New insights on null and timelike warped symmetric spacetime splittings

TL;DR

This paper provides a detailed analysis of 2+2 and 1+1+2 formalisms in spherically symmetric spacetimes, deriving new equations, interpreting key scalars, and relating different formalisms to enhance understanding of spacetime structures and redshift phenomena.

Contribution

It introduces the first derivation of cross-focusing equations, interprets key scalars physically, and establishes relationships between various spacetime formalisms, extending results to other symmetric spacetimes.

Findings

Derived the first cross-focusing equations for null rays.

Identified $ ext{B}$ as the sole source of redshift in spherically symmetric spacetimes.

Established correspondence between 1+1+2 and 1+3 scalars, and extended results to planar and hyperbolic symmetries.

Abstract

We explore in detail the 2+2 and 1+1+2 formalism in spherically symmetric spacetimes, spanning from deducing the dynamical equations to relating them to the well-known generalised Painlev\'e-Gullstrand (GPG) coordinate system. The evolution equations are the Raychaudhuri equations for null rays, including those also known as cross-focusing equations whose derivation, to the best of our knowledge, we present for the first time. We physically interpret the scalars that arise in this scenario, namely the flow 2-expansion $\Theta_{n}$, the flow acceleration $\mathcal{A}$, and the radial extrinsic curvature $\mathcal{B}$. We derive a coordinate independent formula for the redshift which shows that $\mathcal{B}$ is the sole source for the redshift in spherically symmetric spacetimes. We also establish the correspondence between the 1+1+2 scalars and the 1+3 splitting scalars, expansion and shear. We further make a comparison with the Newman-Penrose formalism, in order to clarify the context where each formalism is more useful, and finally, we extend our results to planar and hyperbolic symmetric warped spacetimes as well, in particular, the relationship between $\mathcal{B}$ and the redshift.