Static spherical perfect fluid stars with finite radius in general relativity: a review




relativistic stars, Tolman-Oppenheimer-Volkoff equation


In this article, we provide a pedagogical review of the Tolman-Oppenheimer-Volkoff (TOV) equation and its solutions which describe static, spherically symmetric gaseous stars in general relativity. Our discussion starts with a systematic derivation of the TOV equation from the Einstein field equations and the relativistic Euler equations. Next, we give a proof for the existence and uniqueness of solutions of the TOV equation describing a star of finite radius, assuming suitable conditions on the equation of state characterizing the gas. We also prove that the compactness of the gas contained inside a sphere centered at the origin satisfies the well-known Buchdahl bound, independent of the radius of the sphere. Further, we derive the equation of state for an ideal, classical monoatomic relativistic gas from statistical mechanics considerations and show that it satisfies our assumptions for the existence of a unique solution describing a finite radius star. Although none of the results discussed in this article are new, they are usually scattered in different articles and books in the literature; hence it is our hope that this article will provide a self-contained and useful introduction to the topic of relativistic stellar models.

Author Biographies

Emmanuel Chávez Nambo, Universidad Michoacana de San Nicolás de Hidalgo

Instituto de Física y Matemáticas

Estudiante de maestría

Olivier Sarbach, Universidad Michoacana de San Nicolás de Hidalgo

Profesor e Investigador Titular "C" de Tiempo Completo Instituto de Física y Matemáticas


Digital library of mathematical functions.

L. Andersson and A.Y. Burtscher. On the asymptotic behavior of static perfect fluids. Annales Henri Poincar ́e, 20(3): 813–857, 2019. doi: 10.1007/s00023-018-00758-z.

H. Andréasson. Sharp bounds on 2m/r of general spherically symmetric static objects. J. Diff. Eq., 245:2243–2266, 2008. doi: 10.1016/j.jde.2008.05.010.

T.M. Apostol. Calculus. John Wiley & Sons, Inc, New York, 1967.

H.A. Buchdahl. General Relativistic Fluid Spheres. Phys. Rev., 116:1027, 1959. doi: 10.1103/PhysRev.116.1027.

S. Chandrasekhar. An Introduction to the Study of Stellar Structure. Dover Publications Inc., New York, 1958.

E. Chaverra, N. Ortiz, and O. Sarbach. Linear perturbations of self-gravitating spherically symmetric configurations. Phys. Rev. D, 87(4):044015, 2013. doi: 10.1103/PhysRevD.87.044015.

F. Jüttner. Maxwell’s law of speed distribution in the theory of relativity. Annal. Phys., 34:856–882, 1911.

N.K. Glendenning. Compact stars: Nuclear physics, particle physics, and general relativity. Springer-Verlag, 1997.

F.S. Guzmán, F.D. Lora-Clavijo, and M.D. Morales. Revisiting spherically symmetric relativistic hydrodynamics. Rev. Mex. Fis. E, 58(2):84–98, 2012.

J.M. Heinzle. (In)finiteness of spherically symmetric static perfect fluids. Class. Quantum Grav., 19:2835–2852, 2002. doi: 10.1088/0264-9381/19/11/307.

J.M. Heinzle, N. Röhr, and C. Uggla. Dynamical systems approach to relativistic spherically symmetric static perfect fluid models. Classical and Quantum Gravity, 20(21):4567–4586, 2003. doi: 10.1088/0264-9381/20/21/004.

K. Huang. Statistical Mechanics. John Wiley and Sons, Inc., New York, 1987.

S.L. Liebling and C. Palenzuela. Dynamical boson stars. Living Rev. Rel., 20:5, 2017. doi:


T. Makino. On spherically symmetric stellar models in general relativity. J. Math. Kyoto Univ., 38:55–69, 1998. doi:


J.R. Oppenheimer and G.M. Volkoff. On Massive neutron cores. Phys. Rev., 55:374–381, 1939. doi: 10.1103/PhysRev.55. 374.

T. Ramming and G. Rein. Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the nonrelativistic and relativistic case—a simple proof for finite extension. SIAM Journal on Mathematical Analysis, 45(2):900–914, 2013. doi: 10.1137/120896712.

M. Reed and B. Simon. Methods of Modern Mathematical Physics, Vol. I. Academic Press, San Diego, 1980.

A.D. Rendall and B.G. Schmidt. Existence and properties of spherically symmetric static fluid bodies with a given equation of state. Classical and Quantum Gravity, 8(5):985–1000, 1991. doi: 10.1088/0264-9381/8/5/022.

O. Sarbach and M. Tiglio. Continuum and discrete initial-boundary-value problems and Einstein’s field equations. Living Rev. Rel., 15:9, 2012. doi: 10.12942/lrr-2012-9.

U.M. Schaudt. On static stars in Newtonian gravity and Lane-Emden type equations. Ann. Henri Poincar ́e, 1:945–976, 2000.

S.L. Shapiro and S.A. Teukolsky. Black holes, white dwarfs, and neutron stars : the physics of compact objects. John Wiley and Sons, Inc., 1983.

W. Simon. Criteria for (in)finite extent of static perfect fluids. Lect. Notes Phys., 604:223–238, 2002.

K.S. Thorne and R.D. Blandford. Modern Classical Physics. Princeton University Press, Princeton, 2017.

R.C. Tolman. Static solutions of Einstein’s field equations for spheres of fluid. Phys. Rev., 55:364–373, 1939. doi:


R.M. Wald. General Relativity. The University of Chicago Press., Chicago, 1984.




How to Cite

E. Chávez Nambo and O. Sarbach, “Static spherical perfect fluid stars with finite radius in general relativity: a review”, Rev. Mex. Fis. E, vol. 18, no. 2 Jul-Dec, pp. 020208 1–20, Jul. 2021.