General Relativistic Polytropic Fluid Spheres.

This paper discusses solutions of the field equations of general relativity for a compressible fluid sphere in gravitational equilibrium under the assumption that the fluid obeys a polytropic equation of state. With suitable transformations the equations of static equilibrium are equivalent to two coupled first- order non-linear differential equations analogous to the Lane-Emden equation in the Newtonian theory of polytropes. Solutions of the equilibrium equations are obtained in terms of the polytropic index n and a parameter whose physical interpretation is the ratio of pressure to energy density at the center of the sphere. The quantity (Tmeasures the deviation from Newtonian gravitational theory. A solution is obtained in closed form for n = 0; this corresponds to the Schwarzschild interior solution for a fluid sphere of uniform density. Solutions for n = 1.0(0.5)3.0 are obtained by numerical integration. The ratio of total mass to invariant radius of a polytropic sphere is found in terms of boundary values of the relativistic Lane-Emden functions. Integrals for the gravitational potential energy and rest energy are obtained and evaluated numerically. Properties of the solutions are tabulated for each n and a range of values of . The distributions of density, pressure, mass, and metric tensor components are shown graphically for some typical cases Plots of the mass-radius relation are given in a form suitable for determination of the internal structure of a polytrope of given mass, radius, and polytropic index. The existence of multiple solutions for some values of mass and radius is a general-relativistic feature. The maximum ratio of half the gravitational radius to the geometrical radius is 0.214 for n = 1.0 and 0.0631 for n = 3.0. These values are smaller than the limiting ratio 0.340 previously known for the Schwarzschild interior solution (n = 0.0). It appears that models with n = 3.0 and ff> 0 5 are energetically unstable. The gravitational collapse of massive ( 10'M0) starlike objects which may exist near the centers of galaxies is discussed as a possible general-relativistic mechanism for producing the large amounts of energy ( 10" ergs) associated with strong radio sources.

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