4.7 Black hole charge and 4 Extensions of the basic 4.5 Critical phenomena and naked

4.6 Beyond spherical symmetry

  Every aspect of the basic scenario: CSS and DSS, universality and scaling applies directly to a critical solution that is not spherically symmetric, but all the models we have described are spherically symmetric. There are only two exceptions to date: A numerical investigation of critical collapse in axisymmetric pure gravity [1Jump To The Next Citation Point In The Article], and studies of the nonspherical perturbations the spherically symmetric perfect fluid [77Jump To The Next Citation Point In The Article] and scalar field [99Jump To The Next Citation Point In The Article] critical solutions. They correspond to two related questions: Are the critical phenomena in the known spherically symmetric examples destroyed already by small deviations from spherical symmetry? And: Are there critical phenomena in gravitational collapse far from spherical symmetry?

4.6.1 Axisymmetric gravitational waves 

The paper of Abrahams and Evans [1] was the first paper on critical collapse to be published after Choptuik's PRL, but it remains the only one to investigate a nonspherically symmetric situation, and therefore also the only one to investigate critical phenomena in the collapse of gravitational waves in vacuum. Because of its importance, we summarize its contents here in some technical detail.

The physical situation under consideration is axisymmetric vacuum gravity. The numerical scheme uses a 3+1 split of the spacetime. The ansatz for the spacetime metric is

  eqnarray1104

parameterized by the lapse tex2html_wrap_inline2323, shift components tex2html_wrap_inline3277 and tex2html_wrap_inline3279, and two independent coefficients tex2html_wrap_inline2325 and tex2html_wrap_inline3283 in the 3-metric. All are functions of r, t and tex2html_wrap_inline3289 . The fact that tex2html_wrap_inline3291 and tex2html_wrap_inline3293 are multiplied by the same coefficient is called quasi-isotropic spatial gauge. The variables for a first-order-in-time version of the Einstein equations are completed by the three independent components of the extrinsic curvature, tex2html_wrap_inline3295, tex2html_wrap_inline3297, and tex2html_wrap_inline3299 . The ansatz limits gravitational waves to one ``polarisation'' out of two, so that there are as many physical degrees of freedom as in a single wave equation. In order to obtain initial data obeying the constraints, tex2html_wrap_inline3283 and tex2html_wrap_inline3295 are given as free data, while the remaining components of the initial data, namely tex2html_wrap_inline2325, tex2html_wrap_inline3297, and tex2html_wrap_inline3299, are determined by solving the Hamiltonian constraint and the two independent components of the momentum constraint respectively. There are five initial data variables, and three gauge variables. Four of the five initial data variables, namely tex2html_wrap_inline3283, tex2html_wrap_inline3295, tex2html_wrap_inline3297, and tex2html_wrap_inline3299, are updated from one time step to the next via evolution equations. As many variables as possible, namely tex2html_wrap_inline2325 and the three gauge variables tex2html_wrap_inline2323, tex2html_wrap_inline3277 and tex2html_wrap_inline3279, are obtained at each new time step by solving elliptic equations. These elliptic equations are the Hamiltonian constraint for tex2html_wrap_inline2325, the gauge condition of maximal slicing (tex2html_wrap_inline3329) for tex2html_wrap_inline2323, and the gauge conditions tex2html_wrap_inline3333 and tex2html_wrap_inline3335 for tex2html_wrap_inline3277 and tex2html_wrap_inline3279 (quasi-isotropic gauge).

For definiteness, the two free functions, tex2html_wrap_inline3283 and tex2html_wrap_inline3295, in the initial data were chosen to have the same functional form they would have in a linearized gravitational wave with pure (l =2, m =0) angular dependence. Of course, depending on the overall amplitude of tex2html_wrap_inline3283 and tex2html_wrap_inline3295, the other functions in the initial data will deviate more or less from their linearized values, as the nonlinear initial value problem is solved exactly. In axisymmetry, only one of the two degrees of freedom of gravitational waves exists. In order to keep their numerical grid as small as possible, Abrahams and Evans chose the pseudo-linear waves to be purely ingoing Popup Footnote . This ansatz (pseudo-linear, ingoing, l =2), reduced the freedom in the initial data to one free function of advanced time, tex2html_wrap_inline3353 Popup Footnote . A suitably peaked function was chosen.

Limited numerical resolution (numerical grids are now two-dimensional, not one-dimensional as in spherical symmetry) allowed Abrahams and Evans to find black holes with masses only down to 0.2 of the ADM mass. Even this far from criticality, they found power-law scaling of the black hole mass, with a critical exponent tex2html_wrap_inline3357 . Determining the black hole mass is not trivial, and was done from the apparent horizon surface area, and the frequencies of the lowest quasi-normal modes of the black hole. There was tentative evidence for scale echoing in the time evolution, with tex2html_wrap_inline3359, with about three echoes seen. This corresponds to a scale range of about one order of magnitude. By a lucky coincidence, tex2html_wrap_inline2377 is much smaller than in all other examples, so that several echoes could be seen without adaptive mesh refinement. The paper states that the function tex2html_wrap_inline3283 has the echoing property tex2html_wrap_inline3365 . If the spacetime is DSS in the sense defined above, the same echoing property is expected to hold also for tex2html_wrap_inline2323, tex2html_wrap_inline2325, tex2html_wrap_inline3277 and tex2html_wrap_inline3373, as one sees by applying the coordinate transformation (12Popup Equation) to (66Popup Equation).

In a subsequent paper [2], universality of the critical solution, echoing period and critical exponent was demonstrated through the evolution of a second family of initial data, one in which tex2html_wrap_inline3375 at the initial time. In this family, black hole masses down to 0.06 of the ADM mass were achieved. Further work on critical collapse far away from spherical symmetry would be desirable, but appears to be held up by numerical difficulty.

4.6.2 Perturbing around spherical symmetry

A different, and technically simpler, approach is to take a known critical solution in spherical symmetry, and perturb it using nonspherical perturbations. Addressing this perturbative question, Gundlach [77Jump To The Next Citation Point In The Article] has studied the generic nonspherical perturbations around the critical solution found by Evans and Coleman [53] for the tex2html_wrap_inline3379 perfect fluid in spherical symmetry. There is exactly one spherical perturbation mode that grows towards the singularity (confirming the previous results [91Jump To The Next Citation Point In The Article, 98]). There are no growing nonspherical modes at all. A corresponding result was established for nonspherical perturbations of the Choptuik solution for the massless scalar field [99].

The main significance of this result, even though it is only perturbative, is to establish one critical solution that really has only one unstable perturbation mode within the full phase space. As the critical solution itself has a naked singularity (see Section  4.5), this means that there is, for this matter model, a set of initial data of codimension one in the full phase space of general relativity that forms a naked singularity. This result also confirms the role of critical collapse as the most ``natural'' way of creating a naked singularity.



4.7 Black hole charge and 4 Extensions of the basic 4.5 Critical phenomena and naked

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
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