arXiv 2009.082212020-09gr-qchep-th

一般零曲面热力学结构的协变方法

Covariant approach to the thermodynamic structure of a generic null surface

Sumit Dey, Bibhas Ranjan Majhi

备注：: Minor corrections, comments added, to appear in Phys. Rev. D
发表信息：: Phys. Rev. D 102, 124044 (2020)
DOI：: 10.1103/PhysRevD.102.124044

摘要：

我们重新讨论了一般零曲面上几何关系的热力学结构。在三个潜在的候选者中，源自于$R_{ab}$沿着表面的零矢量的不同分量（即$R_{ab}q^a_cl^b$、$R_{ab}l^al^b$和$R_{ab}l^ak^b$，其中$q_{ab}$是零表面上的投影器，而$l^a$、$k^a$分别是零法线和其对应的辅助矢量），第一个导致Navier-Stokes类方程。在这里，我们把我们的调查对其他两个成员。我们发现，$R_{ab}l^al^b$，它产生相应的$l^a$沿其本身的膨胀参数的演化方程，可以解释为一个热力学关系时，集成的二维横向子空间的零超曲面沿$l^a$方向的虚拟位移。此外，对于一个平稳的背景下，它的集成版本产生的Smarr公式的一般形式。虽然这在文献中或多或少是已知的，但是由$R_{ab}l^ak^b$提供的对应于$k^a$沿$l^a$的膨胀参数的演化方程的类似论证导致了更方便的热力学关系形式。在这种分析中，与早期的方法相反，确定的热力学实体是协变形式，也是叶理独立的。因此，这些可以应用于任何适用于零超曲面的坐标系。此外，这些结果不限于任何特定的参数化的$k^a$和$k^a$也不需要超曲面正交。此外，这里没有明确使用任何特定的度规动力学方程，因此我们认为我们的结果仅仅是基于零曲面的性质。

英文摘要：

We readdress the thermodynamic structure of geometrical relations on a generic null surface. Among three potential candidates, originated from different components of $R_{ab}$ along the null vectors for the surface (i.e. $R_{ab}q^a_cl^b$, $R_{ab}l^al^b$ and $R_{ab}l^ak^b$ where $q_{ab}$ is the projector on the null surface and $l^a$, $k^a$ are null normal and corresponding auxiliary vector of it, respectively), the first one leads to Navier-Stokes like equation. Here we devote our investigation on the other two members. We find that $R_{ab}l^al^b$, which yields the evolution equation for expansion parameter corresponding to $l^a$ along itself, can be interpreted as a thermodynamic relation when integrated on the two dimensional transverse subspace of the null hypersurface along with a virtual displacement in the direction of $l^a$. Moreover for a stationary background the integrated version of it yields the general form of Smarr formula. Although this is more or less known in literature, but a similar argument for the evolution equation of the expansion parameter corresponding to $k^a$ along $l^a$, provided by $R_{ab}l^ak^b$, leads to a more convenient form of thermodynamic relation. In this analysis, contrary to earlier approaches, the identified thermodynamic entities come out to be in covariant forms and also are foliation independent. Hence these can be applied to any coordinate system adapted to the null hypersurface. Moreover, these results are not restricted to any specific parametrisation of $k^a$ and also $k^a$ need not be hypersurface orthogonal. In addition, here any particular dynamical equation for metric is not being explicitly used and therefore we feel that our results are solely based on the properties of the null surface.

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