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–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.
Highlights
Independent of the inquiry into suitable matter field models, the following is clear: For the establishment of the energy–momentum conservation law (1.5) when G > 0 that could pave the ground toward a well-posed joint initial value problem for the spacelike slices of spacetime, the electromagnetic and perhaps other matter fields in it, and their charged point singularities, along similar lines as in the special-relativistic formulation mentioned above, it is necessary that the second Bianchi identity (1.2) holds in a weak sense
While a number of ideas developed below are clearly adaptable to more general situations, in the present paper we focus our efforts on static spherically symmetric spacetimes that feature a single timelike singularity, with special emphasis given to electrostatic spacetimes of a single point charge at the center of symmetry
In this paper we have considered the following question: under what conditions is the twicecontracted second Bianchi identity satisfied in a weak sense in a neighborhood of a singular line of a spacetime M with the metric g? We were able to answer this question in case (M, g) is both static and spherically symmetric, by finding sufficient conditions on the metric that, if satisfied, guarantee that the weak second Bianchi identity holds everywhere, the location of a timelike singularity included
Summary
In a series of influential papers, [14,15,16], Einstein and Infeld (EI), originally joined by Hoffmann (EIH), claimed that the field equations of general relativity theory, (1.1), coupled with the Maxwell–Lorentz evolution equations for the electromagnetic fields, determine the equations of motion of matter modeled atomistically as composed of charged point particles, which they identified with point singularities in spacelike slices of a spacetime. Independent of the inquiry into suitable matter field models, the following is clear: For the establishment of the energy–momentum conservation law (1.5) when G > 0 that could pave the ground toward a well-posed joint initial value problem for the spacelike slices of spacetime, the electromagnetic and perhaps other matter fields in it, and their charged point singularities, along similar lines as in the special-relativistic formulation mentioned above, it is necessary that the second Bianchi identity (1.2) holds in a weak sense. We are interested in ‘electromagnetic matter’, whose electromagnetic field satisfies the pre-metric Maxwell’s equations, complemented with a suitable electromagnetic vacuum law, and with charged sources given by a finite number of one-dimensional timelike singularities that are assigned an energy–momentum–stress tensor in the spirit of EIH, and Wallace
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