Gödel metric - Wikipedia

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<span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Einstein_tensor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Einstein_tensor"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Einstein tensor</span> </div> </a> <ul id="toc-Einstein_tensor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Topology</span> </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curvature_invariants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curvature_invariants"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Curvature invariants</span> </div> </a> <ul id="toc-Curvature_invariants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Killing_vectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Killing_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Killing vectors</span> </div> </a> <ul id="toc-Killing_vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Petrov_type_and_Bel_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Petrov_type_and_Bel_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Petrov type and Bel decomposition</span> </div> </a> <ul id="toc-Petrov_type_and_Bel_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigid_rotation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rigid_rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Rigid rotation</span> </div> </a> <ul id="toc-Rigid_rotation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optical_effects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optical_effects"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Optical effects</span> </div> </a> <ul id="toc-Optical_effects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shape_of_absolute_future" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Shape_of_absolute_future"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Shape of absolute future</span> </div> </a> <ul id="toc-Shape_of_absolute_future-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed_timelike_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_timelike_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.9</span> <span>Closed timelike curves</span> </div> </a> <ul id="toc-Closed_timelike_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Globally_nonhyperbolic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Globally_nonhyperbolic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.10</span> <span>Globally nonhyperbolic</span> </div> </a> <ul id="toc-Globally_nonhyperbolic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-A_cylindrical_chart" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_cylindrical_chart"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>A cylindrical chart</span> </div> </a> <button aria-controls="toc-A_cylindrical_chart-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle A cylindrical chart subsection</span> </button> <ul id="toc-A_cylindrical_chart-sublist" class="vector-toc-list"> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Appearance_of_the_light_cones" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Appearance_of_the_light_cones"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Appearance of the light cones</span> </div> </a> <ul id="toc-Appearance_of_the_light_cones-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_congruence_of_closed_timelike_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_congruence_of_closed_timelike_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>A congruence of closed timelike curves</span> </div> </a> <ul id="toc-A_congruence_of_closed_timelike_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Null_geodesics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Null_geodesics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Null geodesics</span> </div> </a> <ul id="toc-Null_geodesics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_absolute_future" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_absolute_future"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>The absolute future</span> </div> </a> <ul id="toc-The_absolute_future-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cosmological_interpretation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cosmological_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Cosmological interpretation</span> </div> </a> <ul id="toc-Cosmological_interpretation-sublist" class="vector-toc-list"> </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">5</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">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " 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href="https://et.wikipedia.org/wiki/G%C3%B6deli_meetrika" title="Gödeli meetrika – Estonian" lang="et" hreflang="et" data-title="Gödeli meetrika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Universo_de_G%C3%B6del" title="Universo de Gödel – Spanish" lang="es" hreflang="es" data-title="Universo de Gödel" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AA%D8%B1%DB%8C%DA%A9_%DA%AF%D9%88%D8%AF%D9%84" title="متریک گودل – Persian" lang="fa" hreflang="fa" data-title="متریک گودل" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Univers_de_G%C3%B6del" title="Univers de Gödel – French" lang="fr" hreflang="fr" data-title="Univers de Gödel" 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/%EA%B4%B4%EB%8D%B8_%EA%B3%84%EB%9F%89" title="괴델 계량 – Korean" lang="ko" hreflang="ko" data-title="괴델 계량" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Metrica_di_G%C3%B6del" title="Metrica di Gödel – Italian" lang="it" hreflang="it" data-title="Metrica di Gödel" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/General_relativity" title="General relativity">General relativity</a></th></tr><tr><td class="sidebar-image"><span class="notpageimage" typeof="mw:File"><a href="/wiki/File:Spacetime_lattice_analogy.svg" class="mw-file-description" title="Spacetime curvature schematic"><img alt="Spacetime curvature schematic" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/220px-Spacetime_lattice_analogy.svg.png" decoding="async" width="220" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/330px-Spacetime_lattice_analogy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/440px-Spacetime_lattice_analogy.svg.png 2x" data-file-width="1260" data-file-height="469" /></a></span><div class="sidebar-caption" style="padding:0.5em 0.2em 0.6em;border-bottom:1px solid #aaa; display:block;margin-bottom:0.1em;"><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 G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124ab80fcb17e2733cc17ff6f93da5e52f355c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"></span></div></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><div class="hlist"><ul><li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History</a></li><li><a href="/wiki/Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline</a></li><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li></ul></div></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Fundamental concepts</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Phenomena</div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Kepler problem</a></li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lensing</a></li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">Gravitational redshift</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">Frame-dragging</a></li> <li><a href="/wiki/Geodetic_effect" title="Geodetic effect">Geodetic effect</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ececff; font-style:italic;font-weight:normal;"> <a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Einstein–Rosen bridge</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li> <li><a href="/wiki/Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a class="mw-selflink selflink">Gödel</a></li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li> <li><a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li> <li><a href="/wiki/Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li> <li><a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Weyl%E2%80%93Lewis%E2%80%93Papapetrou_coordinates" title="Weyl–Lewis–Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hans_Reissner" title="Hans Reissner">Reissner</a></li> <li><a href="/wiki/Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li> <li><a href="/wiki/Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li> <li><a href="/wiki/Abraham_H._Taub" title="Abraham H. Taub">Taub</a></li> <li><a href="/wiki/Ezra_T._Newman" title="Ezra T. Newman">Newman</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" 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a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_relativity_sidebar" title="Template:General relativity sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_relativity_sidebar" title="Template talk:General relativity sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_relativity_sidebar" title="Special:EditPage/Template:General relativity sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>Gödel metric</b>, also known as the <b>Gödel solution</b> or <b>Gödel universe</b>, is an <a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">exact solution</a>, found in 1949 by <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>,<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> of