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Derivation and stability determination of black hole metrics

Svensson, Emma LU (2018) FYTK02 20181
Theoretical Particle Physics
Department of Astronomy and Theoretical Physics
Abstract
The Schwarzschild metric and the Kerr metric describe the gravitational fields around static and rotating black holes, respectively. Here, we derive the Kerr metric in a simpler way than how it was derived originally and determine the stability of the Schwarzschild metric. The Kerr metric was derived by using the Einstein-Hilbert action as well as directly from the Einstein field equations. In order to do this, we first made an anzats with the help of the Weyl method. By using the same method as Chandrasekhar, we found that the Schwarzschild metric is stable.
Popular Abstract
Black holes are not only graves of older stars in the universe; they are also potential time machines. Because of their incredibly high gravitation, they are able to bend space-time so much that if you were to travel to one and then come back, hundreds of years could have passed of Earth. Many would think that traveling into a black hole would mean the traveler's demise, but if the black hole was rotating, this may not be the case.

Black holes are some of the most fascinating objects that we know about today. With the first announcement of gravitational waves given in 2016 by the LIGO observation, by taking data from colliding black holes, they have opened up a large range of studies regarding the cosmos. There is data pointing towards... (More)
Black holes are not only graves of older stars in the universe; they are also potential time machines. Because of their incredibly high gravitation, they are able to bend space-time so much that if you were to travel to one and then come back, hundreds of years could have passed of Earth. Many would think that traveling into a black hole would mean the traveler's demise, but if the black hole was rotating, this may not be the case.

Black holes are some of the most fascinating objects that we know about today. With the first announcement of gravitational waves given in 2016 by the LIGO observation, by taking data from colliding black holes, they have opened up a large range of studies regarding the cosmos. There is data pointing towards the fact that there may exist a super-massive black hole at the center of the Milky Way and there have been discussions about the possibility that the universe may have been created out of one. They have also opened up discussions about the possible existence of worm holes and parallel universes.

The idea that an object with an escape velocity larger than the speed of light was proposed as early as in 1783 by John Michell, but at that time they were called ”dark stars”. It took more than 100 years, when Einstein had published his theory of general relativity, to give further calculations pointing to the existence of such objects. A solution of Einstein's equations pointing towards the existence of black holes was done by Karl Schwarzschild and independently by Johannes Droste, less than a year after Einstein published his theory of general relativity.

A black hole is formed as a result of the decomposition of a massive star. The force of gravity is so strong that the pressure preserving the structure of the star eventually fails, which means that the star collapses into itself. As a result, something with a gravitational force so strong that not even light can escape it is formed. This new object is called a \emph{black hole}, namely because it is "black" in the sense that we can not see it and "hole" since everything that gets too close to it falls into it and can not escape, like a deep hole in the ground.

One of the things that makes black holes so interesting is their incredible density. The corresponding radius of a black hole for a given mass can be simply calculated by using an equation derived by Schwarzschild. This calculation shows that if the Earth were to collapse into a black hole, it would be as small as a grape!



One of the many characteristics of a black hole is that it can rotate. It took almost 50 years after Einstein published his theory of relativity to correctly describe this rotation, due to the calculations being so complicated. Before the solution done by Kerr, there had been discussions regarding the fact that rotation could perhaps slow down the process of a collapsing star. With the picture given by Schwarzschild, traveling to a black hole would mean that you would end up in a so-called singularity, but with Kerr this may not be the case.

My bachelor thesis deals with the rotation and the stability of black holes. The solution obtained by Kerr is first derived in a more pedestrian fashion and then the stability of the Schwarzschild solution is determined. Could the existence of stable, rotating black holes mean a possibility to travel into black holes? Could they be the doors to possible parallel universes? It remains to be seen. (Less)
Please use this url to cite or link to this publication:
author
Svensson, Emma LU
supervisor
organization
course
FYTK02 20181
year
type
M2 - Bachelor Degree
subject
keywords
black holes, relativity, general relativity, kerr metric, schwarzschild metric, derivation, stability
language
English
id
8959364
date added to LUP
2018-09-28 16:15:06
date last changed
2018-09-28 16:15:06
@misc{8959364,
  abstract     = {The Schwarzschild metric and the Kerr metric describe the gravitational fields around static and rotating black holes, respectively. Here, we derive the Kerr metric in a simpler way than how it was derived originally and determine the stability of the Schwarzschild metric. The Kerr metric was derived by using the Einstein-Hilbert action as well as directly from the Einstein field equations. In order to do this, we first made an anzats with the help of the Weyl method. By using the same method as Chandrasekhar, we found that the Schwarzschild metric is stable.},
  author       = {Svensson, Emma},
  keyword      = {black holes,relativity,general relativity,kerr metric,schwarzschild metric,derivation,stability},
  language     = {eng},
  note         = {Student Paper},
  title        = {Derivation and stability determination of black hole metrics},
  year         = {2018},
}