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A simple one is:</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a> (without <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>) is called <strong>globally hyperbolic</strong> if it contains a <a class="existingWikiWord" href="/nlab/show/Cauchy+surface">Cauchy surface</a>.</p> </div> <p>In this form the characterization of global hyperbolicity appears for instance in the paragraph at the bottom of page 211 in (<a href="#LargeScale">HE</a>). The equivalence of this to more traditional definitions is (<a href="#LargeScale">HE, prop. 6.6.3</a>) together with (<a href="#LargeScale">HE, prop. 6.6.8</a>), due to (<a href="#Geroch">Geroch1970</a>). The latter in fact implies the following stronger statement:</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a> (without <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>) is <strong>globally hyperbolic</strong> if it admits a <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> by <a class="existingWikiWord" href="/nlab/show/Cauchy+surfaces">Cauchy surfaces</a>.</p> </div> <p>See also (<a href="#BaerGinouxPfaeffle07">Baer-Ginoux-Pfaeffle 07, theorem 1.3.10</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>So in particular for a globally hyperbolic spacetime <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> there is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>ℝ</mi><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \phi\colon \mathbb{R} \times \Sigma \to X </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/product">product</a> of the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>dim</mi><mi>X</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(dim X - 1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">t \in \mathbb{R}</annotation></semantics></math> the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi(t, \Sigma) \subset X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Cauchy+surface">Cauchy surface</a> of <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>.</p> </div> <p>A <em>time orientation</em> of a globally hyperbolic Lorentzian spacetime is a choice of <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> of the factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> under the above homeomorphism.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+quantum+field+theory">locally covariant perturbative quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cauchy+problem">Cauchy problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+partial+differential+equation">Green hyperbolic partial differential equation</a></p> </li> </ul> <h2 id="references">References</h2> <p>Concise collection of definitions:</p> <ul> <li id="MinguzziSánchez">E. Minguzzi, <a class="existingWikiWord" href="/nlab/show/Miguel+S%C3%A1nchez">Miguel Sánchez</a>, §3.11 in: <em>The causal hierarchy of spacetimes</em>, in: <em>Recent Developments in Pseudo-Riemannian Geometry</em>, EMS ESI Lectures in Mathematics and Physics <strong>4</strong> (2008) 299-358 [<a href="https://arxiv.org/abs/gr-qc/0609119">arXiv:gr-qc/0609119</a>, <a href="https://bookstore.ams.org/view?ProductCode=EMSESILEC/4">ISBN:978-3-03719-051-7</a>]</li> </ul> <p>Survey:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Miguel+S%C3%A1nchez">Miguel Sánchez</a>, <em>Globally hyperbolic spacetimes: slicings, boundaries and counterexamples</em>, Gen Relativ Gravit <strong>54</strong> 124 (2022) [<a href="https://arxiv.org/abs/2110.13672">arXiv:2110.13672</a>, <a href="https://doi.org/10.1007/s10714-022-03002-6">doi:10.1007/s10714-022-03002-6</a>]</li> </ul> <p>See also:</p> <ul> <li id="LargeScale"> <p>Hawking, Ellis, section 6.6 of <em>The large-scale structure of Space-Time</em> Cambridge (1973)</p> </li> <li id="BaerGinouxPfaeffle07"> <p><a class="existingWikiWord" href="/nlab/show/Christian+B%C3%A4r">Christian Bär</a>, <a class="existingWikiWord" href="/nlab/show/Nicolas+Ginoux">Nicolas Ginoux</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Pf%C3%A4ffle">Frank Pfäffle</a>, <em>Wave Equations on Lorentzian Manifolds and Quantization</em>, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (<a href="https://arxiv.org/abs/0806.1036">arXiv:0806.1036</a>)</p> </li> </ul> <p>The fact that a single Cauchy surface implies a foliation by Cauchy surfaces is due to:</p> <ul> <li id="Geroch"><a class="existingWikiWord" href="/nlab/show/Robert+Geroch">Robert Geroch</a>, §5, Thm. 11 in: <em>Domain of Dependence</em>, J. Math. Phys. <strong>11</strong> (1970) 437–449 [<a href="https://doi.org/10.1063/1.1665157">doi:10.1063/1.1665157</a>]</li> </ul> <p>The refinement of this statement to a smooth splitting:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Antonio+N.+Bernal">Antonio N. Bernal</a>, <a class="existingWikiWord" href="/nlab/show/Miguel+S%C3%A1nchez">Miguel Sánchez</a>, <em>On smooth Cauchy hypersurfaces and Geroch’s splitting theorem</em>, Commun. Math. Phys. <strong>243</strong> (2003) 461-470 [<a href="http://arxiv.org/abs/gr-qc/0306108">arXiv:gr-qc/0306108v2</a>, <a href="https://doi.org/10.1007/s00220-003-0982-6">doi:10.1007/s00220-003-0982-6</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Antonio+N.+Bernal">Antonio N. Bernal</a>, <a class="existingWikiWord" href="/nlab/show/Miguel+S%C3%A1nchez">Miguel Sánchez</a>, <em>Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes</em>, Commun. Math. Phys. <strong>257</strong> (2005) 43-50 [<a href="https://arxiv.org/abs/gr-qc/0401112">arXiv:gr-qc/0401112</a>, <a href="https://doi.org/10.1007/s00220-005-1346-1">doi:10.1007/s00220-005-1346-1</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Antonio+N.+Bernal">Antonio N. Bernal</a>, <a class="existingWikiWord" href="/nlab/show/Miguel+S%C3%A1nchez">Miguel Sánchez</a>, <em>Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions</em>, Lett. Math. Phys. <strong>77</strong> (2006) 183-197 [<a href="https://arxiv.org/abs/gr-qc/0512095">arXiv:gr-qc/0512095</a>, <a href="https://doi.org/10.1007/s11005-006-0091-5">doi:10.1007/s11005-006-0091-5</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 21, 2023 at 13:31:53. 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