space can be defined as a <a href="/wiki/Submanifold" title="Submanifold">submanifold</a> of a generalized <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> of one higher <a href="/wiki/Dimension" title="Dimension">dimension</a>, including the induced metric. Take Minkowski space <b>R</b><sup>1,<i>n</i></sup> with the standard <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=-dx_{0}^{2}+\sum _{i=1}^{n}dx_{i}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=-dx_{0}^{2}+\sum _{i=1}^{n}dx_{i}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60a2f98c1a3d6da5b4142fc4c832450e1a221a1b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.696ex; height:6.843ex;" alt="{\displaystyle ds^{2}=-dx_{0}^{2}+\sum _{i=1}^{n}dx_{i}^{2}.}"></span> </p><p>The <i>n</i>-dimensional de Sitter space is the submanifold described by the <a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid</a> of one sheet <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x_{0}^{2}+\sum _{i=1}^{n}x_{i}^{2}=\alpha ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x_{0}^{2}+\sum _{i=1}^{n}x_{i}^{2}=\alpha ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b10cc418b2866908f82ed2999f1de4be1f7802a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.446ex; height:6.843ex;" alt="{\displaystyle -x_{0}^{2}+\sum _{i=1}^{n}x_{i}^{2}=\alpha ^{2},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is some nonzero constant with its dimension being that of length. The induced <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a> on the de Sitter space induced from the ambient Minkowski metric. It is <a href="/wiki/Nondegenerate" class="mw-redirect" title="Nondegenerate">nondegenerate</a> and has Lorentzian signature. (If one replaces <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 \alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9850832395fd0deec6790b4b2265df6656f08ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.676ex;" alt="{\displaystyle \alpha ^{2}}"></span> with <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 -\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2973234387b55778ce29262d829f47261068a82d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.35ex; height:2.843ex;" alt="{\displaystyle -\alpha ^{2}}"></span> in the above definition, one obtains a <a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid</a> of two sheets. The induced metric in this case is <a href="/wiki/Definite_quadratic_form" title="Definite quadratic form">positive-definite</a>, and each sheet is a copy of <a href="/wiki/Hyperbolic_space" title="Hyperbolic space">hyperbolic <i>n</i>-space</a>. See <i><a href="/wiki/Minkowski_space#Geometry" title="Minkowski space">Minkowski space § Geometry</a></i>.) </p><p>The de Sitter space can also be defined as the <a href="/wiki/Homogeneous_space" title="Homogeneous space">quotient</a> <span class="nowrap">O(1, <i>n</i>) / O(1, <i>n</i> − 1)</span> of two <a href="/wiki/Indefinite_orthogonal_group" title="Indefinite orthogonal group">indefinite orthogonal groups</a>, which shows that it is a non-Riemannian <a href="/wiki/Symmetric_space" title="Symmetric space">symmetric space</a>. </p><p><a href="/wiki/Topology" title="Topology">Topologically</a>, dS<sub><i>n</i></sub> is <span class="nowrap"><b>R</b> × <i>S</i><sup><i>n</i>−1</sup></span> (which is <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> if <span class="nowrap"><i>n</i> ≥ 3</span>). </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Isometry_group" title="Isometry group">isometry group</a> of de Sitter space is the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> <span class="nowrap">O(1, <i>n</i>)</span>. The metric therefore then has <span class="nowrap"><i>n</i>(<i>n</i> + 1)/2</span> independent <a href="/wiki/Killing_vector_field" title="Killing vector field">Killing vector fields</a> and is maximally symmetric. Every maximally symmetric space has constant curvature. The <a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a> of de Sitter is given by<sup id="cite_ref-zee-p626_4-0" class="reference"><a href="#cite_note-zee-p626-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </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 R_{\rho \sigma \mu \nu }={1 \over \alpha ^{2}}\left(g_{\rho \mu }g_{\sigma \nu }-g_{\rho \nu }g_{\sigma \mu }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>μ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>ν</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\rho \sigma \mu \nu }={1 \over \alpha ^{2}}\left(g_{\rho \mu }g_{\sigma \nu }-g_{\rho \nu }g_{\sigma \mu }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d10ca765173e6a344a83940cf0f433cce38735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.832ex; height:5.509ex;" alt="{\displaystyle R_{\rho \sigma \mu \nu }={1 \over \alpha ^{2}}\left(g_{\rho \mu }g_{\sigma \nu }-g_{\rho \nu }g_{\sigma \mu }\right)}"></span></dd></dl> <p>(using the sign convention <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^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msubsup> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64fddf066c60e94ac0a24b7d09bd2412846f6e3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:43.671ex; height:3.509ex;" alt="{\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }}"></span> for the Riemann curvature tensor). De Sitter space is an <a href="/wiki/Einstein_manifold" title="Einstein