the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a> in which the <a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a> contains two terms: the first representing the matter density of a homogeneous distribution of swirling dust particles (see <a href="/wiki/Dust_solution" title="Dust solution">dust solution</a>), and the second associated with a negative <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a> (see <a href="/wiki/Lambdavacuum_solution" title="Lambdavacuum solution">Lambdavacuum solution</a>). </p><p>This solution has many unusual properties—in particular, the existence of <a href="/wiki/Closed_timelike_curve" title="Closed timelike curve">closed time-like curves</a> that would allow <a href="/wiki/Time_travel" title="Time travel">time travel</a> in a universe described by the solution. Its definition is somewhat artificial, since the value of the cosmological constant must be carefully chosen to correspond to the density of the dust grains, but this <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> is an important pedagogical example. </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=G%C3%B6del_metric&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like any other <a href="/wiki/Pseudo-Riemannian_manifold#Lorentzian_manifold" title="Pseudo-Riemannian manifold">Lorentzian spacetime</a>, the Gödel solution represents the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> in terms of a local <a href="/wiki/Coordinate_chart" class="mw-redirect" title="Coordinate chart">coordinate chart</a>. It may be easiest to understand the Gödel universe using the cylindrical coordinate system (see below), but this article uses the chart originally used by Gödel. In this chart, the metric (or, equivalently, the <a href="/wiki/Line_element" title="Line element">line element</a>) is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\frac {1}{2\omega ^{2}}}\left[-(dt+e^{x}\,dy)^{2}+dx^{2}+{\tfrac {1}{2}}e^{2x}\,dy^{2}+dz^{2}\right],\qquad -\infty <t,x,y,z<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>t</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo><</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\frac {1}{2\omega ^{2}}}\left[-(dt+e^{x}\,dy)^{2}+dx^{2}+{\tfrac {1}{2}}e^{2x}\,dy^{2}+dz^{2}\right],\qquad -\infty <t,x,y,z<\infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301d821977ccf5c359ab8eeb33b068abadc1312c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:76.537ex; height:5.509ex;" alt="{\displaystyle g={\frac {1}{2\omega ^{2}}}\left[-(dt+e^{x}\,dy)^{2}+dx^{2}+{\tfrac {1}{2}}e^{2x}\,dy^{2}+dz^{2}\right],\qquad -\infty <t,x,y,z<\infty ,}"></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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is a non-zero real constant that gives the angular velocity of the surrounding dust grains about the <i>y</i>-axis, measured by a "non-spinning" observer riding on one of the dust grains. "Non-spinning" means that the observer does not feel centrifugal forces, but in this coordinate system, it would rotate about an axis parallel to the <i>y</i>-axis. In this rotating frame, the dust grains remain at constant values of <i>x</i>, <i>y</i>, and <i>z</i>. Their density in this coordinate diagram increases with <i>x</i>, but their density in their own frames of reference is the same everywhere. </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=G%C3%B6del_metric&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To investigate the properties of the Gödel solution, the <a href="/wiki/Frame_fields_in_general_relativity" title="Frame fields in general relativity">frame field</a> can be assumed (dual to the co-frame read from the metric as given above), </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 {\vec {e}}_{0}={\sqrt {2}}\omega \,\partial _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0}={\sqrt {2}}\omega \,\partial _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c471c66affd74b67ee39d966925c09ae6faba8f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.367ex; height:3.009ex;" alt="{\displaystyle {\vec {e}}_{0}={\sqrt {2}}\omega \,\partial _{t}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1}={\sqrt {2}}\omega \,\partial _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1}={\sqrt {2}}\omega \,\partial _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2767ab2e4b7f93de7955a9cfb4cf72ee6c545577" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.714ex; height:3.009ex;" alt="{\displaystyle {\vec {e}}_{1}={\sqrt {2}}\omega \,\partial _{x}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{2}={\sqrt {2}}\omega \,\partial _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{2}={\sqrt {2}}\omega \,\partial _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e8a4a9a45c08a6b185b5fd59a8117b771d68ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.591ex; height:3.343ex;" alt="{\displaystyle {\vec {e}}_{2}={\sqrt {2}}\omega \,\partial _{y}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{3}=2\omega \,\left(\exp(-x)\,\partial _{z}-\partial _{t}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>ω</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{3}=2\omega \,\left(\exp(-x)\,\partial _{z}-\partial _{t}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3464efb1ae346a085c044f9e0e47467c710e4ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.625ex; height:2.843ex;" alt="{\displaystyle {\vec {e}}_{3}=2\omega \,\left(\exp(-x)\,\partial _{z}-\partial _{t}\right).}"></span></dd></dl> <p>This framework defines a family of inertial observers that are 'comoving with the dust grains'. The computation of the <a href="/wiki/Fermi%E2%80%93Walker_differentiation" class="mw-redirect" title="Fermi–Walker differentiation">Fermi–Walker derivatives</a> with respect to <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 {\vec {e}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686ab9780c489d02208a20dfdccb4c0365f415e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{0}}"></span> shows that the spatial frames are <i>spinning</i> about <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 {\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c09c5782e951663cb81ed18fcfe715315d4e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{2}}"></span> with the angular velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1f5c70d9a04a6c3983b4ebe398367f56c46b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.254ex; height:2.176ex;" alt="{\displaystyle -\omega }"></span>. It follows that the 'non spinning inertial frame' comoving with the dust particles is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}={\vec {e}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}={\vec {e}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f74d2018f55a503d4f2179575f5926c152f21264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.095ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}={\vec {e}}_{0}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{1}=\cos(\omega t)\,{\vec {e}}_{1}-\sin(\omega t)\,{\vec {e}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{1}=\cos(\omega t)\,{\vec {e}}_{1}-\sin(\omega t)\,{\vec {e}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d162ee84e0fc18c1720154af5c7450e4dc99c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.143ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{1}=\cos(\omega t)\,{\vec {e}}_{1}-\sin(\omega t)\,{\vec {e}}_{3}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{2}={\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{2}={\vec {e}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbbf79d0fb2efbe188f58aba95c8a5be7f5b7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.095ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{2}={\vec {e}}_{2}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{3}=\sin(\omega t)\,{\vec {e}}_{1}+\cos(\omega t)\,{\vec {e}}_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{3}=\sin(\omega t)\,{\vec {e}}_{1}+\cos(\omega t)\,{\vec {e}}_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf2a121eb03a799449f319486b93bf7731ea043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.79ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{3}=\sin(\omega t)\,{\vec {e}}_{1}+\cos(\omega t)\,{\vec {e}}_{3}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Einstein_tensor">Einstein tensor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=3" title="Edit section: Einstein tensor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The components of the <a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a> (with respect to either frame above) are </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 G^{{\hat {a}}{\hat {b}}}=\omega ^{2}\operatorname {diag} (-1,1,1,1)+2\omega ^{2}\operatorname {diag} (1,0,0,0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{{\hat {a}}{\hat {b}}}=\omega ^{2}\operatorname {diag} (-1,1,1,1)+2\omega ^{2}\operatorname {diag} (1,0,0,0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92dd01ebdea896dac74e42080f986a8e6bf83ec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.732ex; height:3.676ex;" alt="{\displaystyle G^{{\hat {a}}{\hat {b}}}=\omega ^{2}\operatorname {diag} (-1,1,1,1)+2\omega ^{2}\operatorname {diag} (1,0,0,0).