Some Exact Solutions in General Relativity

<p><b>In this thesis four separate problems in general relativity are considered, dividedinto two separate themes: coordinate conditions and perfect fluid spheres. Regardingcoordinate conditions we present a pedagogical discussion of how the appropriateuse of coordinate conditions can lead to simplifications in the form of the spacetimecurvature — such tricks are often helpful when seeking specific exact solutions of theEinstein equations. Regarding perfect fluid spheres we present several methods oftransforming any given perfect fluid sphere into a possibly new perfect fluid sphere.</b></p> <p>This is done in three qualitatively distinct manners: The first set of solution generatingtheorems apply in Schwarzschild curvature coordinates, and are phrased in termsof the metric components: they show how to transform one static spherical perfectfluid spacetime geometry into another. A second set of solution generating theoremsextends these ideas to other coordinate systems (such as isotropic, Gaussian polar,Buchdahl, Synge, and exponential coordinates), again working directly in terms of themetric components. Finally, the solution generating theorems are rephrased in termsof the TOV equation and density and pressure profiles. Most of the relevant calculationsare carried out analytically, though some numerical explorations are also carriedout.</p>

Some Exact Solutions in General Relativity

<p><b>In this thesis four separate problems in general relativity are considered, dividedinto two separate themes: coordinate conditions and perfect fluid spheres. Regardingcoordinate conditions we present a pedagogical discussion of how the appropriateuse of coordinate conditions can lead to simplifications in the form of the spacetimecurvature — such tricks are often helpful when seeking specific exact solutions of theEinstein equations. Regarding perfect fluid spheres we present several methods oftransforming any given perfect fluid sphere into a possibly new perfect fluid sphere.</b></p> <p>This is done in three qualitatively distinct manners: The first set of solution generatingtheorems apply in Schwarzschild curvature coordinates, and are phrased in termsof the metric components: they show how to transform one static spherical perfectfluid spacetime geometry into another. A second set of solution generating theoremsextends these ideas to other coordinate systems (such as isotropic, Gaussian polar,Buchdahl, Synge, and exponential coordinates), again working directly in terms of themetric components. Finally, the solution generating theorems are rephrased in termsof the TOV equation and density and pressure profiles. Most of the relevant calculationsare carried out analytically, though some numerical explorations are also carriedout.</p>

SCHWARZSCHILD–DE SITTER METRIC AND INERTIAL BELTRAMI COORDINATES

Under consideration of coordinate conditions, we get the Schwarzschild–Beltrami–de Sitter (S-BdS) metric solution of the Einstein field equations with a cosmological constant Λ. A brief review to the de Sitter invariant special relativity (dS-SR), and de Sitter general relativity (dS-GR, or GR with a Λ) is presented. The Beltrami metric Bμν provides inertial reference frame for the dS-spacetime. By examining the Schwarzschild–de Sitter (S-dS) metric [Formula: see text] existed in literatures since 1918, we find that the existed S-dS metric [Formula: see text] describes some mixing effects of gravity and inertial-force, instead of a pure gravity effect arisen from "solar mass" M in dS-GR. In this paper, we solve the vacuum Einstein equation of dS-GR, with the requirement of gravity-free metric [Formula: see text]. In this way we find S-BdS solution of dS-GR, written in inertial Beltrami coordinates. This is a new form of S-dS metric. Its physical meaning and possible applications are discussed.

GENERAL SPHERICAL SYMMETRY SOLUTION TO EINSTEIN EQUATIONS AND NEWTONIAN GRAVITY MODIFICATION AT SMALL LENGTH SCALE

In this paper, we first we present a general spherical symmetry solution, independent of any coordinate condition, to the Einstein equations. By using this solution, we obtain different cases in different coordinate conditions, such as Schwarzschild solution, Fock solution, which have different forms of f(r). Then, we obtain a modified formula with respect to f(r) of the Newtonian gravity. This formula may describe the gravity at small length scale.

QUANTUM PLANCK SIZE BLACK HOLE STATES WITHOUT A HORIZON

A minisuperspace method is used to define a Wheeler–DeWitt equation. By imposing coordinate conditions, the space-like coordinate r is chosen as the foliation parameter. This procedure makes it possible to find "quantum black hole" states. These turn out to be naked singularities, quantum Planck size states without a horizon.