Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities
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–Nordström 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.