}"></span></dd></dl> <p>Here, the first term is characteristic of a <a href="/wiki/Lambdavacuum_solution" title="Lambdavacuum solution">Lambdavacuum solution</a> and the second term is characteristic of a pressureless <a href="/wiki/Perfect_fluid" title="Perfect fluid">perfect fluid</a> or dust solution. The cosmological constant is carefully chosen to partially cancel the matter density of the dust. </p> <div class="mw-heading mw-heading3"><h3 id="Topology">Topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=4" title="Edit section: Topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Gödel spacetime is a rare example of a regular (singularity-free) solution of the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>. Gödel's original chart is <a href="/wiki/Riemannian_manifold#Geodesic_completeness" title="Riemannian manifold">geodesically complete</a> and free of singularities. Therefore, it is a global chart, and the spacetime is <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> to <b>R</b><sup>4</sup>, and therefore, simply connected. </p> <div class="mw-heading mw-heading3"><h3 id="Curvature_invariants">Curvature invariants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=5" title="Edit section: Curvature invariants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In any Lorentzian spacetime, the fourth rank <a href="/wiki/Riemann_tensor" class="mw-redirect" title="Riemann tensor">Riemann tensor</a> is a multilinear operator on the four-dimensional space of <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vectors</a> (at some event), but a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> on the six-dimensional space of <a href="/wiki/Bivector" title="Bivector">bivectors</a> at that event. Accordingly, it has a <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a>, whose roots are the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>. In Gödelian spacetime, these eigenvalues are very simple: </p> <ul><li>triple eigenvalue zero,</li> <li>double eigenvalue <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 -\omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d00f89efd25a772265b82458cb95d4528c2164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.308ex; height:2.843ex;" alt="{\displaystyle -\omega ^{2}}"></span>,</li> <li>single eigenvalue <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 \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc60ab391d9835017f0778767fb25a54402d20f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.5ex; height:2.676ex;" alt="{\displaystyle \omega ^{2}}"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Killing_vectors">Killing vectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=6" title="Edit section: Killing vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This spacetime admits a five-dimensional <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of <a href="/wiki/Killing_vector" class="mw-redirect" title="Killing vector">Killing vectors</a>, which can be generated by '<a href="/wiki/Time_translation" class="mw-redirect" title="Time translation">time translation</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 \partial _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec987d5a79787a5086aa23e31e1d46663f276e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.06ex; height:2.509ex;" alt="{\displaystyle \partial _{t}}"></span>, two 'spatial translations' <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 \partial _{y},\;\partial _{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{y},\;\partial _{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3976318cb8a7de2e2b0073b180d89257068327b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.199ex; height:2.843ex;" alt="{\displaystyle \partial _{y},\;\partial _{z}}"></span>, plus two further Killing vector fields: </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 \partial _{x}-y\,\partial _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mi>y</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{x}-y\,\partial _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e350198564e61cde0b74411f70a78e749d1451a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.074ex; height:2.843ex;" alt="{\displaystyle \partial _{x}-y\,\partial _{y}}"></span></dd></dl> <p>and </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 -2\exp(-x)\,\partial _{t}+y\,\partial _{x}+\left(\exp(-2x)-y^{2}/2\right)\,\partial _{y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>y</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2\exp(-x)\,\partial _{t}+y\,\partial _{x}+\left(\exp(-2x)-y^{2}/2\right)\,\partial _{y}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/932c76b6e7257c4ccd7c8f9e9bf5d30cd478c1e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.812ex; height:3.343ex;" alt="{\displaystyle -2\exp(-x)\,\partial _{t}+y\,\partial _{x}+\left(\exp(-2x)-y^{2}/2\right)\,\partial _{y}.}"></span></dd></dl> <p>The isometry group acts 'transitively' (since we can translate into <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,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c0096c1e5d7aafa845035064fc54ee7b21df03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.151ex; height:2.343ex;" alt="{\displaystyle t,y,z}"></span>, and with the fourth vector we can move along <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>), so spacetime is 'homogeneous'. However, it is not 'isotropic', as can be seen. </p><p>The given demonstrators show that the slices <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=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span> admit a <a href="/wiki/Transitive_group_action" class="mw-redirect" title="Transitive group action">transitive</a> <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> three-dimensional <a href="/wiki/Transformation_group" class="mw-redirect" title="Transformation group">transformation group</a>, so that a quotient of the solution can be reinterpreted as a stationary cylindrically symmetric solution. The slices <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=y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4bd08f4d455f973f4ec698ec878ec050246278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.447ex; height:2.009ex;" alt="{\displaystyle y=y_{0}}"></span> allow for an <a href="/wiki/SL2(R)" title="SL2(R)">SL(2,<b>R</b>)</a> action, and the slices <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=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}"></span> admit a Bianchi III (c.f. the fourth Killing vector field). This can be rewritten as the symmetry group containing three-dimensional subgroups with examples of Bianchi types I, III, and VIII. Four of the five Killing vectors, as well as the curvature tensor do not depend on the coordinate y. The Gödel solution is the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of a factor <b>R</b> with a three-dimensional Lorentzian manifold (<a href="/wiki/Signature" title="Signature">signature</a> −++). </p><p>It can be shown that, except for the <a href="/wiki/Local_isometry" class="mw-redirect" title="Local isometry">local isometry</a>, the Gödel solution is the only perfect fluid solution of the Einstein field equation which admits a five-dimensional Lie algebra of the Killing vectors. </p> <div class="mw-heading mw-heading3"><h3 id="Petrov_type_and_Bel_decomposition">Petrov type and Bel decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=7" title="Edit section: Petrov type and Bel decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a> of the Gödel solution has <a href="/wiki/Petrov_type_D" class="mw-redirect" title="Petrov type D">Petrov type <b>D</b></a>. This means that for an appropriately chosen observer, the tidal forces are very close to those that would be felt from a point mass in <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newtonian gravity.</a> </p><p>To study the tidal forces in more detail, the <a href="/wiki/Bel_decomposition" title="Bel decomposition">Bel decomposition</a> of the Riemann tensor can be computed into three pieces, the tidal or electrogravitic tensor (which represents tidal forces), the magnetogravitic tensor (which represents spin-spin forces on spinning test particles and other gravitational effects analogous to magnetism), and the topogravitic tensor (which represents the spatial sectional curvatures). </p><p>Observers comoving with the dust particles would observe that the tidal tensor (with respect to <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 {\vec {u}}={\vec {e}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}={\vec {e}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df4120401614b0f9e5ce15eaf7ce95f3562cb67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.705ex; height:2.676ex;" alt="{\displaystyle {\vec {u}}={\vec {e}}_{0}}"></span>, which components evaluated in our frame) has 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 {E\left[{\vec {u}}\right]}_{{\hat {m}}{\hat {n}}}=\omega ^{2}\operatorname {diag} (1,0,1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {E\left[{\vec {u}}\right]}_{{\hat {m}}{\hat {n}}}=\omega ^{2}\operatorname {diag} (1,0,1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf3ae4c3b5e15374e49f947aa770dd2ed1f486b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.709ex; height:3.343ex;" alt="{\displaystyle {E\left[{\vec {u}}\right]}_{{\hat {m}}{\hat {n}}}=\omega ^{2}\operatorname {diag} (1,0,1).