On Maximal Extensions of the Vaidya Metric

Abstract

The common ``Eddington-Finkelstein-like" coordinates are not complete, thus not covering the entirety of the Vaidya manifold. The main aim of this thesis is to create and analyze three maximal extensions of the Vaidya metric, which is a spherically symmetric solution to the Einstein field equations for the energy momentum tensor of pure radiation in the high-frequency approximation. This metric is necessary for various applications, such as describing the exterior geometry of a radiating star in astrophysics and studying possible formation of naked singularities in the geometry of spacetime. We look into the Israel metric, which was an important step in finding maximal extensions of the Vaidya manifold, though it was given via coordinate transformations. We obtain the Israel coordinates in a constructive manner without recourse to coordinate transformations. By recognizing that each maximal extension can be differentiated by the mass function, we develop three mass functions, one for each extension. We then inspect the qualitative characteristics of the three mass models and the surfaces of constant radius. The first extension is formed by starting with a Schwarzschild vacuum solution, which then has outflux' of radiation for a period of time before the outflux' is turned off to get another (with a less mass) Schwarzschild vacuum solution. The second extension is accomplished by adding `influx' of radiation to an already existing Schwarzschild vacuum region, and the final spacetime is essentially Schwarzschild vacuum (the mass now being larger than the initial Schwarzschild). The third extension is constructed by choosing a mass function that creates an exploding Vaidya metric that eventually changes to an imploding Vaidya metric. The most noteworthy element of these maximal extensions is that they are null geodesically complete, which we assess by solving the radial null geodesics equation and forming the Penrose conformal diagram for each extension.