Abstract
In their classic 1939 paper Oppenheimer and Snyder introduced the first mathematical model for gravitational collapse of stars based on spherically symmetric solutions of the Einstein gravitational field equations. In this exact solution of the Einstein equations, the boundary surface of a massive fluid sphere falls continuously into a black hole and the dynamics is described by exact formulas. This provided the first solid evidence for the idea that black holes could form from gravitational collapse in massive stars. The Oppenheimer—Snyder paper also provided the first example in which a solution of the Einstein equations having interesting dynamics was constructed by using the covariance of the equations to match two simpler solutions across an interface. The Oppenheimer—Snyder model requires the simplifying assumption that the pressure be identically zero. In this article we construct shock wave generalizations of the Oppenheimer—Snyder model that apply to the case when the pressure is nonzero. A general characterization of shock wave interfaces in solutions of the Einstein equations is presented in Section 2, and examples are derived and discussed in detail in the later sections.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S.K. Blau and A.H. Guth, Inflationary cosmology. In: Three Hundred Years of Gravitation ed. by S.W. Hawking and W. Israel, Cambridge University Press, 1987, pp. 524–603.
(Private Communication).
R. Courant and K. Friedrichs Supersonic Flow and Shock—Waves Wiley-Interscience,20 1948. 1972.
A. Einstein, Die Feldgleichungen der Gravitation Preuss. Akad. Wiss. Berlin, Sitzber. 1915b, pp. 844–847.
S.W. Hawking and G.F.R. Ellis The Large Scale Structure of Spacetime Cambridge University Press, 1973.
S.W. Hawking and R. Penrose, The singularities of gravitational collapse and cosmology Proc. Roy. Soc. Lond. A, 314(1970), pp. 529–548.
W. Israel, Singular hypersurfaces and thin shells in general relativity Il Nuovo Cimento Vol. XLIV B, N. 1, 1966, pp. 1–14.
W. Israel, Dark Stars: The Evolution of an Idea, in: 300 years of Gravitation edited by S. W. Hawking and W. Israel, Cambridge University Press, 1987, pp. 199–276.
P.D. Lax, Hyperbolic systems of conservation laws, II Comm. Pure Appl. Math. 10(1957), pp. 537–566.
P.D. Lax, Shock waves and entropy. In: Contributions to Nonlinear Functional Analysis ed. by E. Zarantonello, Academic Press, 1971, pp. 603–634.
M.S. Longair Our Evolving Universe Cambridge University Press, 1996.
G.C. McVittie, Gravitational collapse to a small volume Astro. Phys. Jour. 140(1964), pp. 401–416.
C. Misner and D. Sharp, Relativistic equations for adiabatic, spherically symmetric gravitational collapse Phys. Rev. 26(1964), pp. 571–576.
C. Misner, K. Thorne, and J. Wheeler Gravitation Freeman, 1973.
J.R. Oppenheimer and J.R. Snyder, On continued gravitational contraction Phys. Rev. 56 (1939), pp. 455–459.
J.R. Oppenheimer and G.M. Volkoff, On massive neutron cores Phys. Rev. 55(1939), pp. 374–381.
P.J.E. Peebles Principles of Physical Cosmology Princeton University Press, 1993.
R. Schoen and S.T. Yau, Proof of the positive mass theorem II Commun. Math. Phys. 79 1981, pp. 231–260.
J. Smoller Shock Waves and Reaction-Diffusion Equations Springer Verlag, 1983.
J. Smoller and B. Temple, Global solutions of the relativistic Euler equations Commun. Math. Phys. 157(1993), pp. 67–99.
J. Smoller and B. Temple, Shock-wave solutions of the Einstein equations: the Oppenheimer—Snyder model of gravitational collapse extended to the case of nonzero pressure Arch. Rat. Mech. Anal. 128 (1994), pp. 249–297.
J. Smoller and B. Temple, Astrophysical shock wave solutions of the Einstein equations Phys. Rev. D 51 6 (1995). 101–200
J. Smoller and B. Temple, General relativistic shock waves that extend the Oppenheimer—Snyder model Arch. Rat. Mech. Anal. 138(1997), pp. 239–277
J. Smoller and B. Temple, Shock-waves near the Schwarzschild radius and the stability limit for stars Phys. Rev. D 55(1997), pp. 7518–7528.
J. Smoller and B. Temple, Shock-wave solutions in closed form and the Oppenheimer—Snyder limit in general relativity Siam J. Appl. Math 58, No. 1, pp. 15–33, February, 1998.
J. Smoller and B. Temple, On the Oppenheimer—Volkov equations in general relativity Arch. Rat. Mech. Anal. 142 (1998), pp. 177–191.
J. Smoller and B. Temple, Solutions of the Oppenheimer—Volkoff equations inside 9/8’ths of the Schwarzschild radius Commun. Math. Phys. 184 (1997), pp. 597–617.
J. Smoller and B. Temple Cosmology with a shock wave Comm. Math. Phys. 210 (2000), pp. 275–308.
J. Groah and B. Temple, A shock wave formulation of the Einstein equations, (in preparation).
R. Tolman, Static Solutions of Einstein’s Field Equations for Spheres of Fluid Phys. Rev. 55(1939), pp. 364–374.
R. Tolman Relativity Thermodynamics and Cosmology Oxford University Press, 1934.
R.M. Wald General Relativity University of Chicago Press, 1984.
S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Smoller, J., Temple, B. (2001). Shock Wave Solutions of the Einstein Equations: A General Theory with Examples. In: Freistühler, H., Szepessy, A. (eds) Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations and Their Applications, vol 47. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0193-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0193-9_3
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6655-6
Online ISBN: 978-1-4612-0193-9
eBook Packages: Springer Book Archive