Topics in Black-Hole Physics: Geometric Constraints on Noncollapsing, Gravitating Systems and Tidal Distortions of a Schwarzschild Black Hole

Citation

Redmount, Ian H. (1984) Topics in Black-Hole Physics: Geometric Constraints on Noncollapsing, Gravitating Systems and Tidal Distortions of a Schwarzschild Black Hole. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/mp3n-4r06. https://resolver.caltech.edu/CaltechTHESIS:08232017-105535039

Abstract

This dissertation consists of two studies on the general-relativistic theory of black holes. The first work concerns the fundamental issue of black-hole forma­tion: in it I seek geometric constraints on gravitating matter systems, in the special case of axial symmetry, which determine whether or not those systems undergo gravitational collapse to form black holes. The second project deals with mechanical behavior of a black hole: specifically, I study the tidal deforma­tion of a static black hole by the gravitational fields of external bodies.

In the first paper I approach the problem of geometric constraints deter­mining gravitational collapse or non-collapse through the initial-value formalism of general relativity. I construct initial-value data representing noncollapsing, nonsingular, axisymmetric matter systems and examine the constraints imposed on this construction by the initial-value equation derived from the Ein­stein field equations. The construction consists of a nonsingular, momentarily static interior geometry with nonnegative mass-energy density, matched smoothly to a static, vacuum exterior geometry (described by a Weyl solution of the Einstein field equations) at a boundary surface. The initial-value equation is found to impose restrictions on the choice of the boundary surface for such a system. Two such constraints are obtained here, appropriate to spherical and toroidal interior-region topologies. These constraints are studied by applying them to simple examples of Weyl exterior geometries. The "hoop conjecture" for the general geometric-constraints problem states that a system must collapse to a black hole unless its circumference in some direction exceeds a lower bound of the order of the system's mass. The examples examined here show, however, that the constraints derived in this study are not generally correlated with any simple measure of system size, and thus that they do not embody the hoop conjecture.

The second paper examines the tidal distortion of a Schwarzschild black hole by bodies ("moons") suspended above the horizon on "ropes." A solution of the Einstein field equations is constructed describing this configuration, using the Weyl formalism for axisymmetric, static, vacuum geometries. The intrinsic geometry of the tidally deformed black-hole horizon is obtained from this solu­tion; I construct embedding diagrams to represent the shape of the horizon and the tidal bulges raised on it for both weak and strong perturbations. The rela­tions among the masses of the hole and moons, the binding energy of the sys­tem, and the rope density and tension are calculated from the solution and shown to be mutually consistent. Also, the Riemann curvature tensor represent­ing the tidal fields near the horizon is calculated. This solution is found to agree with a previous calculation by Hartle of black-hole tides, in the limit of perturb­ing moons far from the horizon. In the opposite case of moons very near the horizon, this solution approaches the static limit of the distorted horizon in Rindler space calculated by Suen and Price. The results of this study thus sup­port the use of the Rindler approximation to Schwarzschild spacetime in calcu­lating static black-hole tides, and its extension to dynamical situations.

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