}"></span></dd></dl> <p>That is, they measure isotropic tidal tension orthogonal to the distinguished direction <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 \partial _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a280dfbf6f38f1af288cc21b0ced05dafebf7f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.284ex; height:2.843ex;" alt="{\displaystyle \partial _{y}}"></span>. </p><p>The gravitomagnetic tensor vanishes identically </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 {B\left[{\vec {u}}\right]}_{{\hat {m}}{\hat {n}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {B\left[{\vec {u}}\right]}_{{\hat {m}}{\hat {n}}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9247f0a8995dbb79014509ee1dc9ea40f69598af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.343ex; height:3.009ex;" alt="{\displaystyle {B\left[{\vec {u}}\right]}_{{\hat {m}}{\hat {n}}}=0.}"></span></dd></dl> <p>This is an artifact of the unusual symmetries of this spacetime, and implies that the putative "rotation" of the dust does not have the gravitomagnetic effects usually associated with the gravitational field produced by rotating matter. </p><p>The principal <a href="/wiki/Lorentz_invariant" class="mw-redirect" title="Lorentz invariant">Lorentz invariants</a> of the Riemann tensor are </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_{abcd}\,R^{abcd}=12\omega ^{4},\;R_{abcd}{{}^{\star }R}^{abcd}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msup> <mo>=</mo> <mn>12</mn> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{abcd}\,R^{abcd}=12\omega ^{4},\;R_{abcd}{{}^{\star }R}^{abcd}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec2d579027a7a7d35ab72246dabd28bfd1ac284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.523ex; height:3.176ex;" alt="{\displaystyle R_{abcd}\,R^{abcd}=12\omega ^{4},\;R_{abcd}{{}^{\star }R}^{abcd}=0.}"></span></dd></dl> <p>The vanishing of the second invariant means that some observers measure no gravitomagnetism, which is consistent with what was just said. The fact that the first invariant (the <a href="/wiki/Kretschmann_invariant" class="mw-redirect" title="Kretschmann invariant">Kretschmann invariant</a>) is constant reflects the homogeneity of the Gödel spacetime. </p> <div class="mw-heading mw-heading3"><h3 id="Rigid_rotation">Rigid rotation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=8" title="Edit section: Rigid rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The frame fields given above are both inertial, <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 \nabla _{{\vec {e}}_{0}}{\vec {e}}_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{{\vec {e}}_{0}}{\vec {e}}_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860add60cfbeae1050ee1631469b15539d077cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.441ex; height:3.343ex;" alt="{\displaystyle \nabla _{{\vec {e}}_{0}}{\vec {e}}_{0}=0}"></span>, but the <i>vorticity vector</i> of the timelike geodesic congruence defined by the timelike unit vectors is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\omega {\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ω</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\omega {\vec {e}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/529177d5ea9a26e3cde3df1bec1ef5d39bafdaa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.531ex; height:2.676ex;" alt="{\displaystyle -\omega {\vec {e}}_{2}}"></span></dd></dl> <p>This means that the world lines of nearby dust particles are twisting about one another. Furthermore, the shear tensor of the congruence <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 {\vec {e}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686ab9780c489d02208a20dfdccb4c0365f415e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{0}}"></span> vanishes, so the dust particles exhibit rigid rotation. </p> <div class="mw-heading mw-heading3"><h3 id="Optical_effects">Optical effects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=9" title="Edit section: Optical effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the past <a href="/wiki/Light_cone" title="Light cone">light cone</a> of a given observer is studied, it can be found that null geodesics moving orthogonally to <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 \partial _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a280dfbf6f38f1af288cc21b0ced05dafebf7f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.284ex; height:2.843ex;" alt="{\displaystyle \partial _{y}}"></span> spiral inwards toward the observer, so that if one looks radially, one sees the other dust grains in progressively time-lagged positions. However, the solution is stationary, so it might seem that an observer riding on a dust grain will not see the other grains rotating about oneself. However, recall that while the first frame given above (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 {\vec {e}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58e88e3fb935200354e129d4e013303279e15449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.133ex; height:3.009ex;" alt="{\displaystyle {\vec {e}}_{j}}"></span>) appears static in the chart, the Fermi–Walker derivatives show that it is spinning with respect to gyroscopes. The second frame (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 {\vec {f}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292e642c38265528a2d23aa5daa61ece2bc543b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.574ex; height:3.843ex;" alt="{\displaystyle {\vec {f}}_{j}}"></span>) appears to be spinning in the chart, but it is gyrostabilized, and a non-spinning inertial observer riding on a dust grain will indeed see the other dust grains rotating clockwise with angular velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> about his axis of symmetry. It turns out that in addition, optical images are expanded and sheared in the direction of rotation. </p><p>If a non-spinning inertial observer looks along his axis of symmetry, one sees one's coaxial non-spinning inertial peers apparently non-spinning with respect to oneself, as would be expected. </p> <div class="mw-heading mw-heading3"><h3 id="Shape_of_absolute_future">Shape of absolute future</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=10" title="Edit section: Shape of absolute future"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to Hawking and Ellis, another remarkable feature of this spacetime is the fact that, if the inessential y coordinate is suppressed, light emitted from an event on the world line of a given dust particle spirals outwards, forms a circular cusp, then spirals inward and reconverges at a subsequent event on the world line of the original dust particle. This means that observers looking orthogonally to 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 {\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c09c5782e951663cb81ed18fcfe715315d4e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{2}}"></span> direction can see only finitely far out, and also see themselves at an earlier time. </p><p>The cusp is a non-geodesic closed null curve. (See the more detailed discussion below using an alternative coordinate chart.) </p> <div class="mw-heading mw-heading3"><h3 id="Closed_timelike_curves">Closed timelike curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=11" title="Edit section: Closed timelike curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because of the homogeneity of the spacetime and the mutual twisting of our family of timelike geodesics, it is more or less inevitable that the Gödel spacetime should have <a href="/wiki/Closed_timelike_curve" title="Closed timelike curve">closed timelike curves</a> (CTCs). Indeed, there are CTCs through every event in the Gödel spacetime. This causal anomaly seems to have been regarded as the whole point of the model by Gödel himself, who was apparently striving to prove that Einstein's equations of spacetime are not consistent with what we intuitively understand time to be (i. e. that it passes and the past no longer exists, the position philosophers call <a href="/wiki/Presentism_(philosophy_of_time)" class="mw-redirect" title="Presentism (philosophy of time)">presentism</a>, whereas Gödel seems to have been arguing for something more like the philosophy of <a href="/wiki/Eternalism_(philosophy_of_time)" title="Eternalism (philosophy of time)">eternalism</a>).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Einstein was aware of Gödel's solution and commented in <i>Albert Einstein: Philosopher-Scientist</i><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> that if there are a series of causally-connected events in which "the series is closed in itself" (in other words, a closed timelike curve), then this suggests that there is no good physical way to define whether a given event in the series happened "earlier" or "later" than another event in the series: </p> <blockquote> <p>In that case the distinction "earlier-later" is abandoned for world-points which lie far apart in a cosmological sense, and those paradoxes, regarding the direction of the causal connection, arise, of which Mr. Gödel has spoken. </p><p>Such cosmological solutions of the gravitation-equations (with not vanishing A-constant) have been found by Mr. Gödel. It will be interesting to weigh whether these are not to be excluded on physical grounds. </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Globally_nonhyperbolic">Globally nonhyperbolic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=12" title="Edit section: Globally nonhyperbolic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the Gödel spacetime admitted any boundary-less temporal hyperslices (e.g. a <a href="/wiki/Cauchy_surface" title="Cauchy surface">Cauchy surface</a>), any such CTC would have to intersect it an odd number of times, contradicting the fact that the spacetime is simply connected. Therefore, this spacetime is not <a href="/wiki/Globally_hyperbolic" class="mw-redirect" title="Globally hyperbolic">globally hyperbolic</a>. </p> <div class="mw-heading mw-heading2"><h2 id="A_cylindrical_chart">A cylindrical chart</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=13" title="Edit section: A cylindrical chart"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this section, we introduce another coordinate chart for the Gödel solution, in which some of the features mentioned above are easier to see. