# Applications of embedding theory in higher dimensional general relativity.

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## Date

2012

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## 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.

## Description

Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2012.

## Keywords

Einstein manifolds., Isometrics (Mathematics), Einstein field equations., General relativity (Physics), Theses--Applied mathematics.