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**Cosmology, black holes and shock waves beyond the Hubble length.**
*(English)*
Zbl 1088.83025

In the standard model of cosmology based on a critically expanding Friedmann-Robertson-Walker metric, the universe is infinite at each instant after the occurrence of the big bang [S. Weinberg, Gravitation and Cosmology. J. Wiley and Sons, New York (1972); P. J. E. Peebles, Principles of Physical Cosmology. Princeton University Press, Princeton, New Jersey (1993); J. A. Peacock, Cosmological Physics. Cambridge University Press, (Cambridge, New York, Melbourne) (1999; Zbl 0952.83002); N. Straumann, General Relativity, with Applications to Astrophysics. Texts and Monographs in Physics. (Berlin, Springer). (2004; Zbl 1059.83001)]. The Hubble constant, which measures the recessional velocity of the galaxies, applies to the entire flat vector space \({\mathbb R}^3\) at each fixed positive time in the standard model. The paper under review presents a new cosmological model in which the expansion of the galaxies is a bounded expansion of finite total mass, and the Hubble law applies only to a bounded region of space-time.

In this vein, the paper under review studies exact, entropy satisfying shock wave solutions of the Einstein field equations for a perfect fluid which extend the Oppenheimer-Snyder model to the case of non-zero pressure inside the black hole [J. R. Oppenheimer, J. R. Snyder, Phys. Rev. 56, 455–459 (1939; Zbl 0022.28104); J. Smoller, B. Temple, Arch. Rat. Mech. Anal. 128, 249–297 (1994; Zbl 0824.53080); J. Smoller, B. Temple, Arch. Rat. Mech. Anal. 138, 239–277 (1997; Zbl 0888.76096); SIAM J. Appl. Math. 58, 15–33 (1998; Zbl 0945.76099)].

These shock wave solutions indicate a cosmological model in which the expanding Friedmann-Robertson-Walker universe emerges from the big bang with a shock wave at the leading edge of the expansion, analogous to a classical shock wave explosion [J. Smoller, Shock-Waves and Reaction-Diffusion. Springer-Verlag, (Berlin, Heidelberg, New York) (1983; Zbl 0508.35002)], [J. Smoller, B. Temple, “Astrophysical shock-wave solutions of the Einstein equations”, Phys. Rev. D 51, 2733–2743 (1995)]. This explosion is powerful enough to account for the enormous scale on which the galaxies and the background radiation appear uniform. In these models, the shock wave must lie beyond one Hubble length from the Friedmann-Robertson-Walker center, this threshold being the boundary across which the bounded mass lies inside its own Schwarschild radius, and in this sense the shock wave solution evolves inside the black hole [J. Smoller, B. Temple, “Shock-waves near the Schwarzschild radius and the stability limit for stars”. Phys. Rev. D 55, 7518–7528 (1997)]. The entropy condition, which breaks the time symmetry by selecting the explosion over the implosion, also implies that the shock wave must weaken until it eventually settles down to a zero pressure Oppenheimer-Snyder surface, bounding a finite total mass, that emerges from the white hole event horizon of an ambient Schwarzschild space-time. However, unlike shock matching outside a black hole, the equation of state \(p = {c^2 \over 3} \rho\), the equation of state at the earliest stage of big bang physics, is distinguished at the instant of the big bang, for this equation of state alone, the shock wave emerges from the big bang at a finite nonzero speed, the speed of light \(c\), decelerating to a subluminous wave from that time onward [J. Smoller, B. Temple, Commun. Math. Phys. 210, 275–308 (2000; Zbl 1009.83067)]. Thus the attempt to incorporate a shock wave beyond one Hubble length leads to unexpected links between big bang cosmology and black hole physics.

In this vein, the paper under review studies exact, entropy satisfying shock wave solutions of the Einstein field equations for a perfect fluid which extend the Oppenheimer-Snyder model to the case of non-zero pressure inside the black hole [J. R. Oppenheimer, J. R. Snyder, Phys. Rev. 56, 455–459 (1939; Zbl 0022.28104); J. Smoller, B. Temple, Arch. Rat. Mech. Anal. 128, 249–297 (1994; Zbl 0824.53080); J. Smoller, B. Temple, Arch. Rat. Mech. Anal. 138, 239–277 (1997; Zbl 0888.76096); SIAM J. Appl. Math. 58, 15–33 (1998; Zbl 0945.76099)].

These shock wave solutions indicate a cosmological model in which the expanding Friedmann-Robertson-Walker universe emerges from the big bang with a shock wave at the leading edge of the expansion, analogous to a classical shock wave explosion [J. Smoller, Shock-Waves and Reaction-Diffusion. Springer-Verlag, (Berlin, Heidelberg, New York) (1983; Zbl 0508.35002)], [J. Smoller, B. Temple, “Astrophysical shock-wave solutions of the Einstein equations”, Phys. Rev. D 51, 2733–2743 (1995)]. This explosion is powerful enough to account for the enormous scale on which the galaxies and the background radiation appear uniform. In these models, the shock wave must lie beyond one Hubble length from the Friedmann-Robertson-Walker center, this threshold being the boundary across which the bounded mass lies inside its own Schwarschild radius, and in this sense the shock wave solution evolves inside the black hole [J. Smoller, B. Temple, “Shock-waves near the Schwarzschild radius and the stability limit for stars”. Phys. Rev. D 55, 7518–7528 (1997)]. The entropy condition, which breaks the time symmetry by selecting the explosion over the implosion, also implies that the shock wave must weaken until it eventually settles down to a zero pressure Oppenheimer-Snyder surface, bounding a finite total mass, that emerges from the white hole event horizon of an ambient Schwarzschild space-time. However, unlike shock matching outside a black hole, the equation of state \(p = {c^2 \over 3} \rho\), the equation of state at the earliest stage of big bang physics, is distinguished at the instant of the big bang, for this equation of state alone, the shock wave emerges from the big bang at a finite nonzero speed, the speed of light \(c\), decelerating to a subluminous wave from that time onward [J. Smoller, B. Temple, Commun. Math. Phys. 210, 275–308 (2000; Zbl 1009.83067)]. Thus the attempt to incorporate a shock wave beyond one Hubble length leads to unexpected links between big bang cosmology and black hole physics.

Reviewer: Walter Schempp (Siegen)