Shock-wave cosmology inside a black hole Joel Smoller† and Blake Temple‡§ †Department of Mathematics, University of Michigan, Ann Arbor, MI 48109; and ‡Department of Mathematics, University of California, Davis, CA 95616 Communicated by S.-T. Yau, Harvard University, Cambridge, MA, June 23, 2003 (received for review October 17, 2002) We construct a class of global exact solutions of the Einstein the total mass M(r៮, t) inside radius r៮ in the FRW metric at fixed equations that extend the Oppeheimer–Snyder model to the case time t increases like r៮,¶ and so at each fixed time t we must have of nonzero pressure, inside the black hole, by incorporating a shock r៮ Ͼ 2M(r៮, t) for small enough r៮, while the reverse inequality holds wave at the leading edge of the expansion of the galaxies, for large r៮.Weletr៮ ϭ r៮crit denote the smallest radius at which arbitrarily far beyond the Hubble length in the Friedmann– r៮crit ϭ 2M(r៮crit, t).] We show that when k ϭ 0, r៮crit equals the Robertson–Walker (FRW) spacetime. Here the expanding FRW Hubble length. Thus, we cannot match a critically expanding universe emerges be-hind a subluminous blast wave that explodes FRW metric to a classical TOV metric beyond one Hubble outward from the FRW center at the instant of the big bang. The length without continuing the TOV solution into a black hole, total mass behind the shock decreases as the shock wave expands, and we showed in ref. 4 that the standard TOV metric cannot be and the entropy condition implies that the shock wave must continued into a black hole. Thus to do shock matching with a weaken to the point where it settles down to an Oppenheimer– k ϭ 0 FRW metric beyond one Hubble length, we introduce the Snyder interface, (bounding a finite total mass), that eventually TOV metric inside the black hole. emerges from the white hole event horizon of an ambient Schwarzschild spacetime. The entropy condition breaks the time 2. The TOV Metric Inside the Black Hole symmetry of the Einstein equations, selecting the explosion over the implosion. These shock-wave solutions indicate a cosmological When the metric anzatz is taken to be of the TOV form model in which the big bang arises from a localized explo- sion occurring inside the black hole of an asymptotically flat ds2 ϭ ϪB͑r៮͒dt៮ 2 ϩ AϪ1͑r៮͒dr៮ 2 ϩ r៮ 2d⍀2, [2.1] Schwarzschild spacetime. and the stress tensor T is taken to be that of a perfect fluid e describe a cosmological model based on matching a comoving with the metric, the Einstein equations G ϭ ␬T, inside Wcritically expanding Friedmann–Robertson–Walker the black hole, take the form (FRW) metric to a metric that we call the Tolman– Oppenheimer–Volkoff (TOV) metric inside the black hole p៮ ϩ ¯␳ NЈ ៮ Ј ϭ across a shock wave that lies beyond one Hubble length from the p Ϫ , [2.2] center of the FRW spacetime. This implies that the spacetime 2 N 1 beyond the shock wave must lie inside a black hole, thus extending the shock matching limit of one Hubble length that we N NЈ ϭ Ϫͭ ϩ ␬p៮r៮ͮ, [2.3] identified in ref. 1. r៮ In the exact solutions constructed in this article, the expanding FRW universe emerges behind a subluminous blast wave that BЈ 1 N explodes outward from the origin r៮ ϭ 0 at the instant of the big ϭ Ϫ ͭ ϩ ␬␳៮ͮ. [2.4] B N Ϫ 1 r៮ bang t ϭ 0, at a distance beyond one Hubble length.¶ The shock wave then continues to weaken as it expands outward until the ␳៮ ៮ ៮ Hubble length eventually catches up to the shock, and this marks We let , p denote the density and pressure, respectively, and r the event horizon of a black hole in the TOV metric beyond the is taken to be the timelike variable because we assume shock. From this time onward, the shock wave is approximated by a zero pressure (k ϭ 0) Oppenheimer–Snyder (OS) interface, 2M͑r៮͒ A͑r៮͒ ϭ 1 Ϫ ϵ 1 Ϫ N͑r៮͒ Ͻ 0. [2.5] and thus the OS solution gives the large time asymptotics of these r៮ solutions. Surprisingly, the equation of state p ϭ 1␳ of early big 3 bang physics is distinguished by the differential equations, and Here, M(r៮) has the interpretation as the total mass inside the ball only for this equation of state does the shock