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

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.

18 pages, 2 figures, The version that Classical and Quantum Gravity accepted on December 24, 2024

References in corpus (9)

• Determining the Type, Redshift, and Age of a Supernova Spectrum
• Time Dilation in Type Ia Supernova Spectra at High Redshift
• Spatially averaged cosmology in an arbitrary coordinate system
• Lagrangian theory of structure formation in relativistic cosmology III: gravitoelectric perturbation and solution schemes at any order
• Introduction to Numerical Relativity
• A viable varying speed of light model in the RW metric
• Exact non-linear equations for cosmological perturbations
• Cosmology in Painleve-Gullstrand coordinates
• Classical and quantum Chaplygin gas Hořava-Lifshitz scalar-metric cosmology

Cited by in corpus (1)

• A Unified Interpretation of Supernova, GRB, and QSO Time Dilation Signals in a Generalized Cosmological Time Framework