On the (1+3) threading of spacetime with respect to an arbitrary timelike vector field

TL;DR

This paper introduces a novel (1+3) threading formalism of spacetime using spatial tensor fields and a Riemannian connection, providing new formulas for Ricci tensor components and applications to black hole dynamics.

Contribution

It develops a new (1+3) threading approach based on spatial tensors and a Riemannian connection, with formulas for Ricci tensors and applications to black hole physics.

Findings

New formulas for Ricci tensor components in (1+3) threading

A novel form of Raychaudhuri's equation

Application to Kerr-Newman black hole dynamics

Abstract

We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial tensor fields and on the Riemannian spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We obtain new formulas for local components of the Ricci tensor field of $(M, g)$ with respect to the threading frame field, in terms of the Ricci tensor field of $\nabla^{\star}$ and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri's equation enables us to prove Lemma 6.2, which completes a well known lemma used in the proof of Penrose-Hawking singularity theorems.Finally, we apply the new $(1+3)$ formalism to the study of the dynamics of a Kerr-Newman black hole.