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
parameterized by the lapse , shift components and , and two independent coefficients and in the 3-metric. All are functions of r, t and . The fact that and 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, , , and . 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, and are given as free data, while the remaining components of the initial data, namely , , and , 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 , , , and , are updated from one time step to the next via evolution equations. As many variables as possible, namely and the three gauge variables , and , are obtained at each new time step by solving elliptic equations. These elliptic equations are the Hamiltonian constraint for , the gauge condition of maximal slicing () for , and the gauge conditions and for and (quasi-isotropic gauge).
For definiteness, the two free functions, and , 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 and , 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 . This ansatz (pseudo-linear, ingoing, l =2), reduced the freedom in the initial data to one free function of advanced time, . 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 . 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 , with about three echoes seen. This corresponds to a scale range of about one order of magnitude. By a lucky coincidence, 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 has the echoing property . If the spacetime is DSS in the sense defined above, the same echoing property is expected to hold also for , , and , as one sees by applying the coordinate transformation (12) to (66).
In a subsequent paper , 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 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.
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.
|Critical Phenomena in Gravitational Collapse
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