Near-critical spacetimes and naked singularities

3.4 Near-critical spacetimes and naked singularities

Choptuik’s results have an obvious bearing on the issue of cosmic censorship (see [203 ] for a general review of cosmic censorship). Roughly speaking, fine-tuning to the black hole threshold provides a set of data which is codimension one in the space of generic, smooth, asymptotically flat initial data, and whose time evolution contains at least the point singularity of the critical solution. The cosmic censorship hypothesis must therefore be formulated as “generic smooth initial data for reasonable matter do not form naked singularities”. Here we look at the relation between fine-tuning and naked singularities in more detail.

Christodoulou [63 ] proves rigorously that naked singularity formation is not generic, but in a rather larger function space, functions of bounded variation, than one would naturally consider. In particular, the instability of the naked singularity found by Christodoulou is not differentiable on the past light cone. This is unnatural in the context of critical collapse, where the naked singularity can arise from generic (up to fine-tuning) smooth initial data, and the intersection of the past light cone of the singularity with the initial data surface is as smooth as the initial data elsewhere. It is therefore not clear how this theorem relates to the numerical and analytical results strongly indicating that naked singularities are codimension-1 generic within the space of smooth initial data.

First, consider the exact critical solution. The lapse $α$ defined by Equation (20 ) is bounded above and below in the critical solution. Therefore the redshift measured between constant $r$ observers located at any two points on an outgoing radial null geodesic in the critical spacetime to the past of the CH is bounded above and below. Within the exact critical solution, a point with arbitrarily high curvature can therefore be observed from a point with arbitrarily low curvature. Next consider a spacetime where the critical solution in a central region is smoothly matched to an asymptotically flat outer region such that the resulting asymptotically flat spacetime contains a part of the critical solution that includes the singularity and a part of the CH. In this spacetime, a point of arbitrarily large curvature can be seen from $ℐ +$ with finite redshift. This is illustrated in Figure 4 .

Now consider the evolution of asymptotically flat initial data that have been fine-tuned to the black hole threshold. The global structure of such spacetimes has been investigated numerically in [111 , 77 , 80 , 182 ]. Empirically, these spacetimes can be approximated near the singularity by Equation (6 ). Almost all perturbations decay as the singularity is approached and the approximation becomes better, until the one growing perturbation (which by the assumption of fine-tuning starts out small) becomes significant. A maximal value of the curvature is then reached which is still visible from $ℐ +$ and which scales as [84 ]

Finally, consider the limit of perfect fine-tuning. The growing mode is then absent, and all other modes decay as the naked singularity is approached. Note that Equation (6

) suggests this is true regardless of the direction (future, past, or spacelike, depending on the value of x) in which the singularity is approached. This seems to be in conflict with causality: The issue is sensitively connected to the completeness of the modes Z_i and to the stability of the CH, and requires more investigation.

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Figure 4: Conformal diagram of the critical solution matched to an asymptotically flat one (RC: regular centre, S: singularity, SSH: self-similarity horizon). Curved lines are lines of constant coordinate $τ$ , while converging straight lines are lines of constant coordinate x. Let the initial data on the Cauchy surface CS be those for the exact critical solution out to the 2-sphere R, and let these data be smoothly extended to some data that are asymptotically flat, so that the future null infinity $ℐ +$ exists. To the past of the matching surface MS the solution coincides with the critical solution. The spacetime cannot be uniquely continued beyond the Cauchy horizon CH. The redshift from point A to point B is finite by self-similarity, and the redshift from B to C is finite by asymptotic flatness.

A complication in the supercritical case has been pointed out in [182 ]. In the literature on supercritical evolutions, what is quoted as the black hole mass is in fact the mass of the first apparent horizon (AH) that appears in the time slicing used by the code (spacelike or null). The black hole mass can and generically will be larger than the AH mass when it is first measured because of matter falling in later, and the region of maximal curvature may well be inside the event horizon, and hidden from observers at $+ ℐ$ (see Figure 5 for an illustration.) The true black hole mass can only be measured at $ℐ +$ , where it is defined to be the limit of the Bondi mass $mB$ as the Bondi time $uB → ∞$ . This was implemented in [182 ]. Only one family of initial data was investigated, but in this family it was found that $m B$ converges to 10^–4 of the initial Bondi mass in the fine tuning limit. More numerical evidence would be helpful, but the result is plausible. As the underlying physics is perfectly scale-invariant in the massless scalar field model, the minimum mass must be determined by the family of initial data through the infall of matter into the black hole. Simulations of critical collapse of a perfect fluid in a cosmological context show a similar lower bound [119 ] due to matter falling back after shock formation, but this may not be true for all initial data [160 ].

Figure 5: The final event horizon of a black hole is only known when the infall of matter has stopped. Radiation at 1 collapses to form a small black hole which settles down, but later more radiation at 2 falls in to give rise to a larger final mass. Fine-tuning of a parameter may result in $m1 ∼ (p − p∗)γ$ , but the final mass $m2$ would be approximately independent of $p$ .