Vector volume and black holes
Vector volume and black holes
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DOI：
10.1103/physrevd.88.104038
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发表时间：
2013-10
影响因子：
5
通讯作者：
William Ballik;K. Lake
中科院分区：
文献类型：
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作者：
William Ballik;K. Lake
通过研究沿着无发散向量场v^\alpha$的某个时空区域的不变体积v\mathcal V\的增长率，我们引入了“向量体积”v\mathcal{V}的概念。该体积可以以各种等效方式定义。例如，它可以表示为$\mathrm d \mathcal V（\mu）/ \mathrm d \mu$，其中$v^\alpha \partial_\alpha = \mathrm d / \mathrm d \mu$，$\mu$是沿$v$的积分曲线沿着的参数距离。等价地，它可以定义为$\int v^\alpha \mathrm d \Sigma_\alpha$，其中$\mathrm d \Sigma_\alpha$是有向曲面元素。我们发现这个体积对于描述黑洞特别有用，但它也可以用于其他情况。此外，该体积具有几个感兴趣的性质。其中之一是向量体积相对于向量v^\alpha$的选择是线性的。因此，例如，在具有类时基灵向量$t^\alpha$和轴对称基灵向量$\phi^\alpha$的静止轴对称时空中，对于向量$t^\alpha + \Omega \phi^\alpha$的轴对称区域的向量体积对于任何$\Omega$都是相等的，$\phi^\alpha$对$\mathcal{V}_v$没有贡献。也许最有趣的事实是，在克尔-席德时空中，完整时空的体积元等于背景时空的体积元。我们讨论了不同的方式使用矢量体积来定义黑洞的体积。最后，我们将我们的工作与最近广泛传播的宇宙学常数$\Lambda$的非零值相关的黑洞的“体积”的研究。
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 $\Lambda$.