Nature of the singularity in charged or rotating black holes

2.3 Nature of the singularity in charged or rotating black holes

2.3.1 Overview

Unlike the simple singularity structure of the Schwarzschild solution, where the event horizon encloses a spacelike singularity at $r = 0$ , charged and/or rotating BH’s have a much richer singularity structure. The extended spacetimes have an inner Cauchy horizon (CH) which is the boundary of predictability. To the future of the CH lies a timelike (ring) singularity [244 ]. Poisson and Israel [212 , 213 ] began an analytic study of the effect of perturbations on the CH. Their goal was to check conjectures that the blue-shifted infalling radiation during collapse would convert the CH into a true singularity and thus prevent an observer’s passage into the rest of the extended regions. By including both ingoing and back-scattered outgoing radiation, they find for the Reissner–Nordström (RN) solution that the mass function (qualitatively $R αβγδ ∝ M ∕r3$ ) diverges at the CH (mass inflation). However, Ori showed both for RN and Kerr [202 , 203 ] that the metric perturbations are finite (even though $R R μνρσ μνρσ$ diverges) so that an observer would not be destroyed by tidal forces (the tidal distortion would be finite) and could survive passage through the CH. A numerical solution of the Einstein–Maxwell–scalar field equations would test these perturbative results.

For an excellent, brief review of the history of this topic see the introduction in [205 ].

2.3.2 Numerical studies

Gnedin and Gnedin [118 ] have numerically evolved the spherically symmetric Einstein–Maxwell with massless scalar field equations in a 2 + 2 formulation. The initial conditions place a scalar field on part of the RN event horizon (with zero field on the rest). An asymptotically null or spacelike singularity whose shape depends on the strength of the initial perturbation replaces the CH. For a sufficiently strong perturbation, the singularity is Schwarzschild-like. Although they claim to have found that the CH evolved to become a spacelike singularity, the diagrams in their paper show at least part of the final singularity to be null or asymptotically null in most cases.

Brady and Smith [56 ] used the Goldwirth–Piran formulation [119 ] to study the same problem. They assume the spacetime is RN for $v < v0$ . They follow the evolution of the CH into a null singularity, demonstrate mass inflation, and support (with observed exponential decay of the metric component $g$ ) the validity of previous analytic results [212 , 213 , 202 , 203 ] including the “weak” nature of the singularity that forms. They find that the observer hits the null CH singularity before falling into the curvature singularity at $r = 0$ . Whether or not these results are in conflict with Gnedin and Gnedin [118 ] is unclear [50 ]. However, it has become clear that Brady and Smith’s conclusions are correct. The collapse of a scalar field in a charged, spherically symmetric spacetime causes an initial RN CH to become a null singularity except at $r = 0$ where it is spacelike. The observer falling into the BH experiences (and potentially survives) the weak, null singularity [202 , 203 , 51 ] before the spacelike singularity forms. This has been confirmed by Droz [92 ] using a plane wave model of the interior and by Burko [63 ] using a collapsing scalar field. See also [65 , 68 ].

Numerical studies of the interiors of non-Abelian black holes have been carried out by Breitenlohner et al. [57 , 58 ] and by Gal’tsov et al. [91 , 104 , 105 , 106 ] (see also [254 ]). Although there appear to be conflicts between the two groups, the differences can be attributed to misunderstandings of each other’s notation [59 ]. The main results include an interesting oscillatory behavior of the metric.

Figure 4: Figure 1 from [66 ] is a schematic diagram of the singularity structure within a spherical charged black hole.

The current status of the topic of singularities within BH’s includes an apparent conflict between the belief [19 ] and numerical evidence [37 ] that the generic singularity is strong, oscillatory, and spacelike, and analytic evidence that the singularity inside a generic (rotating) BH is weak, oscillatory (but in a different way), and null [206 ]. See the discussion at the end of [206 ].

Various recent perturbative results reinforce the belief that the singularity within a “realistic” (i.e. one which results from collapse) black hole is of the weak, null type described by Ori [202 , 203 ]. Brady et al. [54 ] performed an analysis in the spirit of Belinskii et al. [19 ] to argue that the singularity is of this type. They constructed an asymptotic expansion about the CH of a black hole formed in gravitational collapse without assuming any symmetry of the perturbed solution. To illustrate their techniques, they also considered a simplified “almost” plane symmetric model. Actual plane symmetric models with weak, null singularities were constructed by Ori [204 ].

The best numerical evidence for the nature of the singularity in realistic collapse is Hod and Piran’s simulation of the gravitational collapse of a spherically symmetric, charged scalar field [154 , 153 ]. Rather than start with (part of) a RN spacetime which already has a singularity (as in, e.g., [56 ]), they begin with a regular spacetime and follow its dynamical evolution. They observe mass inflation, the formation of a null singularity, and the eventual formation of a spacelike singularity. Ori argues [206 ] that the rotating black hole case is different and that the spacelike singularity will never form. No numerical studies beyond perturbation theory have yet been made for the rotating BH.

Some insight into the conflict between the cosmological results and those from BH interiors may be found by comparing the approach to the singularity in Gowdy [121 ] spatially inhomogeneous cosmologies (see Section 3.4.2) with $T 3$ [36 ] and $S2 × S1$ [111 ] spatial topologies. Early in the collapse, the boundary conditions associated with the $2 1 S × S$ topology influence the gravitational waveforms. Eventually, however, the local behavior of the two spacetimes becomes qualitatively indistinguishable and characteristic of a (non-oscillatory in this case) spacelike singularity. This may be relevant because the BH environment imposes effective boundary conditions on the metric just as topology does. Unfortunately, no systematic study of the relationship between the cosmological and BH interior results yet exists.

2.3.3 Going further

Replacing the AF boundary conditions with Schwarzchild–de Sitter and RN–de Sitter BH’s was long believed to yield a counterexample to strong cosmic censorship. (See [185 , 186 , 211 , 70 ] and references therein for background and extended discussions.) The stability of the CH is related to the decay tails of the radiating scalar field. Numerical studies have determined these to be exponential [53 , 70 , 72 ] rather than power law as in AF spacetimes [67 ]. The decay tails of the radiation are necessary initial data for numerical study of CH stability [56 ] and are crucial to the development of the null singularity. Analytic studies had indicated that the CH is stable in RN–de Sitter BH’s for a restricted range of parameters near extremality. However, Brady et al. [55 ] have shown (using linear perturbation theory) that, if one includes the backscattering of outgoing modes which originate near the event horizon, the CH is always unstable for all ranges of parameters. Thus RN–de Sitter BH’s appear not to be a counterexample to strong cosmic censorship. Numerical studies are needed to demonstrate the existence of a null singularity at the CH in nonlinear evolution.

Extension of these studies to AF rotating BH’s has yielded the surprising result that the tails are not necessarily power law and differ for different fields. Frame dragging effects appear to be responsible [150 ].

As a potentially useful approach to the numerical study of singularities, we consider Hübner’s [155 , 156 , 157 ] numerical scheme to evolve on a conformal compactified grid using Friedrich’s formalism [103 ]. He considers the spherically symmetric scalar field model in a 2 + 2 formulation. So far, this code has been used in this context to locate singularities. It was also used to search for Choptuik’s scaling [77 ] and failed to produce agreement with Choptuik’s results [155 ]. This was probably due to limitations of the code rather than inherent problems with the conformal method.