This among the most interesting solutions as Einstein’s

This means that there is an amount of redundancy when
treating it in four dimensions, as for instance nothing changes in time. The
same hold true for the angular coordinates and one can reduce the dimension of
the problem by mathematically removing one dimension. The dimensional reduction
in this paper was developed by the German mathematician Theodore Kaluza and the
Swedish physicist Oskar Klein. Reducing one dimension is not as easy as it may
sound, and we do not discuss in detail how this works. The action in four
dimensions is replaced by a corresponding action in three dimensions. Solving
this problem and then performing a decompactification gives the solution in
four dimensions. The solution generating techniques described in this thesis are
useful when developing a theory of quantum gravity, the combination of quantum
mechanics and general relativity. To construct quantum gravity, it is necesary
to understand the solutions predicted by gravity. Black holes are among the
most interesting solutions as Einstein’s theories break down into a
singularity. To describe black holes in a completely satisfactory way, quantum
gravity is needed. Black hole solutions are therefore of great interest, and
dimensional reduction is a powerful tool when obtaining these solutions since
hidden symmetries in four dimensions may revealed in three or two dimensions.
Using these symmetries, it is possible to classify black holes and derive
entire families of black holes from one solution. Dimensional reduction is useful,
not only to derive solutions of black holes, but also in constructing the
theory of quantum gravity itself. These theories, such as supersymmetry and
string theory, describe a world of ten or eleven dimensions and dimensional
reduction is therefore necessary to describe our four-dimensional world. In
dimensional reduction, more terms are added to the action as the number of
dimensions is reduced. Physical theories that seem different in four dimensions
can be unified in ten or eleven dimensions. On the other hand, the different
action obtained when four-dimensional Einstein gravity is reduced to three or
two dimensions reveal hidden symmetries and can be analyzed in the framework of
group theory to obtain the four-dimensional solution. The purpose of this
thesis was to show how it is possible to derive the Schwarzschild solution with
the hidden symmetries of a black hole revealed with the dimensional reduction
from four to three dimensions. To do this we first present a short introduction
to general relativity and group theory and then combine the two to arrive at
the Schwarzschild solution. We succeed in reaching our goal by looking at the
action of the given system, then performing a Kaluza-Klein compactification on
the four-dimensional spacetime to solve the problem in three dimensions. By
performing a decompactification we then obtain the solution in four dimensions.
We also reach beyond Schwarzschild and derive the solution of a charged black
hole, the Reissner-Nordstrom solution. To conclude, in this thesis we first
study the Schwarzschild solution with Einstein’s theory. Then we perform a
dimensional reduction to three dimensions to derive the same solution using
group theory. After that we go beyond the Schwarzschild solution and look at
the Reissner-Nordstrom solution as well as other solutions in the
Schwarzschild family.