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Bianchi type I solutions to Einstein's vacuum equations.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2013 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

A natural question in general relativity is whether there exist singularities, like the Big Bang and

black holes, in the universe. Albert Einstein did not in the beginning believe that singularities

in general relativity are generic ([2], [3]). He claimed that the existence of singularities is due

to symmetry assumptions. The symmetry assumptions are usually spatial isotropy and spatial

homogeneity. Spatial isotropy means intuitively that, for a xed time, universe looks the same at

all points and in all spatial directions.

In the present paper, we will show the following: If we solve Einstein's vacuum equations with a

certain type of initial data, called the Bianchi type I, the resulting space-time will either be the

Minkowski space or an anisotropic space-time equipped with a so called Kasner metric. We show

that, in the anisotropic case, the space-time will contain a certain singularity: the Big Bang.

We distinguish between two dierent classes of a Kasner metrics; the Flat Kasner metric and

the Non-at Kasner metric. In the case of a Flat Kasner metric, we show that it is possible to

isometrically embed the entire space-time into Minkowski space. In the case of the Non-at Kasner

metric, the space-time is not extendible and the gravity goes to innity approaching the time of

the Big Bang.

In addition we show, using any Kasner metric, that the universe expands proportional to the

time passed since the Big Bang. This happens even though some directions will shrink or not


The conclusion is: We have found two natural classes of anisotropic space-times, that include

a Big Bang and expand. These results supports the idea that singularities are generic, i.e. are not

due to the assumptions of symmetry of the universe.


Place, publisher, year, edition, pages
2013. , 36 p.
National Category
Engineering and Technology
URN: urn:nbn:se:kth:diva-129194OAI: diva2:650621
Available from: 2013-09-23 Created: 2013-09-23 Last updated: 2013-09-23Bibliographically approved

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