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=14" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gödel did not explain how he found his solution, but there are in fact many possible derivations. We will sketch one here, and at the same time verify some of the claims made above. </p><p>Start with a simple frame in a <i>cylindrical</i> type chart, featuring two undetermined functions of the radial coordinate: </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 {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=\partial _{z},\;{\vec {e}}_{2}=\partial _{r},\,{\vec {e}}_{3}={\frac {1}{b(r)}}\,\left(-a(r)\,\partial _{t}+\partial _{\varphi }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>b</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=\partial _{z},\;{\vec {e}}_{2}=\partial _{r},\,{\vec {e}}_{3}={\frac {1}{b(r)}}\,\left(-a(r)\,\partial _{t}+\partial _{\varphi }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a25b15b07dae52ee4cdde4b73fbf726abab4044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.788ex; height:6.009ex;" alt="{\displaystyle {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=\partial _{z},\;{\vec {e}}_{2}=\partial _{r},\,{\vec {e}}_{3}={\frac {1}{b(r)}}\,\left(-a(r)\,\partial _{t}+\partial _{\varphi }\right)}"></span></dd></dl> <p>Here, we think of the timelike unit vector field <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 {\vec {e}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686ab9780c489d02208a20dfdccb4c0365f415e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{0}}"></span> as tangent to the world lines of the dust particles, and their world lines will in general exhibit nonzero vorticity but vanishing expansion and shear. Let us demand that the Einstein tensor match a dust term plus a vacuum energy term. This is equivalent to requiring that it match a perfect fluid; i.e., we require that the components of the Einstein tensor, computed with respect to our frame, take 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 G^{{\hat {i}}{\hat {j}}}=\mu \operatorname {diag} (1,0,0,0)+p\operatorname {diag} (0,1,1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msup> <mo>=</mo> <mi>μ</mi> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>p</mi> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{{\hat {i}}{\hat {j}}}=\mu \operatorname {diag} (1,0,0,0)+p\operatorname {diag} (0,1,1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507a3b0b3d1db8c932c0e8f730e86281287edd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.638ex; height:3.676ex;" alt="{\displaystyle G^{{\hat {i}}{\hat {j}}}=\mu \operatorname {diag} (1,0,0,0)+p\operatorname {diag} (0,1,1,1)}"></span></dd></dl> <p>This gives the conditions </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 b^{\prime \prime \prime }={\frac {b^{\prime \prime }\,b^{\prime }}{b}},\;\left(a^{\prime }\right)^{2}=2\,b^{\prime \prime }\,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{\prime \prime \prime }={\frac {b^{\prime \prime }\,b^{\prime }}{b}},\;\left(a^{\prime }\right)^{2}=2\,b^{\prime \prime }\,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e9728bca98562a6d8dbbea6c51241da3677386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.35ex; height:5.676ex;" alt="{\displaystyle b^{\prime \prime \prime }={\frac {b^{\prime \prime }\,b^{\prime }}{b}},\;\left(a^{\prime }\right)^{2}=2\,b^{\prime \prime }\,b}"></span></dd></dl> <p>Plugging these into the Einstein tensor, we see that in fact we now have <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 \mu =p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19b2786cdd68d182461bc265f4f3517152d2eabd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle \mu =p}"></span>. The simplest nontrivial spacetime we can construct in this way evidently would have this coefficient be some nonzero but <i>constant</i> function of the radial coordinate. Specifically, with a bit of foresight, let us choose <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 \mu =\omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e499d2363830f9ae4e7b521997e5e2a424d0faae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7ex; height:3.176ex;" alt="{\displaystyle \mu =\omega ^{2}}"></span>. This gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(r)={\frac {\sinh({\sqrt {2}}\omega \,r)}{{\sqrt {2}}\omega }},\;a(r)={\frac {\cosh({\sqrt {2}}\omega r)}{\omega }}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> <mspace width="thinmathspace" /> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mi>ω</mi> </mfrac> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(r)={\frac {\sinh({\sqrt {2}}\omega \,r)}{{\sqrt {2}}\omega }},\;a(r)={\frac {\cosh({\sqrt {2}}\omega r)}{\omega }}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4e726ce36651618fda6e304461381079a4da33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:45.082ex; height:7.009ex;" alt="{\displaystyle b(r)={\frac {\sinh({\sqrt {2}}\omega \,r)}{{\sqrt {2}}\omega }},\;a(r)={\frac {\cosh({\sqrt {2}}\omega r)}{\omega }}+c}"></span></dd></dl> <p>Finally, let us demand that this frame satisfy </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 {\vec {e}}_{3}={\frac {1}{r}}\,\partial _{\varphi }+O\left({\frac {1}{r^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{3}={\frac {1}{r}}\,\partial _{\varphi }+O\left({\frac {1}{r^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a58e7c497b29d0fe4f82aaecedffa2bd703840a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.664ex; height:6.176ex;" alt="{\displaystyle {\vec {e}}_{3}={\frac {1}{r}}\,\partial _{\varphi }+O\left({\frac {1}{r^{2}}}\right)}"></span></dd></dl> <p>This gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=-1/\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=-1/\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372c5c6741afad939651fd93f774dea2b040061b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.684ex; height:2.843ex;" alt="{\displaystyle c=-1/\omega }"></span>, and our frame becomes </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 {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=\partial _{z},\;{\vec {e}}_{2}=\partial _{r},\;{\vec {e}}_{3}={\frac {{\sqrt {2}}\omega }{\sinh({\sqrt {2}}\omega r)}}\,\partial _{\varphi }-{\frac {{\sqrt {2}}\sinh({\sqrt {2}}\omega r)}{1+\cosh({\sqrt {2}}\omega r)}}\,\partial _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> </mrow> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=\partial _{z},\;{\vec {e}}_{2}=\partial _{r},\;{\vec {e}}_{3}={\frac {{\sqrt {2}}\omega }{\sinh({\sqrt {2}}\omega r)}}\,\partial _{\varphi }-{\frac {{\sqrt {2}}\sinh({\sqrt {2}}\omega r)}{1+\cosh({\sqrt {2}}\omega r)}}\,\partial _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b1321980cbf9bcf77dd8159002c963610d74bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.293ex; height:7.176ex;" alt="{\displaystyle {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=\partial _{z},\;{\vec {e}}_{2}=\partial _{r},\;{\vec {e}}_{3}={\frac {{\sqrt {2}}\omega }{\sinh({\sqrt {2}}\omega r)}}\,\partial _{\varphi }-{\frac {{\sqrt {2}}\sinh({\sqrt {2}}\omega r)}{1+\cosh({\sqrt {2}}\omega r)}}\,\partial _{t}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Appearance_of_the_light_cones">Appearance of the light cones</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=15" title="Edit section: Appearance of the light cones"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the metric tensor we find that the vector field <span class="nowrap"><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 \partial _{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/388ac11e50ceea5dfa01db387ae33d0a1b4b2071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.542ex; height:2.843ex;" alt="{\displaystyle \partial _{\varphi }}"></span>,</span> which is <i>spacelike</i> for small radii, becomes <i>null</i> 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=r_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f8080446d3967ee12d16b0d0d19ff1ba79f509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.14ex; height:2.009ex;" alt="{\displaystyle r=r_{c}}"></span> 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 r_{c}={\frac {\operatorname {arccosh} (3)}{{\sqrt {2}}\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>arccosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ω</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{c}={\frac {\operatorname {arccosh} (3)}{{\sqrt {2}}\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8ff43e6396d78ab8c984097cc9de5a25a4641a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.409ex; height:6.676ex;" alt="{\displaystyle r_{c}={\frac {\operatorname {arccosh} (3)}{{\sqrt {2}}\omega }}}"></span></dd></dl> <p>This is because at that radius we find that <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 {\vec {e}}_{3}={\tfrac {\omega }{2}}\,\partial _{\varphi }-\partial _{t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>ω</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{3}={\tfrac {\omega }{2}}\,\partial _{\varphi }-\partial _{t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f3cad38127febb6ca5879bf9c13e21cc250f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.711ex; height:3.176ex;" alt="{\displaystyle {\vec {e}}_{3}={\tfrac {\omega }{2}}\,\partial _{\varphi }-\partial _{t},}"></span> so <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 {\tfrac {\omega }{2}}\,\partial _{\varphi }={\vec {e}}_{3}+{\vec {e}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>ω</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\omega }{2}}\,\partial _{\varphi }={\vec {e}}_{3}+{\vec {e}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9188be7bec5ebc298236805f7207a3b833fed9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.281ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\omega }{2}}\,\partial _{\varphi }={\vec {e}}_{3}+{\vec {e}}_{0}}"></span> and is therefore null. The circle <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=r_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f8080446d3967ee12d16b0d0d19ff1ba79f509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.14ex; height:2.009ex;" alt="{\displaystyle r=r_{c}}"></span> at a given <i>t</i> is a closed null curve, but not a null geodesic. </p><p>Examining the frame above, we can see that the coordinate <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> is inessential; our spacetime is the direct product of a factor <b>R</b> with a signature −++ three-manifold. Suppressing <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> in order to focus our attention on this three-manifold, let us examine how the appearance of the light cones changes as we travel out from the axis of symmetry <span class="nowrap"><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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></span>:</span> </p> <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Goedel_lambdadust_samplecones.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c7/Goedel_lambdadust_samplecones.gif" decoding="async" width="400" height="400" class="mw-file-element" data-file-width="400" data-file-height="400" /></a><figcaption>Two light cones (with their accompanying frame vectors) in the cylindrical chart for the Gödel lambda dust solution. As we move outwards from the nominal symmetry axis, the cones <i>tip forward</i> and <i>widen</i>. Vertical coordinate lines (representing the world lines of the dust particles) are <i>timelike</i>.</figcaption></figure> <p>When we get to the critical radius, the cones become tangent to the closed null curve. </p> <div class="mw-heading mw-heading3"><h3 id="A_congruence_of_closed_timelike_curves">A congruence of closed timelike curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=16" title="Edit section: A congruence of closed timelike curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At the critical radius <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=r_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f8080446d3967ee12d16b0d0d19ff1ba79f509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.14ex; height:2.009ex;" alt="{\displaystyle r=r_{c}}"></span>, the vector field <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 \partial _{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/388ac11e50ceea5dfa01db387ae33d0a1b4b2071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.542ex; height:2.843ex;" alt="{\displaystyle \partial _{\varphi }}"></span> becomes null. For larger radii, it is <i>timelike</i>. Thus, corresponding to our symmetry axis we have a timelike <a href="/wiki/Congruence_(general_relativity)" title="Congruence (general relativity)">congruence</a> made up of <i>circles</i> and corresponding to certain observers. This congruence is however <i>only defined outside the cylinder</i> <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=r_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f8080446d3967ee12d16b0d0d19ff1ba79f509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.14ex; height:2.009ex;" alt="{\displaystyle r=r_{c}}"></span>. </p><p>This is not a geodesic congruence; rather, each observer in this family must maintain a <i>constant acceleration</i> in order to hold his course. Observers with smaller radii must accelerate harder; as <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\rightarrow r_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\rightarrow r_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82cbd7b2b35b75efbb42839f48f2c78025fa0b5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.656ex; height:2.176ex;" alt="{\displaystyle r\rightarrow r_{c}}"></span> the magnitude of acceleration diverges, which is just what is expected, given that <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=r_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f8080446d3967ee12d16b0d0d19ff1ba79f509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.14ex; height:2.009ex;" alt="{\displaystyle r=r_{c}}"></span> is a null curve. </p> <div class="mw-heading mw-heading3"><h3 id="Null_geodesics">Null geodesics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=17" title="Edit section: Null geodesics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we examine the past light cone of an event on the axis of symmetry, we find the following picture: </p> <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Goedel_lambdadust_centralcone_inspiral.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/65/Goedel_lambdadust_centralcone_inspiral.gif" decoding="async" width="240" height="240" class="mw-file-element" data-file-width="240" data-file-height="240" /></a><figcaption>The null geodesics spiral counterclockwise toward an observer on the axis of symmetry. This shows them from "above".</figcaption></figure> <p>Recall that vertical coordinate lines in our chart represent the world lines of the dust particles, but <i>despite their straight appearance in our chart</i>, the congruence formed by these curves has nonzero vorticity, so the world lines are actually <i>twisting about each other</i>. The fact that the null geodesics spiral inwards in the manner shown above means that when our observer, when looking <i>radially outwards</i>, sees nearby dust particles not at their current locations, but at their earlier locations. This is what we would expect if the dust particles are in fact rotating about one another. </p><p>The null geodesics are <i>geometrically straight</i>; in the figure, they appear to be spirals only because the coordinates are "rotating" in order to permit the dust particles to appear stationary. </p> <div class="mw-heading mw-heading3"><h3 id="The_absolute_future">The absolute future</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=18" title="Edit section: The absolute future"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to Hawking and Ellis (see monograph cited below), all light rays emitted from an event on the symmetry axis reconverge at a later event on the axis, with the null geodesics forming a circular cusp (which is a null curve, but not a null geodesic): </p> <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Goedel_lambdadust_centralcone.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/48/Goedel_lambdadust_centralcone.gif" decoding="async" width="240" height="240" class="mw-file-element" data-file-width="240" data-file-height="240" /></a><figcaption>Hawking and Ellis picture of expansion and reconvergence of light emitted by an observer on the axis of symmetry.</figcaption></figure> <p>This implies that in the Gödel lambda dust solution, the <a href="/wiki/Absolute_future" class="mw-redirect" title="Absolute future">absolute future</a> of each event has a character very different from what we might naively expect. </p> <div class="mw-heading mw-heading2"><h2 id="Cosmological_interpretation">Cosmological interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=19" title="Edit section: Cosmological interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Following Gödel, we can interpret the dust particles as galaxies, so that the Gödel solution becomes a <i>cosmological model of a rotating universe</i>. Besides rotating, this model exhibits no <a href="/wiki/Hubble_expansion" class="mw-redirect" title="Hubble expansion">Hubble expansion</a>, so it is not a realistic model of the universe in which we live, but can be taken as illustrating an alternative universe, which would in principle be allowed by general relativity (if one admits the legitimacy of a negative cosmological constant). Less well known solutions of Gödel's exhibit both rotation and Hubble expansion and have other qualities of his first model, but traveling into the past is not possible. According to <a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Stephen Hawking</a>, <i>these models could well be a reasonable description of the universe that we observe</i>, however observational data are compatible only with a very low rate of rotation.