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

A novel probe of Einstein-Hilbert action: Dynamic upgradation of metric parameters