5.2 Axisymmetric vacuum gravity

Abrahams and Evans [1] have numerically investigated black hole formation in axisymmetric vacuum gravity. They write the metric as
ds2 = − α2dt2 + ϕ4[e2η∕3(dr + βrdt )2 + r2e2η∕3(d 𝜃 + β𝜃dt)2 + e− 4η∕3r2sin2𝜃d φ2], (42 )
where the lapse α, shift components βr and β𝜃, and 3-metric coefficients ϕ and η are functions of r, t, and 𝜃. Axisymmetry 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. On the initial slice, η and Kr 𝜃 are given as free data, and ϕ, r K r, and φ K φ are determined by solving the Hamiltonian constraint and the two independent components of the momentum constraint. Afterwards, η, Kr𝜃, Krr, and K φφ are evolved, and only ϕ is obtained by solving the Hamiltonian constraint. α is obtained by solving the maximal slicing condition K i = 0 i for α, and βr and β 𝜃 are obtained from the time derivatives of the quasi-isotropic spatial gauge conditions 2 g𝜃𝜃 = r grr and gr𝜃 = 0.

In order to keep their numerical grid as small as possible, Abrahams and Evans chose their initial data to be mostly ingoing. The two free functions η and Kr 𝜃 in the initial data were chosen to have the same functional form they would have in a linearised gravitational wave with pure l = 2, m = 0 angular dependence. This ansatz reduced the freedom in the initial data to one free function of advanced time. A specific peaked function was chosen, and only the overall amplitude was varied.

Limited numerical resolution 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 γ ≃ 0.36. The black hole mass was determined 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 Δ â‰ƒ 0.6, with about three echos seen. Here η has the echoing property η(eΔr,eΔt ) = η (r,t), and the same echoing property is expected to hold also for α, ϕ, r β, and − 1 𝜃 r β. In a subsequent paper [2], some evidence for universality of the critical solution, echoing period and critical exponent was given in the evolution of a second family of initial data, one in which η = 0 at the initial time. In this family, black hole masses down to 0.06 of the ADM mass were achieved.

It is striking that at 14 years later, these results have not yet been independently verified. An attempt with a 3-dimensional numerical relativity code (but axisymmetric initial data), using free evolution in the BSSN formulation with maximal slicing and zero shift, to repeat the results of Abrahams and Evans was not successful [4]. The reason could be a combination of rather lower resolution than that of Abrahams and Evans, growing constraint violations, and an inappropriate choice of gauge. (It is now known that BSSN with maximal slicing and zero shift is an ill-posed system [108].)

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