manifold">Einstein manifold</a> since the <a href="/wiki/Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci tensor</a> is proportional to the metric: </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 R_{\mu \nu }=R^{\lambda }{}_{\mu \lambda \nu }={\frac {n-1}{\alpha ^{2}}}g_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>λ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }=R^{\lambda }{}_{\mu \lambda \nu }={\frac {n-1}{\alpha ^{2}}}g_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296aedc3591818e656648f5487365656dcf8e7dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.5ex; height:5.509ex;" alt="{\displaystyle R_{\mu \nu }=R^{\lambda }{}_{\mu \lambda \nu }={\frac {n-1}{\alpha ^{2}}}g_{\mu \nu }}"></span></dd></dl> <p>This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by </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 \Lambda ={\frac {(n-1)(n-2)}{2\alpha ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda ={\frac {(n-1)(n-2)}{2\alpha ^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fde480d3503cead2a43fe1b3595124e767f9a4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.608ex; height:6.009ex;" alt="{\displaystyle \Lambda ={\frac {(n-1)(n-2)}{2\alpha ^{2}}}.}"></span></dd></dl> <p>The <a href="/wiki/Scalar_curvature" title="Scalar curvature">scalar curvature</a> of de Sitter space is given by<sup id="cite_ref-zee-p626_4-1" class="reference"><a href="#cite_note-zee-p626-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </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 R={\frac {n(n-1)}{\alpha ^{2}}}={\frac {2n}{n-2}}\Lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {n(n-1)}{\alpha ^{2}}}={\frac {2n}{n-2}}\Lambda .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6535e0395f7ecfd03c30aad7e7cd9475f79d2f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.892ex; height:6.009ex;" alt="{\displaystyle R={\frac {n(n-1)}{\alpha ^{2}}}={\frac {2n}{n-2}}\Lambda .}"></span></dd></dl> <p>For the case <span class="nowrap"><i>n</i> = 4</span>, we have <span class="nowrap">Λ = 3/<i>α</i><sup>2</sup></span> and <span class="nowrap"><i>R</i> = 4Λ = 12/<i>α</i><sup>2</sup></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Coordinates">Coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=3" title="Edit section: Coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Static_coordinates">Static coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=4" title="Edit section: Static coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We can introduce <a href="/wiki/Static_spacetime" title="Static spacetime">static coordinates</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 (t,r,\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,r,\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16de62036356627e91764255c62e920b98b05b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.489ex; height:2.843ex;" alt="{\displaystyle (t,r,\ldots )}"></span> for de Sitter as follows: </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 {\begin{aligned}x_{0}&={\sqrt {\alpha ^{2}-r^{2}}}\sinh \left({\frac {1}{\alpha }}t\right)\\x_{1}&={\sqrt {\alpha ^{2}-r^{2}}}\cosh \left({\frac {1}{\alpha }}t\right)\\x_{i}&=rz_{i}\qquad \qquad \qquad \qquad \qquad 2\leq i\leq n.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>cosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mn>2</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{0}&={\sqrt {\alpha ^{2}-r^{2}}}\sinh \left({\frac {1}{\alpha }}t\right)\\x_{1}&={\sqrt {\alpha ^{2}-r^{2}}}\cosh \left({\frac {1}{\alpha }}t\right)\\x_{i}&=rz_{i}\qquad \qquad \qquad \qquad \qquad 2\leq i\leq n.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89ba32b16481a89fc3e82dc3f4675a34a9a1af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:42.593ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}x_{0}&={\sqrt {\alpha ^{2}-r^{2}}}\sinh \left({\frac {1}{\alpha }}t\right)\\x_{1}&={\sqrt {\alpha ^{2}-r^{2}}}\cosh \left({\frac {1}{\alpha }}t\right)\\x_{i}&=rz_{i}\qquad \qquad \qquad \qquad \qquad 2\leq i\leq n.\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}"></span> gives the standard embedding the <span class="nowrap">(<i>n</i> − 2)</span>-sphere in <b>R</b><sup><i>n</i>−1</sup>. In these coordinates the de Sitter metric takes the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=-\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)dt^{2}+\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega _{n-2}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=-\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)dt^{2}+\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega _{n-2}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/846e157859cf53d3bb2c1549b0b44830a0e859ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.05ex; height:6.676ex;" alt="{\displaystyle ds^{2}=-\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)dt^{2}+\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega _{n-2}^{2}.