Ideal stars and General Relativity |
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Authors: | Christian Frønsdal |
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Institution: | (1) Physics Department, University of California, Los Angeles, CA 90095-1547, USA |
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Abstract: | We study a system of differential equations that governs the distribution of matter in the theory of General Relativity. The
new element in this paper is the use of a dynamical action principle that includes all the degrees of freedom, matter as well
as metric. The matter lagrangian defines a relativistic version of non-viscous, isentropic hydrodynamics. The matter fields
are a scalar density and a velocity potential; the conventional, four-vector velocity field is replaced by the gradient of
the potential and its scale is fixed by one of the Eulerian equations of motion, an innovation that significantly affects
the imposition of boundary conditions. If the density is integrable at infinity, then the metric approaches the Schwarzschild
metric at large distances. There are stars without boundary and with finite total mass; the metric shows rapid variation in
the neighbourhood of the Schwarzschild radius and there is a very small core where a singularity indicates that the gas laws
break down. For stars with boundary there emerges a new, critical relation between the radius and the gravitational mass,
a consequence of the stronger boundary conditions. Tentative applications are suggested, to certain Red Giants, and to neutron
stars, but the investigation reported here was limited to homogeneous polytropes. Comparison with the results of Oppenheimer
and Volkoff on neutron cores shows a close agreement of numerical results. However, in the model the boundary of the star
is fixed uniquely by the required matching of the interior metric to the external Schwarzschild metric, which is not the case
in the traditional approach. There are solutions for which the metric is very close to the Schwarzshild metric everywhere
outside the horizon, where the source is concentrated. The Schwarzschild metric is interpreted as the metric of an ideal,
limiting configuration of matter, not as the metric of empty space. |
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