Paper detail

Determination of varying speed of light from Black hole

The Generalized Cosmological Time (GCT) framework offers an alternative phenomenological approach to addressing the Hubble tension and the observed time dilation of Type Ia supernovae, characterized by a background parameter b \simeq 0.04 and an associated cosmological scaling of fundamental constants. A key conceptual question is whether such a background evolution is compatible with the stability of local, gravitationally bound systems, in particular black holes. This work examines black hole thermodynamics within the GCT framework, focusing on the geometric compatibility between a locally static region and a time-dependent cosmological background. By matching a static interior spacetime to a GCT-FLRW exterior across a timelike boundary, it is shown that the Israel junction conditions allow for the coexistence of distinct time normalizations without introducing surface stresses. In this setting, the local interior naturally admits a unit lapse function, while the background evolution is encoded in the cosmological time gauge. The resulting separation of time normalizations implies that the effective GCT parameter governing local physics is observationally indistinguishable from b_{\mathrm{local}} \simeq 0. Under this geometric shielding, black hole thermodynamics reduces to its standard general relativistic form, and the Generalized Second Law is satisfied without imposing additional constraints on the background parameter b. These results indicate that the empirical stability of black hole thermodynamics does not directly constrain the global GCT evolution but instead reflects a geometric decoupling between local and cosmological time gauges. Black hole stability thus emerges as a consistency condition for geometric shielding, rather than as independent evidence for or against the underlying cosmological model.