wave emerge from of radius r៮ when r៮ Ͼ 2M, but M does not have the same the big bang at a finite nonzero speed, the speed of light. This ࿣ 1 interpretation inside the black hole because r៮ Ͻ 2M. The system is surprising because the equation of state p ϭ ␳ played no 3 2.2–2.4 defines the simplest class of gravitational metrics that special role in shock matching outside the black hole (2). We find it interesting that such a shock wave emerging from the big bang contain matter and evolve inside the black hole. beyond the Hubble length would account for the thermalization of radiation in a region well beyond the light cone of an observer Abbreviations: FRW, Friedmann–Robertson–Walker; TOV, Tolman–Oppenheimer–Volkoff; positioned at the FRW center at present time, even though the OS, Oppenheimer–Snyder. FRW expansion is finite, and the model does not invoke §To whom correspondence should be addressed. E-mail: [email protected]. inflation. Details will appear in our forthcoming article (3); here, ¶We let (t, r) denote standard FRW coordinates, so that r៮ ϭ rR(t) measures arclength we summarize this work and describe its physical interpretation. distance at each fixed value of the FRW time t, where R denotes the cosmological scale factor. Barred coordinates also refer to TOV standard coordinates, in which case r៮ ϭ rR(t) 1. Statement of the Problem also holds as a consequence of shock matching. If there is a shock wave at the leading edge of the expansion of ࿣System 2.2–2.4 for A Ͻ 0 differs substantially from the TOV equations for A Ͼ 0 because, ៮ for example, the energy density T00 is equated with the timelike component Grr when A Ͻ the galaxies, then we can ask what is the critical radius rcrit at each 0, but with Gtt when A Ͼ 0. In particular, this implies that MЈϭϪ4␲␳៮r៮2 when A Ͻ 0, versus fixed time t in a k ϭ 0 FRW metric such that the total mass inside MЈϭ4␲␳៮r៮2 when A Ͼ 0, the latter being the equation that gives the mass function its a shock wave positioned beyond that radius puts the universe physical interpretation outside the black hole. inside a black hole? [There must be such a critical radius because © 2003 by The National Academy of Sciences of the USA 11216–11218 ͉ PNAS ͉ September 30, 2003 ͉ vol. 100 ͉ no. 20 www.pnas.org͞cgi͞doi͞10.1073͞pnas.1833875100 Downloaded by guest on September 25, 2021 3. Shock Matching Inside the Black Hole Eq. 3.1 uncouples from 3.2, and thus solutions of system 3.1–3.3 Matching a given k ϭ 0 FRW metric to a TOV metric inside the are determined by the scalar nonautonomous Eq. 3.1. Making ϭ ͞ black hole across a shock interface, leads to the system of the change of variable S 1 N, which transforms the big bang 3 ϱ 3 ordinary differential equations N over to rest point at S 0, cf. ref. 1, we obtain, du ͑1 ϩ u͒ ͑3u Ϫ 1͒͑␴ Ϫ u͒ ϩ 6u͑1 ϩ u͒S du ͑1 ϩ u͒ ͑3u Ϫ 1͒͑␴ Ϫ u͒N ϩ 6u͑1 ϩ u͒ ϭ ͭ ͮͭ ͮ ϭ Ϫͭ ͮͭ ͮ, ͑ ϩ ͒ ͑␴ Ϫ ͒ ϩ ͑ ϩ ͒ . dN 2͑1 ϩ 3u͒N ͑␴ Ϫ u͒N ϩ ͑1 ϩ u͒ dS 2 1 3u S u 1 u S [3.1] [4.4] dr៮ 1 r៮ We take as entropy condition 3.6, and in addition, we require ϭ Ϫ , [3.2] that the TOV equation of state meet the physical bound dN 1 ϩ 3u N 0 Ͻ p៮ Ͻ ␳៮ . [4.5] with conservation constraint The conditions N Ͼ 1 and 0 Ͻ p៮ Ͻ p restrict the domain of 4.4 to Ϫ␴͑1 ϩ u͒ ϩ ͑␴ Ϫ u͒N Ͻ Ͻ ␴ Ͻ Ͻ Ͻ ␷ ϭ . [3.3] the region 0 u 1, 0 S 1. Inequalities 3.6 and 4.5 ͑1 ϩ u͒ ϩ ͑␴ Ϫ u͒N are both implied by the single condition Here 1 Ϫ u ␴ Ϫ u S Ͻ ͩ ͪͩ ͪ. [4.6] 1 ϩ u ␴ ϩ u p៮ ␳៮ p u ϭ , v ϭ , ␴ ϭ , [3.4] ␳ ␳ ␳ We prove the following theorem: ␳ ␴ ϭ ͞␳ and p denote the (known) FRW density and pressure, p , Theorem 1. For every ␴,0Ͻ ␴ Ͻ 1, there exists a unique solution and all variables are evaluated at the shock. Solutions of 3.1–3.3 u␴(S) of 4.4, such that 4.6 holds on the solution for all S,0Ͻ S Ͻ determine the (unknown) TOV metrics that match the given 1, and on this solution,0Ͻ u␴(S) Ͻ u៮, limS30 u␴(S) ϭ u៮, where FRW metric Lipschitz continuously across a shock interface, u៮ ϭ Min{1͞3, ␴}, and APPLIED such that conservation of energy and momemtum hold across the MATHEMATICS shock, and such that there are no delta function sources at the lim p៮ ϭ 0 ϭ lim p៮.