Stability and Dynamics of Spherically Symmetric Masses in General Relativity

Citation

Bardeen, James Maxwell (1965) Stability and Dynamics of Spherically Symmetric Masses in General Relativity. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/HQ2N-0J27. https://resolver.caltech.edu/CaltechETD:etd-04152003-164152

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The Einstein equations for a spherically symmetric distribution of matter are recast in comoving (Lagrangian) coordinates in a form similar to that of the classical hydrodynamic equations, thus facilitating the physical interpretation of the equations. The jump conditions for a shock wave in an ideal fluid are found in these coordinates. The equation of radiative transfer in general relativity is derived, and analyzed with respect to the non-gravitational interaction of the radiation and the matter as reflected in the equations arising from the zero covariant divergence of the energy-momentum tensor. The radiative transfer equation is solved assuming the radiation is in local thermodynamic equilibrium with the matter. We present some examples of numerical calculations of the equilibrium, stability, and dynamics near equilibrium of spherically symmetric masses for a simple, although physically reasonable, type of equation of state, in which the thermal energy density is given solely in terms of the pressure and an adiabatic index [...] that is independent of density and. pressure. One numerical method follows the growth of instabilities to all orders in the fractional change in radius away from equilibrium. Our formulation of the Einstein equations is applied to the analytical study of the stability, and we prove that, regardless of the equation of state, a maximum. or minimum of the binding energy as a function of central density along a sequence of masses in hydrostatic and convective equilibrium with constant number of baryons and constant rest mass per baryon implies some mode of radial oscillation has zero frequency there. As a result of this theorem, the stability properties of models in convective equilibrium can be read off a plot of fractional binding energy against the ratio of gravitational radius to radius. One example is presented of a numerical difference equation calculation of the collapse of a large mass inside the gravitational radius.

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