}"></span></dd></dl> <p>Note that there is a <a href="/wiki/Cosmological_horizon" title="Cosmological horizon">cosmological horizon</a> at <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=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb987f55c5c289fac8a055582f5b5f1ed6894a1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.635ex; height:1.676ex;" alt="{\displaystyle r=\alpha }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Flat_slicing">Flat slicing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=5" title="Edit section: Flat slicing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let </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 {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)+{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)-{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{i}&=e^{{\frac {1}{\alpha }}t}y_{i},\qquad 2\leq i\leq n\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>cosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <mn>2</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)+{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)-{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{i}&=e^{{\frac {1}{\alpha }}t}y_{i},\qquad 2\leq i\leq n\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/349a3603f826f690930778ef5b13171604bd0e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:31.773ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)+{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)-{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{i}&=e^{{\frac {1}{\alpha }}t}y_{i},\qquad 2\leq i\leq n\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r^{2}=\sum _{i}y_{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r^{2}=\sum _{i}y_{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad269bf011069128394bffbf21a32f52ae88d6d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.057ex; height:3.176ex;" alt="{\textstyle r^{2}=\sum _{i}y_{i}^{2}}"></span>. Then in the <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 \left(t,y_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(t,y_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/947cef8697c3651bc7d6e3b9e48603fdfc91b64d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.622ex; height:2.843ex;" alt="{\displaystyle \left(t,y_{i}\right)}"></span> coordinates metric reads: </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 ds^{2}=-dt^{2}+e^{2{\frac {1}{\alpha }}t}dy^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=-dt^{2}+e^{2{\frac {1}{\alpha }}t}dy^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de241bd7100a85251b1062ae303df0a1b6771663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.069ex; height:3.676ex;" alt="{\displaystyle ds^{2}=-dt^{2}+e^{2{\frac {1}{\alpha }}t}dy^{2}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle dy^{2}=\sum _{i}dy_{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>d</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle dy^{2}=\sum _{i}dy_{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e3f131c1bba62dbb0e1d0c8f060640da079b71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.6ex; height:3.176ex;" alt="{\textstyle dy^{2}=\sum _{i}dy_{i}^{2}}"></span> is the flat metric on <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 y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d30d30b6c2dbe4d6f150d699de040937ecc95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}"></span>'s. </p><p>Setting <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 =\zeta _{\infty }-\alpha e^{-{\frac {1}{\alpha }}t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>−</mo> <mi>α</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta =\zeta _{\infty }-\alpha e^{-{\frac {1}{\alpha }}t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde1535ca0c807fb59b37ec59b60bb0bed015e68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.294ex; height:3.676ex;" alt="{\displaystyle \zeta =\zeta _{\infty }-\alpha e^{-{\frac {1}{\alpha }}t}}"></span>, we obtain the conformally flat metric: </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 ds^{2}={\frac {\alpha ^{2}}{(\zeta _{\infty }-\zeta )^{2}}}\left(dy^{2}-d\zeta ^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>−</mo> <mi>ζ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}={\frac {\alpha ^{2}}{(\zeta _{\infty }-\zeta )^{2}}}\left(dy^{2}-d\zeta ^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e53b06dfb2ea8474c52cffbb00b4be2d86ca2c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.164ex; height:6.509ex;" alt="{\displaystyle ds^{2}={\frac {\alpha ^{2}}{(\zeta _{\infty }-\zeta )^{2}}}\left(dy^{2}-d\zeta ^{2}\right)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Open_slicing">Open slicing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=6" title="Edit section: Open slicing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let </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 {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 2\leq i\leq n\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mi>cosh</mi> <mo>⁡</mo> <mi>ξ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>cosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mi>ξ</mi> <mo>,</mo> <mspace width="2em" /> <mn>2</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 2\leq i\leq n\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da32cf7fa2b73c3e5083336aa1766e27db198994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:41.91ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 2\leq i\leq n\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}z_{i}^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}z_{i}^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f185e0f49658021df9193388998da9db01beaca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.046ex; height:3.176ex;" alt="{\textstyle \sum _{i}z_{i}^{2}=1}"></span> forming 