3+1 formalism of the minimally extended varying speed of light model

TL;DR

This paper applies the 3+1 formalism to a minimally extended varying speed of light cosmological model, deriving key equations and demonstrating its consistency with Einstein's equations and physical interpretation.

Contribution

It introduces a 3+1 formalism for the varying speed of light model, deriving lapse and shift functions, and clarifies the physical significance of the model's parameters.

Findings

Derived the lapse function and shift vector for the model.

Showed the model's equations are consistent with Einstein's equations.

Interpreted the varying speed of light as a cosmological time dilation effect.

Abstract

The $3+1$ formalism provides a structured approach to analyzing spacetime by separating it into spatial and temporal components. When applied to the Robertson-Walker metric, it simplifies the analysis of cosmological evolution by dividing the Einstein field equations into constraint and evolution equations. It introduces the lapse function $N$ and the shift vector $N^i$, which control how time and spatial coordinates evolve between hypersurfaces. In standard model cosmology, $N = 1$ and $N^i = 0$ for the Robertson-Walker metric. However, the $N$ becomes a function of time when we apply the metric to the minimally extended varying speed of light model. This approach allows for a more direct examination of the evolution of spatial geometry and offers flexibility in handling scenarios where the lapse function and shift vector vary. In this manuscript, we derive the model's $N$ and $N^i$, along with the constraint and evolution equations, and demonstrate their consistency with the existing Einstein equations. We have shown in a previous paper that the possibility of changes in the speed of light in the Robertson-Walker metric is due to cosmological time dilation. Through the $3+1$ formalism, we can make the physical significance more explicit and demonstrate that it can be interpreted as the lapse function. From this, we show that the minimally extended varying speed of light model is consistent.