### 5.2 Axisymmetric vacuum gravity

Abrahams and Evans [1] have numerically investigated black hole formation in axisymmetric vacuum
gravity. They write the metric as
where the lapse , shift components and , and 3-metric coefficients and are functions of
, , and . 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, and are
given as free data, and , , and are determined by solving the Hamiltonian constraint and
the two independent components of the momentum constraint. Afterwards, , , ,
and are evolved, and only is obtained by solving the Hamiltonian constraint. is
obtained by solving the maximal slicing condition for , and and are
obtained from the time derivatives of the quasi-isotropic spatial gauge conditions and
.
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 and in the initial data were chosen to have the same
functional form they would have in a linearised gravitational wave with pure
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 . 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
, with about three echos seen. Here has the echoing property ,
and the same echoing property is expected to hold also for , , , and . 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
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].)