<sup id="cite_ref-Hawking_4-0" class="reference"><a href="#cite_note-Hawking-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The quality of these observations improved continually up until Gödel's death, and he would always ask "Is the universe rotating yet?" and be told "No, it isn't".<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><p>We have seen that observers lying on the <i>y</i> axis (in the original chart) see the rest of the universe rotating clockwise about that axis. However, the homogeneity of the spacetime shows that the <i>direction</i> but not the <i>position</i> of this "axis" is distinguished. </p><p>Some have interpreted the Gödel universe as a counterexample to Einstein's hopes that general relativity should exhibit some kind of <a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a>,<sup id="cite_ref-Hawking_4-1" class="reference"><a href="#cite_note-Hawking-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> citing the fact that the matter is rotating (world lines twisting about each other) in a manner sufficient to pick out a preferred direction, although with no distinguished axis of rotation. </p><p>Others<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2015)">citation needed</span></a></i>]</sup> take Mach principle to mean some physical law tying the definition of non-spinning inertial frames at each event to the global distribution and motion of matter everywhere in the universe, and say that because the non-spinning inertial frames are precisely tied to the rotation of the dust in just the way such a Mach principle would suggest, this model <i>does</i> accord with Mach's ideas. </p><p>Many other exact solutions that can be interpreted as cosmological models of rotating universes are known.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </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=G%C3%B6del_metric&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a>, for another rotating dust solution with (true) cylindrical symmetry,</li> <li><a href="/wiki/Dust_solution" title="Dust solution">Dust solution</a>, an article about dust solutions in general relativity.</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=G%C3%B6del_metric&action=edit&section=21" 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"> <div class="mw-references-wrap"><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="CITEREFGödel1949" class="citation journal cs1">Gödel, Kurt (1949-07-01). <a rel="nofollow" class="external text" href="http://cds.cern.ch/record/430019/files/RevModPhys.21.447.pdf">"An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation"</a> <span class="cs1-format">(PDF)</span>. <i>Reviews of Modern Physics</i>. <b>21</b> (3): <span class="nowrap">447–</span>450. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.21.447">10.1103/RevModPhys.21.447</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6861">0034-6861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=An+Example+of+a+New+Type+of+Cosmological+Solutions+of+Einstein%27s+Field+Equations+of+Gravitation&rft.volume=21&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E447-%3C%2Fspan%3E450&rft.date=1949-07-01&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.21.447&rft.issn=0034-6861&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rft_id=http%3A%2F%2Fcds.cern.ch%2Frecord%2F430019%2Ffiles%2FRevModPhys.21.447.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" 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="CITEREFYourgrau2005" class="citation book cs1">Yourgrau, Palle (2005). <i>A world without time: the forgotten legacy of Gödel and Einstein</i>. New York: Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0465092942" title="Special:BookSources/0465092942"><bdi>0465092942</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+world+without+time%3A+the+forgotten+legacy+of+G%C3%B6del+and+Einstein&rft.place=New+York&rft.pub=Basic+Books&rft.date=2005&rft.isbn=0465092942&rft.aulast=Yourgrau&rft.aufirst=Palle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" 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="CITEREFEinstein1949" class="citation web cs1">Einstein, Albert (1949). <a rel="nofollow" class="external text" href="https://www.marxists.org/reference/archive/einstein/works/1940s/reply.htm">"Einstein's Reply to Criticisms"</a>. <i>Albert Einstein: Philosopher-Scientist</i>. Cambridge University Press<span class="reference-accessdate">. Retrieved <span class="nowrap">29 November</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Albert+Einstein%3A+Philosopher-Scientist&rft.atitle=Einstein%27s+Reply+to+Criticisms&rft.date=1949&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=http%3A%2F%2Fwww.marxists.org%2Freference%2Farchive%2Feinstein%2Fworks%2F1940s%2Freply.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" class="Z3988"></span></span> </li> <li id="cite_note-Hawking-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hawking_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hawking_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödelFeferman1986" class="citation book cs1 cs1-prop-foreign-lang-source">Gödel, Kurt; Feferman, Solomon (1986). "Gödel 1949: Introductory note to 1949 and 1952, by S. W. Hawking". <i>Collected works</i> (in English and German). Oxford [Oxfordshire] : New York: Clarendon Press ; Oxford University Press. p. 189. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-503964-1" title="Special:BookSources/978-0-19-503964-1"><bdi>978-0-19-503964-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=G%C3%B6del+1949%3A+Introductory+note+to+1949+and+1952%2C+by+S.+W.+Hawking&rft.btitle=Collected+works&rft.place=Oxford+%5BOxfordshire%5D+%3A+New+York&rft.pages=189&rft.pub=Clarendon+Press+%3B+Oxford+University+Press&rft.date=1986&rft.isbn=978-0-19-503964-1&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rft.au=Feferman%2C+Solomon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWang2002" class="citation book cs1">Wang, Hao (2002). <i>Reflections on Kurt Gödel</i>. A Bradford book (6. print ed.). Cambridge, Mass.: MIT Press. p. 183. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-73087-7" title="Special:BookSources/978-0-262-73087-7"><bdi>978-0-262-73087-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reflections+on+Kurt+G%C3%B6del&rft.place=Cambridge%2C+Mass.&rft.series=A+Bradford+book&rft.pages=183&rft.edition=6.+print&rft.pub=MIT+Press&rft.date=2002&rft.isbn=978-0-262-73087-7&rft.aulast=Wang&rft.aufirst=Hao&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyanShepley1975" class="citation book cs1">Ryan, Michael P.; Shepley, Lawrence C. (1975). <i>Homogeneous relativistic cosmologies</i>. Princeton series in physics. Princeton, N.J: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08146-5" title="Special:BookSources/978-0-691-08146-5"><bdi>978-0-691-08146-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Homogeneous+relativistic+cosmologies&rft.place=Princeton%2C+N.J&rft.series=Princeton+series+in+physics&rft.pub=Princeton+University+Press&rft.date=1975&rft.isbn=978-0-691-08146-5&rft.aulast=Ryan&rft.aufirst=Michael+P.&rft.au=Shepley%2C+Lawrence+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del_metric&action=edit&section=22" title="Edit section: Notes"><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="CITEREFG._Dautcourt_and_M._Abdel-Megied2006" class="citation journal cs1">G. Dautcourt and M. Abdel-Megied (2006). "Revisiting the Light Cone of the Goedel Universe". <i>Classical and Quantum Gravity</i>. <b>23</b> (4): <span class="nowrap">1269–</span>1288. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0511015">gr-qc/0511015</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006CQGra..23.1269D">2006CQGra..23.1269D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F23%2F4%2F013">10.1088/0264-9381/23/4/013</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14666907">14666907</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Classical+and+Quantum+Gravity&rft.atitle=Revisiting+the+Light+Cone+of+the+Goedel+Universe&rft.volume=23&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1269-%3C%2Fspan%3E1288&rft.date=2006&rft_id=info%3Aarxiv%2Fgr-qc%2F0511015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14666907%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F23%2F4%2F013&rft_id=info%3Abibcode%2F2006CQGra..23.1269D&rft.au=G.+Dautcourt+and+M.+Abdel-Megied&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephani,_HansKramer,_DietrichMacCallum,_MalcolmHoenselaers,_Cornelius2003" class="citation book cs1">Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003). <i>Exact Solutions to Einstein's Field Equations</i> (2nd ed.). Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-46136-7" title="Special:BookSources/0-521-46136-7"><bdi>0-521-46136-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exact+Solutions+to+Einstein%27s+Field+Equations&rft.place=Cambridge&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=0-521-46136-7&rft.au=Stephani%2C+Hans&rft.au=Kramer%2C+Dietrich&rft.au=MacCallum%2C+Malcolm&rft.au=Hoenselaers%2C+Cornelius&rft.au=Herlt%2C+Eduard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" class="Z3988"></span> See <i>section 12.4</i> for the uniqueness theorem.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHawking,_StephenEllis,_G._F._R.1973" class="citation book cs1">Hawking, Stephen; Ellis, G. F. R. (1973). <i>The Large Scale Structure of Space-Time</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-09906-4" title="Special:BookSources/0-521-09906-4"><bdi>0-521-09906-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Large+Scale+Structure+of+Space-Time&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1973&rft.isbn=0-521-09906-4&rft.au=Hawking%2C+Stephen&rft.au=Ellis%2C+G.