The Vector Volume and Black Holes

By examining the rate of growth of an invariant volume $\mathcal V$ of some spacetime region along a divergence-free vector field $v^\alpha$, we introduce the concept of a "vector volume" $\mathcal{V}_v$. This volume can be defined in various equivalent ways. For example, it can be given as $\mathrm d \mathcal V(\mu) / \mathrm d \mu$, where $v^\alpha \partial_\alpha = \mathrm d / \mathrm d \mu$, and $\mu$ is a parameter distance along the integral curve of $v$. Equivalently, it can be defined as $\int v^\alpha \mathrm d \Sigma_\alpha$, where $\mathrm d \Sigma_\alpha$ is the directed surface element. We find that this volume is especially useful for the description of black holes, but it can be used in other contexts as well. Moreover, this volume has several properties of interest. Among these is the fact that the vector volume is linear with respect to the the choice of vector $v^\alpha$. As a result, for example, in stationary axially symmetric spacetimes with timelike Killing vectors $t^\alpha$ and axial symmetric Killing vectors $\phi^\alpha$, the vector volume of an axially symmetric region with respect to the vector $t^\alpha + \Omega \phi^\alpha$ is equal for any value of $\Omega$, a consequence of the additional result that $\phi^\alpha$ does not contribute to $\mathcal{V}_v$. Perhaps of most interest is the fact that in Kerr-Schild spacetimes the volume element for the full spacetime is equal to that of the background spacetime. We discuss different ways of using the vector volume to define volumes for black holes. Finally, we relate our work to the recent wide-spread thermodynamically motivated study of the "volumes" of black holes associated with non-zero values of the cosmological constant $\Lambda$.