Physics
Scientific paper
Jan 1999
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999phdt........47b&link_type=abstract
Thesis (PhD). STANFORD UNIVERSITY, Source DAI-B 61/01, p. 310, Jul 2000, 173 pages.
Physics
Scientific paper
A new framework is presented for analysing the spherically symmetric Einstein field equations for a zero-mass scalar field. The framework consists of a coordinate system (p, q), where the coordinate p is the scalar field, and q is a coordinate chosen to be orthogonal to p. This idea allows for a reduction of the field equations into a system of two first order partial differential equations for the areal metric function gqq and a mass function m . The metric coefficients in this coordinate system then take on values which are simply related to the scalars of the problem: 1->f˙1 ->f,gq q and-via the field equations-the scalar curvature R as well. The scalar field coordinate system is shown to have many advantages. Many of the known exact solutions (e.g. static, Roberts) are represented simply, and new self- similar solutions are derived. The framework is then applied to the problem of matching spherically symmetric scalar-tensor vacuum solutions to a homogeneous and isotropic dust solution (e.g. scalar- tensor Einstein-Straus swiss cheese solutions, scalar- tensor Oppenheimer-Snyder dust ball collapse). Scalar field coordinates are shown to be ideal for such an application. We derive the necessary matching conditions in scalar field coordinates, and show how they imply a natural extension of the Schücking condition for spherically symmetric vacuum in general relativity. The problem of finding a vacuum solution which matches a given homogeneous and isotropic solution is examined. It is found that the matching conditions are sufficient to guarantee local existence and uniqueness of the vacuum solution if it is assumed that the scalar field has neither maxima nor minima on the matching interface. In order to find explicit matched solutions, criteria are developed to screen known exact vacuum solutions for matchability, and procedures are given for determining the details of the homogeneous and isotropic solution (curvature constant, comoving radial coordinate of the matching interface, and scalar-tensor coupling function) to which they can be joined. The screening procedure is independent of the particular spherically symmetric coordinate system in which the candidate solution is expressed. Scalar field coordinates are also used to examine succinctly the problem of perturbative expansion of scalar-tensor vacuum solutions about the vacuum solution of general relativity. All solutions are shown to fall into three categories: perturbation about the Schwarzschild solution, perturbation about a two dimensional metric, and non-perturbative solutions. In the process of classification, the Birkhoff theorem for vacuum solutions of general relativity is explicitly shown. In the final section, future directions are discussed. They are: exploiting Lie-theoretic methods to find new solution families which may lead to an explicit matched solution, searching for the Choptuik critical solution, and application of scalar field coordinates to relativistic perfect fluids (T = 0) in order to find exact solutions. Finally, a re-examination of classical gravitational collapse of a scalar field is advocated using matched solutions.
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