论文标题
通用无效表面热力学结构的协变量方法
Covariant approach to the thermodynamic structure of a generic null surface
论文作者
论文摘要
我们在通用空表面上读取几何关系的热力学结构。 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它的无效正常和相应的辅助向量分别是第一个导致等式之类的Navier-Stokes。在这里，我们对其他两个成员进行了调查。我们发现，$ r_ {ab} l^al^b $，它得出与$ l^a $沿本身相对应的扩展参数的演变方程，当在零体超曲面的二维横向子空间中集成时，可以将其解释为一种热力学关系，以及$ l^a $ a $ $ l^a $的虚拟位置。此外，对于固定背景，它的集成版本产生了Smarr公式的一般形式。尽管这在文献中或多或少是众所周知的，但是与$ r_ {ab} l^ak^b $提供的扩展参数的演化方程相似，与$ k^a $相对应，导致更方便的热力学关系形式。在此分析中，与较早的方法相反，已确定的热力学实体以协变形形式出现，并且独立于叶子。因此，这些可以应用于适用于零超出表面的任何坐标系。此外，这些结果不仅限于$ k^a $的任何特定参数化，而且$ k^a $不必是hypersurface Orthoconal。此外，这里没有明确使用任何特定的度量动力学方程，因此我们认为我们的结果仅基于无效表面的性质。
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.
