# Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities

## TL;DR

This paper investigates whether the twice-contracted second Bianchi identity, which relates to energy-momentum conservation, can hold in a weak sense for static, spherically symmetric spacetimes with timelike curvature singularities.

## Contribution

The work establishes sufficient conditions for a distributional version of the Bianchi identity to hold in certain static, spherically symmetric spacetimes with timelike singularities, identifying specific metrics where this applies.

## Key findings

- The Bianchi identity does not hold for Reissner-Weyl-Nordström spacetime.
- It does hold for Hoffmann's spacetime with negative bare mass.
- Conditions for the identity to hold are explicitly characterized.

## Abstract

The (twice-contracted) second Bianchi identity is a differential curvature identity that holds on any smooth manifold with a metric. In the case when such a metric is Lorentzian and solves Einstein's equations with an (in this case inevitably smooth) energy-momentum-stress tensor of a "matter field" as the source of spacetime curvature, this identity implies the physical laws of energy and momentum conservation for the "matter field". The present work inquires into whether such a Bianchi identity can still hold in a weak sense for spacetimes with curvature singularities associated with timelike singularities in the "matter field". Sufficient conditions that establish a distributional version of the twice-contracted second Bianchi identity are found. In our main theorem, a large class of spherically symmetric static Lorentzian metrics with timelike one-dimensional singularities is identified, for which this identity holds. As an important first application we show that the well-known Reissner-Weyl-Nordstr\"om spacetime of a point charge does not belong to this class, but that Hoffmann's spacetime of a point charge with negative bare mass in the Born-Infeld electromagnetic vacuum does.