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Title:  The Volume of Black Holes 
Authors:  Ballik, William John Victor 

Keywords:  black holes general relativity 
Issue Date:  6Jun2012 
Series/Report no.:  Canadian theses 
Abstract:  The invariant fourvolume ($\mathcal V$) of a complete fourdimensional 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 welldefined and finite. We show that for nondegenerate 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 ($\kappa$).
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 KerrNewman 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 KerrNewman 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. 
Description:  Thesis (Master, Physics, Engineering Physics and Astronomy)  Queen's University, 20120604 15:58:03.984 
URI:  http://hdl.handle.net/1974/7270 
Appears in Collections:  Queen's Graduate Theses and Dissertations Department of Physics, Engineering Physics and Astronomy Graduate Theses

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