dc.contributor.author Ballik, William John Victor en dc.date 2012-05-30 14:25:04.211 dc.date 2012-06-04 15:58:03.984 dc.date.accessioned 2012-06-06T18:30:32Z dc.date.issued 2012-06-06 dc.identifier.uri http://hdl.handle.net/1974/7270 dc.description Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2012-06-04 15:58:03.984 en dc.description.abstract The invariant four-volume ($\mathcal V$) of a complete four-dimensional black hole (the volume of the spacetime at and interior to the horizon) diverges. However, if one considers the black hole resulting from the gravitational collapse of an object and integrates only a finite time to the future of the collapse, the resultant volume is well-defined and finite. We show that for non-degenerate black holes, the volume in this case can be written as $\mathcal V \propto \ln|\lambda|$, where lambda is the affine generator of the horizon and we define our volume $\mathcal V^*$ to be the constant of proportionality. In spherical symmetry, this is the Euclidean volume divided by the surface gravity ($\kappa$). en More generally, it turns out that $\mathcal V^*$ is the Parikh volume $({}^3 \mathcal V^*)$ divided by $\kappa$. This allows us to define an alternative local and invariant definition of the surface gravity of a stationary black hole. It also encourages us to find a generalization of the Parikh volume (which depends on the existence of an asymptotically timelike Killing vector) to any region of space or spacetime of arbitrary dimension, provided that this space or spacetime contains a Killing vector. We find some properties of this generalized Killing volume'' and rewrite our volume as a Killing volume for a particular Killing vector. We revisit the laws of black hole mechanics, considering them in terms of volumes rather than areas, by writing out our volume and the Parikh volume of Kerr-Newman black holes and then considering their variation with respect to the parameters $M$, $J$ and $Q$ to find a modified BH mechanics first law. We also use our new definition of $\kappa$ to develop an alternate demonstration of the BH mechanics third law. We note that the Parikh volume of a Kerr-Newman black hole is equal to $A r_+/3$, where $A$ is the horizon surface area and $r_+$ the value of the radius at the horizon, and we offer some interpretations of this relationship. We review some other relevant work by Parikh as well as some by Cveti\v{c} et al. and by Hayward. We point out some possible next steps to follow up on the work in this thesis. dc.language.iso eng en dc.relation.ispartofseries Canadian theses en dc.rights This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner. en dc.subject black holes en dc.subject general relativity en dc.title The Volume of Black Holes en dc.type thesis en dc.description.restricted-thesis My supervisor and I are going to write a paper for publication on this information and do not wish this thesis to be publicly available until such time as our publication is in print. en dc.description.degree M.Sc. en dc.contributor.supervisor Lake, Kayll en dc.contributor.department Physics, Engineering Physics and Astronomy en dc.embargo.terms 1825 en dc.embargo.liftdate 2017-06-05 dc.degree.grantor Queen's University at Kingston en
﻿