+F.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" class="Z3988"></span> See <i>section 5.7</i> for a classic discussion of CTCs in the Gödel spacetime. <i>Warning:</i> in Fig. 31, the light cones do indeed tip over, but they also widen, so that vertical coordinate lines are always timelike; indeed, these represent the world lines of the dust particles, so they are timelike geodesics.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1949" class="citation journal cs1">Gödel, Kurt (1949-07-01). <a rel="nofollow" class="external text" href="http://cds.cern.ch/record/430019/files/RevModPhys.21.447.pdf">"An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation"</a> <span class="cs1-format">(PDF)</span>. <i>Reviews of Modern Physics</i>. <b>21</b> (3): <span class="nowrap">447–</span>450. <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/1949RvMP...21..447G">1949RvMP...21..447G</a>. <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/10.1103%2FRevModPhys.21.447">10.1103/RevModPhys.21.447</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6861">0034-6861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=An+Example+of+a+New+Type+of+Cosmological+Solutions+of+Einstein%27s+Field+Equations+of+Gravitation&rft.volume=21&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E447-%3C%2Fspan%3E450&rft.date=1949-07-01&rft.issn=0034-6861&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.21.447&rft_id=info%3Abibcode%2F1949RvMP...21..447G&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rft_id=http%3A%2F%2Fcds.cern.ch%2Frecord%2F430019%2Ffiles%2FRevModPhys.21.447.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del+metric" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: date and year (<a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">link</a>)</span></li> <li><a rel="nofollow" class="external text" href="http://www.elemenat.com/eng/docs/RotatingUniverse.pdf">Vukovic R. (2014): <i>Tensor Model of the Rotating Universe</i>, Exercise in Special Relativity</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141113154232/http://www.elemenat.com/eng/docs/RotatingUniverse.pdf">Archived</a> 2014-11-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Time_travel" title="Template:Time travel"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_travel" title="Template talk:Time travel"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_travel" title="Special:EditPage/Template:Time travel"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Time_travel73" style="font-size:114%;margin:0 4em"><a href="/wiki/Time_travel" title="Time travel">Time travel</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General terms and concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronology_protection_conjecture" title="Chronology protection conjecture">Chronology protection conjecture</a></li> <li><a href="/wiki/Closed_timelike_curve" title="Closed timelike curve">Closed timelike curve</a></li> <li><a href="/wiki/Novikov_self-consistency_principle" title="Novikov self-consistency principle">Novikov self-consistency principle</a></li> <li><a href="/wiki/Self-fulfilling_prophecy#Causal_loop" title="Self-fulfilling prophecy">Self-fulfilling prophecy</a></li> <li><a href="/wiki/Quantum_mechanics_of_time_travel" title="Quantum mechanics of time travel">Quantum mechanics of time travel</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Philosophy_of_space_and_time" title="Philosophy of space and time">Philosophy of space and time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Butterfly_effect" title="Butterfly effect">Butterfly effect</a></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a></li> <li><a href="/wiki/Eternalism_(philosophy_of_time)" title="Eternalism (philosophy of time)">Eternalism</a></li> <li><a href="/wiki/Fatalism" title="Fatalism">Fatalism</a></li> <li><a href="/wiki/Free_will" title="Free will">Free will</a></li> <li><a href="/wiki/Predestination" title="Predestination">Predestination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Causality</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Causal_loop" class="mw-redirect" title="Causal loop">Causal loop</a></li> <li><a href="/wiki/Causality_(physics)" title="Causality (physics)">Causality (physics)</a></li> <li><a href="/wiki/Chronology_protection_conjecture" title="Chronology protection conjecture">Chronology protection conjecture</a></li> <li><a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship hypothesis</a></li> <li><a href="/wiki/The_chicken_or_the_egg" class="mw-redirect" title="The chicken or the egg">The chicken or the egg</a></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds interpretation</a></li> <li><a href="/wiki/Grandfather_paradox" class="mw-redirect" title="Grandfather paradox">Grandfather paradox</a></li> <li><a href="/wiki/Quantum_mechanics_of_time_travel" title="Quantum mechanics of time travel">Quantum mechanics of time travel</a></li> <li><a href="/wiki/Time_viewer" title="Time viewer">Time viewer</a></li> <li><a href="/wiki/Temporal_paradox" title="Temporal paradox">Temporal paradox</a></li> <li><a href="/wiki/Temporal_paradox#Grandfather_paradox" title="Temporal paradox">Grandfather paradox</a></li> <li><a href="/wiki/Time_loop" title="Time loop">Time loop</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_travel_in_fiction" title="Time travel in fiction">Time travel in fiction</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_timelines#Fictional" title="List of timelines">Timelines in fiction</a></li> <li><a href="/wiki/List_of_time_travel_works_of_fiction" title="List of time travel works of fiction">Time travel in fiction</a></li> <li><a href="/wiki/List_of_films_featuring_time_loops" title="List of films featuring time loops">Time loops in film</a></li> <li><i>Major works:</i></li> <li><a href="/wiki/The_Time_Machine" title="The Time Machine">The Time Machine</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Parallel timelines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_travel_in_fiction#Alternative_future,_history,_timelines,_and_dimensions" title="Time travel in fiction">Alternative future</a></li> <li><a href="/wiki/Alternate_history" title="Alternate history">Alternate history</a></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds interpretation</a></li> <li><a href="/wiki/Multiverse" title="Multiverse">Multiverse</a></li> <li><a href="/wiki/Parallel_universes_in_fiction" title="Parallel universes in fiction">Parallel universes in fiction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spacetime" title="Spacetime">Spacetimes</a> in <a href="/wiki/General_relativity" title="General relativity">general relativity</a> that<br /> can contain <a href="/wiki/Closed_timelike_curve" title="Closed timelike curve">closed timelike curves</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre metric</a></li> <li><a href="/wiki/BTZ_black_hole" title="BTZ black hole">BTZ black hole</a></li> <li><a class="mw-selflink selflink">Gödel metric</a></li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr metric</a></li> <li><a href="/wiki/Krasnikov_tube" title="Krasnikov tube">Krasnikov tube</a></li> <li><a href="/wiki/Misner_space" title="Misner space">Misner space</a></li> <li><a href="/wiki/Tipler_cylinder" title="Tipler cylinder">Tipler cylinder</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Wormhole#Traversable_wormholes" title="Wormhole">Traversable wormholes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Science_fiction" title="Science fiction">Science fiction</a></li> <li><a href="/wiki/Time_machine" class="mw-redirect" title="Time machine">Time machine</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Relativity254" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity254" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Special_relativity" title="Special relativity">Special<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br />theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a class="mw-selflink selflink">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Weiss</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Category</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Gödel_metric&oldid=1264699918">https://en.wikipedia.org/w/index.php?title=Gödel_metric&oldid=1264699918</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Exact_solutions_in_general_relativity" title="Category:Exact solutions in general relativity">Exact solutions in general relativity</a></li><li><a href="/wiki/Category:Metric_tensors" title="Category:Metric tensors">Metric tensors</a></li><li><a href="/wiki/Category:Works_by_Kurt_G%C3%B6del" title="Category:Works by Kurt Gödel">Works by Kurt Gödel</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_July_2015" title="Category:Articles with unsourced statements from July 2015">Articles with unsourced statements from July 2015</a></li><li><a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">CS1 maint: date and year</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 23 December 2024, at 02:37<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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