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 S^{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9599199d8a358396c8f887c37aa926f026b0eb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-2}}"></span> with the standard metric <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 \sum _{i}dz_{i}^{2}=d\Omega _{n-2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>d</mi> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>d</mi> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}dz_{i}^{2}=d\Omega _{n-2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/337ee28147a28890c300700c46204f9a49d730a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.312ex; height:3.343ex;" alt="{\textstyle \sum _{i}dz_{i}^{2}=d\Omega _{n-2}^{2}}"></span>. Then the metric of the de Sitter space reads </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 ds^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-1}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-1}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98268adb83c510a11971a202d348d9067b8a225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.046ex; height:6.176ex;" alt="{\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-1}^{2},}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dH_{n-1}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>d</mi> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dH_{n-1}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b17a53842cde125476531343676d998a25b699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.963ex; height:3.509ex;" alt="{\displaystyle dH_{n-1}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-2}^{2}}"></span></dd></dl> <p>is the standard hyperbolic metric. </p> <div class="mw-heading mw-heading3"><h3 id="Closed_slicing">Closed slicing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=7" title="Edit section: Closed slicing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let </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 {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)z_{i},\qquad 1\leq i\leq n\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>cosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)z_{i},\qquad 1\leq i\leq n\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c47d0dfba421e730e34101b86abb003e06e614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:36.213ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)z_{i},\qquad 1\leq i\leq n\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}"></span>s describe 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 S^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742bebb03fe630674b18823a59d2c75efd0066e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-1}}"></span>. Then the metric reads: </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 ds^{2}=-dt^{2}+\alpha ^{2}\cosh ^{2}\left({\frac {1}{\alpha }}t\right)d\Omega _{n-1}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cosh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\cosh ^{2}\left({\frac {1}{\alpha }}t\right)d\Omega _{n-1}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0e00852fb01ab49ba2a9ba942a0c0eb1c5a17b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.048ex; height:6.176ex;" alt="{\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\cosh ^{2}\left({\frac {1}{\alpha }}t\right)d\Omega _{n-1}^{2}.}"></span></dd></dl> <p>Changing the time variable to the conformal time via <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 \tan \left({\frac {1}{2}}\eta \right)=\tanh \left({\frac {1}{2\alpha }}t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>η</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \tan \left({\frac {1}{2}}\eta \right)=\tanh \left({\frac {1}{2\alpha }}t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5d47c90e54eb0815c32f5fef6dea3bdaf7f6cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.393ex; height:4.843ex;" alt="{\textstyle \tan \left({\frac {1}{2}}\eta \right)=\tanh \left({\frac {1}{2\alpha }}t\right)}"></span> we obtain a metric conformally equivalent to Einstein static universe: </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 ds^{2}={\frac {\alpha ^{2}}{\cos ^{2}\eta }}\left(-d\eta ^{2}+d\Omega _{n-1}^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>η</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>d</mi> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}={\frac {\alpha ^{2}}{\cos ^{2}\eta }}\left(-d\eta ^{2}+d\Omega _{n-1}^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32266687ff58bb723adc898cd0baa1f92ff3e66f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.873ex; height:6.343ex;" alt="{\displaystyle ds^{2}={\frac {\alpha ^{2}}{\cos ^{2}\eta }}\left(-d\eta ^{2}+d\Omega _{n-1}^{2}\right).