Abstract
The Schwarzschild solution has played a fundamental conceptual role in general relativity, and beyond, for instance, regarding event horizons, spacetime singularities and aspects of quantum field theory in curved spacetimes. However, one still encounters the existence of misconceptions and a certain ambiguity inherent in the Schwarzschild solution in the literature. By taking into account the point of view of an observer in the interior of the event horizon, one verifies that new conceptual difficulties arise. In this work, besides providing a very brief pedagogical review, we further analyze the interior Schwarzschild black hole solution. Firstly, by deducing the interior metric by considering time-dependent metric coefficients, the interior region is analyzed without the prejudices inherited from the exterior geometry. We also pay close attention to several respective cosmological interpretations, and briefly address some of the difficulties associated to spacetime singularities. Secondly, we deduce the conserved quantities of null and timelike geodesics, and discuss several particular cases in some detail. Thirdly, we examine the Eddington–Finkelstein and Kruskal coordinates directly from the interior solution. In concluding, it is important to emphasize that the interior structure of realistic black holes has not been satisfactorily determined, and is still open to considerable debate.
Similar content being viewed by others
References
Oppenheimer, J.R., Snyder, H.: On continued gravitational contraction. Phys. Rev. 56, 455 (1939)
Schwarzschild, K.: Uber das Gravitationsfeld einer Kugel aus inkompressibler Flussigkeit nach der Einsteinschen Theorie, Sitz.ber. Dtsch. Akad. Wiss. Berl. Kl. Math. Phys. Tech. 424–434 (1916)
Tolman, R.C.: Static solutions of Einstein’s field equations for spheres of fluid. Phys. Rev. 55, 364 (1939)
Oppenheimer, J.R., Volkoff, G.: On massive neutron cores. Phys. Rev. 55, 374 (1939)
Thorne, K.S., Price, R.H., Macdonald, D.A. (eds.): Black Holes: The Membrane Paradigm. Yale University Press, New Haven (1986)
Zel’dovich, Y.B., Novikov, I.D.: Relativistic Astrophysics. University of Chicago Press, Chicago (1971)
Penrose, R.: Gravitational collapse: the role of general relativity. Riv. Nuovo Soc. Ital. Fis. 1, 252 (1969)
Narlikar, J.V., Padmanabhan, T.: The Schwarzschild solution: some conceptual difficulties. Found. Phys. 18, 659–668 (1988)
Sabbata, V., Pavsic, M., Recami, E.: Black holes and tachyons. Lett. Nuovo Cimento 19, 441–451 (1977)
Goldoni, R.: Black and white holes. Gen. Relativ. Gravit. 1, 103–113 (1975)
Janis, A.I.: Note on motion in the Schwarschild field. Phys. Rev. D 8, 2360 (1973)
Janis, A.I.: Motion in the Schwarschild field: A reply. Phys. Rev. D 15, 3068 (1977)
Crawford, P., Tereno, I.: Generalized observers and velocity measurements in general relativity. Gen. Relativ. Gravit. 34, 2075 (2002). arXiv:gr-qc/0111073
Bolos, V.J.: Lightlike simultaneity, comoving observers and distances in general relativity. J. Geom. Phys. 56, 813–829 (2006). arXiv:gr-qc/0501085
Bolos, V.J.: Intrinsic definitions of ‘relative velocity’ in general relativity. arXiv:gr-qc/0506032 (2005)
Earman, J.: Tolerance for spacetime singularities. Found. Phys. 26, 623–640 (1996)
Misner, C.: Absolute zero of time. Phys. Rev. 186, 1328 (1969)
Mazur, P.O., Mottola, E.: Gravitational condensate stars: an alternative to black holes. arXiv:gr-qc/0109035 (2001)
Mazur, P.O., Mottola, E.: Gravitational vacuum condensate stars. Proc. Natl. Acad. Sci. U.S.A. 111, 9545 (2004). arXiv:gr-qc/0407075
Chapline, G., Hohlfeld, E., Laughlin, R.B., Santiago, D.I.: Quantum phase transitions and the breakdown of classical general relativity. Int. J. Mod. Phys. A 18, 3587–3590 (2003). arXiv:gr-qc/0012094
Visser, M., Wiltshire, D.L.: Stable gravastars—an alternative to black holes? Class. Quantum Gravity 21, 1135 (2004). arXiv:gr-qc/0310107
Cattoen, C., Faber, T., Visser, M.: Gravastars must have anisotropic pressures. Class. Quantum Gravity 22, 4189–4202 (2005). arXiv:gr-qc/0505137
Bilić, N., Tupper, G.B., Viollier, R.D.: Born–Infeld phantom gravastars. JCAP 02(2006)(013) (2006). arXiv:astro-ph/0503427
Dymnikova, I.: Vacuum nonsingular black hole. Gen. Relativ. Gravit. 24, 235 (1992)
Dymnikova, I.: Spherically symmetric space-time with the regular de Sitter center. Int. J. Mod. Phys. D 12, 1015–1034 (2003). arXiv:gr-qc/0304110
Dymnikova, I., Galaktionov, E.: Stability of a vacuum nonsingular black hole. Class. Quantum Gravity 22, 2331–2358 (2005). arXiv:gr-qc/0409049
Chapline, G.: Dark energy stars. arXiv:astro-ph/0503200
Lobo, F.S.N.: Stable dark energy stars. Class. Quantum Gravity 23, 1525 (2006). arXiv:astro-ph/0508115
Abramowicz, M.A., Kluzniak, W., Lasota, J.P.: No observational proof of the black-hole event-horizon. Astron. Astrophys. 396, L31 (2002). arXiv:astro-ph/0207270
Kantowski, R., Sachs, R.K.: Some spatially homogeneous anisotropic relativistic cosmological models. J. Math. Phys. 7, 443–445 (1966)
Collins, C.B.: Global structure of the ‘Kantowski–Sachs’ cosmological models. J. Math. Phys. 18, 2116–2124 (1977)
Vollick, D.N.: Anisotropic Born–Infeld cosmologies. Gen. Relativ. Gravit. 35, 1511 (2003). arXiv:hep-th/0102187
Vollick, D.N.: Black hole and cosmological space-times in Born–Infeld–Einstein theory. arXiv:gr-qc/0601136 (2006)
Doran, R., Lobo, F.S.N., Crawford, P.: Cosmological applications of the interior Schwarzschild solution (in preparation)
Easson, D.A., Brandenberger, R.H.: Universe generation from black hole interiors. J. High Energy Phys. 0106, 024 (2001). arXiv:hep-th/0103019
Brehme, R.W.: Inside the black hole. Am. J. Phys. 45, 423 (1977)
Karlhede, A., Lindström, U., Aman, J.E.: A note on a local effect at the Schwarzschild sphere. Gen. Relativ. Gravit. 14, 569 (1982)
Landau, L., Lifschitz, E.: The Classical Theory of Fields, 4th edn. Pergamon, New York (1975)
Kiselev, V.V.: Radial geodesics as a microscopic origin of black hole entropy. I. Confined under the Schwarzschild horizon. Phys. Rev. D 72, 124011 (2005). arXiv:gr-qc/0412090
Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Doran, R., Lobo, F.S.N. & Crawford, P. Interior of a Schwarzschild Black Hole Revisited. Found Phys 38, 160–187 (2008). https://doi.org/10.1007/s10701-007-9197-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-007-9197-6