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

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.

Initial data and black holes for matter models

“…The density of this solution blows up at the center, hence the name "singular solution". In [9] we have shown that this solution is, although singular, still surprisingly well-behaved in a way that it satisfies the second Bianchi identity weakly. The stability of this solution may be studied using metric convergence, e.g., in the sense of Gromov-Hausdorff convergence or Sormani-Wenger intrinsic flat convergence [1,6,22,29,30].…”

The Einstein–Infeld–Hoffmann legacy in mathematical relativity II. Quantum laws of motion for singularities of spacetime

“…a solution of the Einstein-Maxwell equations. As was shown in [7], the case where the electromagnetic vacuum law is the standard, linear law of Maxwell (D = E, H = B) yields a spacetime that is too singular for weak Bianchi identity to be satisfied at the singularity, so that our third working hypothesis is not satisfied. We showed however, that there are nonlinear vacuum laws, including the one proposed by Born & Infeld [5], for which the corresponding static spherically symmetric solution of the Einstein-Maxwell equations (which was fully analyzed in [23]) does satisfy the weak Bianchi identity.…”

Properties of the Null Distance and Spacetime Convergence

“…where h is bounded function on Σ. Such static metrics appear, for instance, when the Einstein equations are coupled to matter fields [10,23] (which are not necessarily asymptotically flat). Using spacetime convergence results, it may be possible to study their spacetime stability using techniques analogous to [18,49,55,58].…”

“…This is most evident in matter models involving compressible fluids since discontinuities occur both at the matter-vacuum boundary and in the form of shocks within the fluid (see, for instance, Barnes, LeFloch, Schmidt and Stewart [13], Burtscher and LeFloch [24], Groah, Smoller and Temple [43], Le Floch and LeFloch [56], Rendall and Ståhl [68]). Also for the vacuum Einstein equations and other matter models weak solutions play a prominent role (see, for instance, Christodoulou [27], Sbierski [72], Dafermos and Luk [54], Burtscher, Kiessling and Tahvildar-Zadeh [23], Tahvildar-Zadeh [79]).…”