}"></span></dd></dl> <p>These coordinates, also known as "global coordinates" cover the maximal extension of de Sitter space, and can therefore be used to find its <a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="dS_slicing">dS slicing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=8" title="Edit section: dS slicing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let </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 {\begin{aligned}x_{0}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cos \left({\frac {1}{\alpha }}\chi \right),\\x_{2}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 3\leq i\leq n\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mi>cosh</mi> <mo>⁡</mo> <mi>ξ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> <mi>cosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mi>ξ</mi> <mo>,</mo> <mspace width="2em" /> <mn>3</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{0}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cos \left({\frac {1}{\alpha }}\chi \right),\\x_{2}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 3\leq i\leq n\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf0ea6478be4fcd7437df3ba797574578ca2435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:52.353ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}x_{0}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cos \left({\frac {1}{\alpha }}\chi \right),\\x_{2}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 3\leq i\leq n\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}"></span>s describe 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 S^{n-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce56b8f274f4f2bd34016bfd34d0938e314af323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-3}}"></span>. Then the metric reads: </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 ds^{2}=d\chi ^{2}+\sin ^{2}\left({\frac {1}{\alpha }}\chi \right)ds_{dS,\alpha ,n-1}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>S</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=d\chi ^{2}+\sin ^{2}\left({\frac {1}{\alpha }}\chi \right)ds_{dS,\alpha ,n-1}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b358741e354d21fa4d27ae81c5c122d2f8f43eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.293ex; height:6.176ex;" alt="{\displaystyle ds^{2}=d\chi ^{2}+\sin ^{2}\left({\frac {1}{\alpha }}\chi \right)ds_{dS,\alpha ,n-1}^{2},}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds_{dS,\alpha ,n-1}^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>S</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds_{dS,\alpha ,n-1}^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231af2b10c76d026a8f46fa5ecf7d4a6afd86967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.551ex; height:6.176ex;" alt="{\displaystyle ds_{dS,\alpha ,n-1}^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-2}^{2}}"></span></dd></dl> <p>is the metric of an <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 n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span> dimensional de Sitter space with radius of curvature <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> in open slicing coordinates. The hyperbolic metric is given by: </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 dH_{n-2}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-3}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>d</mi> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dH_{n-2}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-3}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb06642a9baa8b6eedf440f4d78d2515bd44a11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.61ex; height:3.509ex;" alt="{\displaystyle dH_{n-2}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-3}^{2}.}"></span></dd></dl> <p>This is the analytic continuation of the open slicing coordinates under <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 \left(t,\xi ,\theta ,\phi _{1},\phi _{2},\ldots ,\phi _{n-3}\right)\to \left(i\chi ,\xi ,it,\theta ,\phi _{1},\ldots ,\phi _{n-4}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mi>ξ</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>χ</mi> <mo>,</mo> <mi>ξ</mi> <mo>,</mo> <mi>i</mi> <mi>t</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(t,\xi ,\theta ,\phi _{1},\phi _{2},\ldots ,\phi _{n-3}\right)\to \left(i\chi ,\xi ,it,\theta ,\phi _{1},\ldots ,\phi _{n-4}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6234f5675a2574a540a6dfdac6bc1bf58442471f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.569ex; height:2.843ex;" alt="{\displaystyle \left(t,\xi ,\theta ,\phi _{1},\phi _{2},\ldots ,\phi _{n-3}\right)\to \left(i\chi ,\xi ,it,\theta ,\phi _{1},\ldots ,\phi _{n-4}\right)}"></span> and also switching <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}"></span> because they change their timelike/spacelike nature. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Anti-de_Sitter_space" title="Anti-de Sitter space">Anti-de Sitter space</a></li> <li><a href="/wiki/De_Sitter_universe" title="De Sitter universe">De Sitter universe</a></li> <li><a href="/wiki/AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li> <li><a href="/wiki/De_Sitter%E2%80%93Schwarzschild_metric" title="De Sitter–Schwarzschild metric">De Sitter–Schwarzschild metric</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFde_Sitter1917" class="citation cs2">de Sitter, W. 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Wet.</i>, <b>19</b>: 1217–1225, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1917KNAB...19.1217D">1917KNAB...19.1217D</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Kon.+Ned.+Acad.+Wet.&rft.atitle=On+the+relativity+of+inertia%3A+Remarks+concerning+Einstein%27s+latest+hypothesis&rft.volume=19&rft.pages=1217-1225&rft.date=1917&rft_id=info%3Abibcode%2F1917KNAB...19.1217D&rft.aulast=de+Sitter&rft.aufirst=W.&rft_id=https%3A%2F%2Fwww.dwc.knaw.nl%2FDL%2Fpublications%2FPU00012455.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Sitter1917" class="citation cs2">de Sitter, W. 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Wet.