The Einstein-Hilbert (EH) action is peculiar in many ways. Some of the Peculiar features have already been highlighted in literature. In the present article, we have discussed some peculiar features of EH action which has not been discussed earlier. It is well-known that there are several ways of decomposing the EH action into the bulk and the surface part with different underlying motivations. We provide a review on all of these decompositions. Then, we attempt to study the static coordinate as a limiting case of a time-dependent coordinate via dynamic upgradation of the constant metric parameters. Firstly, we study the consequences when the constant parameters, present in a static and spherically symmetric (SSS) metric, are promoted to the time dependent variables, which allows us to incorporate the time-dependence in the static coordinate. We find that, in every sets of decomposition, the expression of the bulk term remains invariant, whereas the surface term changes by a total derivative term. Finally, when we obliterate the time dependence of the metric parameters, we find that the expression of the Ricci-scalar (or the EH action) does not go back to its original value. Instead, we find that the curvature becomes singular on the horizon, which implies a topological change from the original spacetime.

爱因斯坦-希尔伯特 (EH) 作用在许多方面都很独特。一些特殊的特征已经在文献中得到强调。在本文中，我们讨论了 EH 作用的一些特殊特征，这些特征之前没有讨论过。众所周知，有几种方法可以将 EH 作用分解为具有不同潜在动机的主体和表面部分。我们对所有这些分解进行了回顾。然后，我们尝试通过动态升级恒定度量参数来研究静态坐标作为时间相关坐标的限制情况。首先，我们研究了将静态和球对称 (SSS) 度量中存在的常数参数提升为时间相关变量时的后果，这使我们能够将时间相关性纳入静态坐标中。我们发现，在每一组分解中，体项的表达式保持不变，而表面项的变化是全导数项。最后，当我们消除度量参数的时间依赖性时，我们发现 Ricci 标量（或 EH 动作）的表达式并没有回到其原始值。相反，我们发现曲率在地平线上变得奇异，这意味着原始时空的拓扑变化。