Minkowski aegruumi geomeetriast
Date
2013
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Tartu Ülikool
Abstract
We consider a four-dimensional real vector space in which three coordinates
describe the space, and the fourth coordinate describes time. Such a vector space,
equipped with Minkowski metric, is said to be a Minkowski spacetime, denotedM
and sometimes referred just as Minkowski space. In the beginning of 20th century
it emerged from the works of Hermann Minkowski and Henry Poincar e that this
mathematical tool can be really useful in areas like the theory of special relativity.
The rst objective of this thesis is to introduce the classical properties of
Minkowski spacetime in a way that doesn't get into the details about the physical
meaning of the topic. The main focus is on the elements of the spacetime and
the orthogonal maps that describe the transformations of these elements. As this
topic, the Minkowski spacetime, has many applications in physics, most of the literature
on it originates from physics, rather than mathematics. In the books that
are written by physicists, strict proofs are not given properly or they are avoided
entirely. Sometimes physical discussion is considered as a proof, but as shown in
proposition 3.1 for example, things are not so obvious after all in a mathematical
point of view. In contrary, the aim of this thesis is to ll this gap by giving rigorous
proof to all the statements that are made.
The last part of the thesis takes a new course and tries to introduce tools from
branch of mathematics called di erential geometry. Apparatus described there
can be useful not just describing the structure of Minkowski spacetime, but also
wide areas of theoretical physics. At last, we generalize classical spacetime and
present the Minkowski superspace { a space with anticommuting coordinates that
has become a useful tool in the eld theory. This is all done in a bit less formal
way.
This thesis consists of four sections.
In the rst section we recall such topics that are not strictly connected to the
thesis but have great impact on further understanding of the paper. For example
Gram{Schmidt orthogonalization process is reminded and we prove one of the
fundamental results related to dot product, the Cauchy{Schwartz{Bunjakowski
inequality. We also describe the rules of the multipication of block matrices as
42
well as nding the inverse of block matrix. Map called matrix exponential is
introduced as it plays a key role in the nal part of the thesis. Considered the
geometrical meaning of the paper we nd ourselves stopping for a bit longer to
study the idea of topological manifold.
Second section sets an objective to describe the elements or events of the
Minkowski spacetime. For that we give a strict de nition of dot product and
move on to the concept of pseudoeucleidic space. After that we describe the dot
product in Minkowski spacetime, which gives us the metric. It appears that by the
metric we can factorize the elements of spacetime into three distinct classes, these
are spacelike, timelike and nullvectors. The last subsection is there to describe the
orthogonal transformations of Minkowski space, referred as Lorentz transformations.
We show how these transformations can be represented in both coordinates
and matrix form and what is the connection between the transformations and the
metric of M. Finally we prove that the set of all Lorentz transformatioins has a
very natural group structure, the Lorentz group.
In the third section we continue with the observation of Lorentz transformations.
In the rts subsection we take a closer look at the properties of such transformations
and concentrate on the most important of them. As a result we nd a
rotation subgroup of Lorentz group. We proceed with noting that Lorentz transformations
remain the distances invariant. That encourages us to look for other
such transformations that share the same property. As it happens translations are
of this kind and by observing translations with Lorentz transformations together,
we get the symmetry group of Minkowski spacetime, the Poincar e group.
The fourth section takes another course. In this section the objective is to introduce,
via more popular approach, the connections between Minkowski spacetime
and di erential geometry, and to uncover the term superspace with the surrounding
mathematics. In the part of di erential geometry we de ne Lie algebra with
its representation and give examples to clear the topic. As expected we then nd
ourselves de ning Lie group which we later connect with Lie algebra by giving an
informal description. The section, as well as the paper is then nished with the
introduction of Minkowski superspace. For that we de ne terms such as superalgebra
and give a well known example, the Grassmann algebra. By avoiding strict
mathematical formalism, we show what are functions on superspace and how to
nd integral or derivative of such functions.
Description
Keywords
Minkowski aegruum, Lorentzi teisendus, ortogonaalteisendus, Lorentzi rühm, Poincaré rühm, superruum, Minkowski superruum, bakalaureusetööd