### 5.4 Neutron star collision in axisymmetry

A first investigation of type I critical collapse in an astrophysically motivated scenario was carried out
in [136, 204]. The matter is a perfect fluid with “Gamma law” EOS , where is a
constant, is the pressure, the rest mass density, and the internal energy per rest mass (so that
is the total energy density). The initial data are constructed with for a constant ,
which corresponds to the “cold” (constant entropy) limit of the Gamma law EOS. The initial data
correspond to two identical stars which have fallen from infinity. (The evolution starts at finite
distance, with an initial velocity calculated in the first post-Newtonian approximation). The entire
solution is axisymmetric with an additional reflection symmetry that maps one star to the
other.
The parameter of the initial data that is varied is the mass of the two stars. Supercritical data form a
single black hole, while subcritical data form a single star. The diagnostics given are plots against time of
the lapse and the Ricci scalar at the symmetry centre of the spacetime, and of outgoing l = 2 gravitational
waves. Collapse of the lapse and blowup of the Ricci scalar are taken as indications of black hole formation.
The limited numerical evidence is compatible with type I critical phenomena, with the putative critical
solution showing oscillations with the same period in the lapse and Ricci scalar. For the critical solution to
be exactly time-periodic, it would have to be spherical in order to not lose energy through
gravitational waves, and there is some evidence that indeed it does not radiate gravitational
waves.

Other 1-parameter families of initial data were obtained by fixing the mass of the stars in the first family
very near its critical value and then varying one of the following: the initial separation, the initial speed,
and . For all approximately the same scaling law for the survival time of the critical solution was found.
However, as all these initial data are very close together, this only confirms the validity of the
general perturbation theory explanation of critical phenomena, but does not provide evidence of
universality.

A key question that has not been answered is how shock heating affects the standard critical collapse
scenario. A priori the existence of a universal spherical critical solution is unlikely in the presence of shock
heating as there is no dynamical mechanism to arrive at a universal spherical temperature
profile, in contrast to non-dissipative matter models or vacuum gravity where all equations are
wave-like.