</i>, <b>20</b>: 229–243</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Kon.+Ned.+Acad.+Wet.&rft.atitle=On+the+curvature+of+space&rft.volume=20&rft.pages=229-243&rft.date=1917&rft.aulast=de+Sitter&rft.aufirst=W.&rft_id=https%3A%2F%2Fwww.dwc.knaw.nl%2FDL%2Fpublications%2FPU00012216.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevi-Civita1917" class="citation cs2"><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Levi-Civita, Tullio</a> (1917), "Realtà fisica di alcuni spazî normali del Bianchi", <i>Rendiconti, Reale Accademia dei Lincei</i>, <b>26</b>: 519–31</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rendiconti%2C+Reale+Accademia+dei+Lincei&rft.atitle=Realt%C3%A0+fisica+di+alcuni+spaz%C3%AE+normali+del+Bianchi&rft.volume=26&rft.pages=519-31&rft.date=1917&rft.aulast=Levi-Civita&rft.aufirst=Tullio&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></span> </li> <li id="cite_note-zee-p626-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-zee-p626_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-zee-p626_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFZee2013">Zee 2013</a>, p. 626</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHawking_&_Ellis" class="citation book cs1">Hawking & Ellis. <i>The large scale structure of space–time</i>. Cambridge Univ. Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+large+scale+structure+of+space%E2%80%93time&rft.pub=Cambridge+Univ.+Press&rft.au=Hawking+%26+Ellis&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></span> </li> </ol></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZee2013" class="citation book cs1"><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee, Anthony</a> (2013). <i>Einstein Gravity in a Nutshell</i>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691145587" title="Special:BookSources/9780691145587"><bdi>9780691145587</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein+Gravity+in+a+Nutshell&rft.pub=Princeton+University+Press&rft.date=2013&rft.isbn=9780691145587&rft.aulast=Zee&rft.aufirst=Anthony&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQingming_Cheng2001" class="citation cs2">Qingming Cheng (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=De_Sitter_space">"De Sitter space"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=De+Sitter+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.au=Qingming+Cheng&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDe_Sitter_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNomizu1982" class="citation cs2"><a href="/wiki/Katsumi_Nomizu" title="Katsumi Nomizu">Nomizu, Katsumi</a> (1982), "The Lorentz–Poincaré metric on the upper half-space and its extension", <i><a href="/w/index.php?title=Hokkaido_Mathematical_Journal&action=edit&redlink=1" class="new" title="Hokkaido Mathematical Journal (page does not exist)">Hokkaido Mathematical Journal</a></i>, <b>11</b> (3): 253–261, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.14492%2Fhokmj%2F1381757803">10.14492/hokmj/1381757803</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Hokkaido+Mathematical+Journal&rft.atitle=The+Lorentz%E2%80%93Poincar%C3%A9+metric+on+the+upper+half-space+and+its+extension&rft.volume=11&rft.issue=3&rft.pages=253-261&rft.date=1982&rft_id=info%3Adoi%2F10.14492%2Fhokmj%2F1381757803&rft.aulast=Nomizu&rft.aufirst=Katsumi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1943" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (1943), "A geometrical background for de Sitter's world", <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>50</b> (4), Mathematical Association of America: 217–228, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2303924">10.2307/2303924</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2303924">2303924</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+geometrical+background+for+de+Sitter%27s+world&rft.volume=50&rft.issue=4&rft.pages=217-228&rft.date=1943&rft_id=info%3Adoi%2F10.2307%2F2303924&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2303924%23id-name%3DJSTOR&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSusskindLindesay2005" class="citation cs2">Susskind, L.; Lindesay, J. (2005), <i>An Introduction to Black Holes, Information and the String Theory Revolution:The Holographic Universe</i>, p. 119(11.5.25)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Black+Holes%2C+Information+and+the+String+Theory+Revolution%3AThe+Holographic+Universe&rft.pages=119%2811.5.25%29&rft.date=2005&rft.aulast=Susskind&rft.aufirst=L.&rft.au=Lindesay%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Sitter+space" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Sitter_space&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.quantumfieldtheory.info/dS_and_AdS_spaces.pdf">Simplified Guide to de Sitter and anti-de Sitter Spaces</a> A pedagogic introduction to de Sitter and anti-de Sitter spaces. The main article is simplified, with almost no math. The appendix is technical and intended for readers with physics or math backgrounds.</li></ul> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 0;column-gap:1em;align-items:baseline;margin:0;list-style:none}.mw-parser-output .portal-bar-content-related{margin:0;list-style:none}.mw-parser-output .portal-bar-item{display:inline-block;margin:0.15em 0.2em;min-height:24px;line-height:24px}@media screen and 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De Sitter space - Wikipedia