Abstract
We discuss the gravitational collapse of a spherically symmetric perfect fluid distribution of uniformly contracting stars. In a uniformly contracting star, the relative volume element (relative distance in the case of spherical symmetry) between any two neighboring fluid particles is preserved irrespective of the radial coordinate. The physical meaning is that during collapse each small volume element of the fluid distribution preserves its spatial position. This new class of gravitational collapse is analogous to the reverse phenomenon of motion of galaxies during the expansion of the Universe. We discuss the shearing solution of a perfect fluid distribution executing the uniform expansion, which is a scalar and obeys the equation of state \(p=p(\rho)\). The field equation is solved in complete generality, such that that the Oppenheimer–Snyder solution with homogeneous density and the Thompson–Whitrow shear-free solution arise as particular cases.
Similar content being viewed by others
References
G. C. McVittie, “Gravitational motions of collapse or of expansion in general relativity,” Ann. Inst. H. Poincaré Sect. A (N. S.), 6, 1–15 (1967).
A. H. Taub, “Restricted motions of gravitating spheres,” Ann. Inst. H. Poincaré Sect. A (N. S.), 9, 153–178 (1968).
M. C. Faulkes, “Non-static fluid spheres in general relativity,” Prog. Theor. Phys., 42, 1139–1142 (1969).
E. N. Glass and B. Mashhoon, “On a spherical star system with a collapsed core,” Astrophys. J., 205, 570–577 (1976).
J. R. Oppenheimer and H. Snyder, “On continued gravitational contraction,” Phys. Rev., 56, 455–459 (1939).
G. C. McVittie, “The mass-particle in an expanding universe,” Mon. Not. R. Astron. Soc., 93, 325–339 (1933).
B. Nolan, “Sources for McVittie’s mass particle in an expanding universe,” J. Math. Phys., 34, 178–185 (1933).
A. Ori and T. Piran, “Naked singularities and other features of self-similar general-relativistic gravitational collapse,” Phys. Rev. D, 42, 1068–1090 (1990).
P. S. Joshi, N. Dadhich, and R. Maartens, “Why do naked singularities form in gravitational collapse?,” Phys. Rev. D, 65, 101501, 4 pp. (2002); arXiv: gr-qc/0109051.
W. B. Bonnor and M. C. Faulkes, “Exact solutions for oscillating spheres in general relativity,” Mon. Not. R. Astron. Soc., 137, 239–251 (1967).
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge Univ. Press, Cambridge (2003).
H. Nariai, “A simple model for gravitational collapse with pressure gradient,” Prog. Theor. Phys., 38, 92–106 (1967).
V. Husain, E. A. Martinez, and D. Núñez, “Exact solution for scalar field collapse,” Phys. Rev. D, 50, 3783–3786 (1994); arXiv: gr-qc/9402021.
C. B. Collins and J. Wainwright, “Role of shear in general-relativistic cosmological and stellar models,” Phys. Rev. D, 27, 1209–1218 (1983).
E. N. Glass, “Shear-free gravitational collapse,” J. Math. Phys., 20, 1508–1513 (1979).
R. Chan, “Collapse of a radiating star with shear,” Mon. Not. R. Astron. Soc., 288, 589–595 (1997); “Erratum: Collapse of a radiating star with shear,” 299, 811–811 (1998).
L. Herrera and N. O. Santos, “Shear-free and homology conditions for self-gravitating dissipative fluids,” Mon. Not. R. Astron. Soc., 343, 1207–1212 (2003).
C. W. Misner and D. H. Sharp, “Relativistic equations for adiabatic, spherically symmetric gravitational collapse,” Phys. Rev. D, 136, B571–B576 (1964).
I. H. Thompson and G. J. Whitrow, “Time-dependent internal solutions for spherically symmetrical bodies in general relativity: I. Adiabatic collapse,” Mon. Not. R. Astron. Soc., 136, 207–217 (1967).
M. Wyman, “Equations of state for radially symmetric distributions of matter,” Phys. Rev. D, 70, 396–400 (1946).
B. Mashhoon and M. Hossein Partovi, “On the gravitational motion of a fluid obeying an equation of state,” Ann. Phys., 130, 99–138 (1980).
J. V. Narlikar, An Introduction to Cosmology, Cambridge Univ. Press, Cambridge (2002).
L. Herrera, N. O. Santos, and A. Wang, “Shearing expansion-free spherical anisotropic fluid evolution,” Phys. Rev. D, 78, 084026, 10 pp. (2008); arXiv: 0810.1083.
V. A. Skripkin, “A point discontinuity in a perfect incompressible fluid in the general theory of relativity,” Sov. Phys. Dokl., 5, 1183–1186 (1961).
Acknowledgments
The authors are thankful to the referee for the valuable suggestions and comments to improve the manuscript.
Funding
R. Kumar is thankful to an UGC-BSR Startup grant, India, for financial assistance. A. Jaiswal is thankful to the Council of Science and Technology, UP, India (vide letter no. CST/D-2289).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflicts of interest.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 211, pp. 136–146 https://doi.org/10.4213/tmf10225.
Rights and permissions
About this article
Cite this article
Kumar, R., Jaiswal, A. A new class of spherically symmetric gravitational collapse. Theor Math Phys 211, 558–566 (2022). https://doi.org/10.1134/S0040577922040092
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577922040092