Applications of embedding theory in higher dimensional general relativity.

Abstract

The study of embeddings is applicable and signicant to higher dimensional theories of

our universe, high-energy physics and classical general relativity. In this thesis we investigate

local and global isometric embeddings of four-dimensional spherically symmetric

spacetimes into five-dimensional Einstein manifolds. Theorems have been established

that guarantee the existence of such embeddings. However, most known explicit results

concern embedded spaces with relatively simple Ricci curvature. We consider the four-dimensional

gravitational field of a global monopole, a simple non-vacuum space with

a more complicated Ricci tensor, which is of theoretical interest in its own right, and

occurs as a limit in Einstein-Gauss-Bonnet Kaluza-Klein black holes, and we obtain

an exact solution for its embedding into Minkowski space. Our local embedding space

can be used to construct global embedding spaces, including a globally

at space and

several types of cosmic strings. We present an analysis of the result and comment on

its signicance in the context of induced matter theory and the Einstein-Gauss-Bonnet

gravity scenario where it can be viewed as a local embedding into a Kaluza-Klein black

hole. Difficulties in solving the five-dimensional equations for given four-dimensional

spaces motivate us to investigate which embedded spaces admit bulks of a specific type.

We show that the general Schwarzschild-de Sitter spacetime and the Einstein Universe

are the only spherically symmetric spacetimes that can be embedded into an Einstein

space with a particular metric form, and we discuss their five-dimensional solutions.

Furthermore, we determine that the only spherically symmetric spacetime in retarded

time coordinates that can be embedded into a particular Einstein bulk is the general

Vaidya-de Sitter solution with constant mass. These analyses help to provide insight to

the general embedding problem. We also consider the conformal Killing geometry of a

five-dimensional Einstein space that embeds a static spherically symmetric spacetime,

and we show how the Killing geometry of the embedded space is inherited by its bulk.

The study of embedding properties such as these enables a deeper mathematical understanding

of higher dimensional cosmological models and is also of physical interest

as conformal symmetries encode conservation laws.