Geometric Matching of Local Static Regions in Cosmological Spacetimes with an Evolving Lapse

Abstract

The generalized cosmological time (GCT) framework introduces a modified lapse function, N(t)\propto a^{b/4}, as a geometric extension of the standard FLRW description. Like other departures from $\Lambda$CDM, such constructions must remain compatible with the observed stability of local gravitational and laboratory physics. In scalar--tensor theories, this compatibility is usually achieved through dynamical screening mechanisms that suppress additional degrees of freedom in dense environments. In this work, we examine whether a locally static spacetime region can be consistently embedded within a cosmological background described by a time-dependent lapse. By embedding a static Schwarzschild interior into an expanding GCT--FLRW exterior, the Israel junction conditions are used to determine the class of background expansions that admit such a matching in the absence of a thin shell. The continuity of the extrinsic curvature yields a Friedmann-type relation that coincides with the GCT background equations of motion. This relation should be interpreted not as a new dynamical equation, but as a geometric consistency condition (GCC) associated with the matching of the two spacetime regions. In this sense, the junction does not introduce new dynamics, but provides a GCC under which a region with fixed local clocks can be embedded in a cosmological spacetime with an evolving lapse. Therefore, the analysis clarifies how locally static gravitational systems can remain compatible with a cosmological time normalization that differs from that of local proper time while preserving the standard description of local physics.

17 pages, 1 figure, Final version accepted for publication in the Classical and Quantum Gravity

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