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class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/isometry+group">isometry group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+Riemannian+metrics">moduli space of Riemannian metrics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo-Riemannian manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a>, <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic">geodesic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic+convexity">geodesic convexity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic+flow">geodesic flow</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Levi-Civita+connection">Levi-Civita connection</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+curvature">Riemann curvature</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+metric+connection">torsion of a metric connection</a></li> </ul> </li> </ul> <h2 id="further_concepts">Further concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+inner+product">Hodge inner product</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gradient">gradient</a>, <a class="existingWikiWord" href="/nlab/show/gradient+flow">gradient flow</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a>-theorem</li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>, <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/Riemannian+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="gravity">Gravity</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></strong></p> <p><strong>Formalism</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/super-Cartan+geometry">super</a>-)<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo</a>-<a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> </li> </ul> </li> </ul> <p><strong>Definition</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>, <a class="existingWikiWord" href="/nlab/show/Einstein%27s+equations">Einstein's equations</a>, <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a>, <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a>, <a class="existingWikiWord" href="/nlab/show/Plebanski+formulation+of+gravity">Plebanski formulation of gravity</a>, <a class="existingWikiWord" href="/nlab/show/teleparallel+gravity">teleparallel gravity</a></p> </li> </ul> </li> </ul> <p><strong>Spacetime configurations</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole+spacetime">black hole spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a></p> </li> </ul> </li> </ul> <p><strong>Properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bekenstein-Hawking+entropy">Bekenstein-Hawking entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+thermodynamics">generalized second law of thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmic+censorship+hypothesis">cosmic censorship hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+gravity+conjecture">weak gravity conjecture</a></p> </li> </ul> <p><strong>Spacetimes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+cosmology">standard model of cosmology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a></p> </li> </ul> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a></strong></th><th>vanishing <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th><th><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th></tr></thead><tbody><tr><td style="text-align: left;"><strong>vanishing <a class="existingWikiWord" href="/nlab/show/charge">charge</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/charge">charge</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Reissner-Nordstrom+spacetime">Reissner-Nordstrom spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr-Newman+spacetime">Kerr-Newman spacetime</a></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/asymptotically+flat+spacetime">asymptotically flat spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitational+wave">gravitational wave</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FRW+spacetime">FRW spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Taub-NUT+spacetime">Taub-NUT spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein+manifold">Einstein manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pp-wave+spacetime">pp-wave spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KK-monopole">KK-monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Malament%E2%80%93Hogarth+spacetime">Malament–Hogarth spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></p> </li> </ul> <p><strong>Quantum theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/graviton">graviton</a>, <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></p> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#intermingling_of_space_and_time'>Intermingling of space and time</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#books'>Books</a></li> <li><a href='#articles'>Articles</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general">General</h3> <p>A <em>spacetime</em> is a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> that models <a class="existingWikiWord" href="/nlab/show/space">space</a> and <a class="existingWikiWord" href="/nlab/show/time">time</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>.</p> <p>This is formalized by saying that a <strong>spacetime</strong> is a <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+space">smooth Lorentzian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\mu)</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/time+orientation">time orientation</a> (see there).</p> <p>Hence a point in a spacetime is called an <em><a class="existingWikiWord" href="/nlab/show/event">event</a></em>.</p> <p>In the context of classical <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> a <strong>spacetime</strong> is usually in addition assumed to be connected and four-dimensional. A connected Lorentzian manifold is either time orientable or it has a two-sheeted covering which is time orientable.</p> <p>In <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a>, notably in <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> and <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> model coordinates where <em>events</em> can take place from the viewpoint of an observer (“points in space and time”) while the metric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> models the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> in <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a>.</p> <h3 id="intermingling_of_space_and_time">Intermingling of space and time</h3> <p>The noun “spacetime” is used in both <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> and <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a>, but is best motivated from the viewpoint of <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a>. <a class="existingWikiWord" href="/nlab/show/special+relativity">Special relativity</a> deals with the <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> only. The <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> allows a <em>canonical</em> choice of <em>global</em> coordinates such that the metric tensor has in every point the form diag(-1, 1, 1, 1), which identifies the first coordinate as representing the time coordinate and the others as representing space coordinates.</p> <p>Given a general spacetime, there is not necessarily a globally defined coordinate system, and therefore not necessarily a globally defined <em>canonical</em> time coordinate. More specifically, there are spacetimes that admit coordinates defined on subsets where the <em>physical</em> interpretation of the coordinates as modelling time and space coordinates <em>changes</em> over the domain of definition.</p> <p>(TODO: references and explanations).</p> <h2 id="examples">Examples</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a></li> </ul> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a></strong></th><th>vanishing <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th><th><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th></tr></thead><tbody><tr><td style="text-align: left;"><strong>vanishing <a class="existingWikiWord" href="/nlab/show/charge">charge</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/charge">charge</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Reissner-Nordstrom+spacetime">Reissner-Nordstrom spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr-Newman+spacetime">Kerr-Newman spacetime</a></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/asymptotically+flat+spacetime">asymptotically flat spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kerr-Schild+spacetime">Kerr-Schild spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/piecewise+flat+spacetime">piecewise flat spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitational+wave">gravitational wave</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FRW+spacetime">FRW spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Taub-NUT+spacetime">Taub-NUT spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein+manifold">Einstein manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pp-wave+spacetime">pp-wave spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KK-monopole">KK-monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Malament%E2%80%93Hogarth+spacetime">Malament–Hogarth spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime+signature">spacetime signature</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/static+spacetime">static spacetime</a>, <a class="existingWikiWord" href="/nlab/show/stationary+spacetime">stationary spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+spacetime">universal spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/cosmology">cosmology</a>, <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a>, <a class="existingWikiWord" href="/nlab/show/matter">matter</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/force">force</a>, <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="books">Books</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stephen+W.+Hawking">Stephen W. Hawking</a>, <a class="existingWikiWord" href="/nlab/show/George+F.+R.+Ellis">George F. R. Ellis</a>: <em>The large scale structure of space-time</em>, Cambridge Univ. Press (1973, 2010) [<a href="https://doi.org/10.1017/CBO9780511524646">doi:10.1017/CBO9780511524646</a>, <a href="https://en.wikipedia.org/wiki/The_Large_Scale_Structure_of_Space%E2%80%93Time">Wikipedia entry</a>]</p> </li> <li> <p>John Beem, Paul Ehrlich, <em>Global Lorentzian geometry</em>, Marcel Dekker 1981 (and Russian, updated translation, Mir 1985)</p> </li> </ul> <h3 id="articles">Articles</h3> <ul> <li> <p>L. Markus, <em>Line element fields and Lorentz structures on differentiable manifolds</em>, Ann. of Math. (2) 62 (1955), 411–417, <a href="http://www.ams.org/mathscinet-getitem?mr=0073169">MR0073169</a> <a href="http://www.jstor.org/stable/1970071">jstor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a>, <em>Gravitational collapse and space-time singularities</em>, Phys. Rev. Lett. <strong>14</strong>, 57–59</p> </li> <li> <p>Paul Ernest, The Mathematics of Space-Time and the Limits of Human Understanding, Philosophy of Mathematics Education Journal, No. 30 (October 2016) ISSN:1465-2978</p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/physics">physics</a>, <a class="category_link" href="/nlab/all_pages/geometry">geometry</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on July 30, 2024 at 09:13:00. 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