Critical Phenomena in General Relativity

Consider the Einstein-scalar field equations for the spherically symmetric metric

$ds^2 = -V(u,\lambda)du^2-2d\lambda du + r^2(u,\lambda)\left(d\theta^2+\sin^2\theta d\phi^2\right)$

for which hypersurfaces $u=const$ are out-going null cones having their vertices at $\lambda=r(u,0)=0$ . In these coordinates, an inertial observer is tracing the vertices of the outgoing null cones. The Minkowski spacetime of such inertial observer is given by $V=1$ and $r/\lambda = 1$ . The illustration below shows how the inertial coordinate system at the origin remains connected with a coordinate system at infinity, e.g. in the limit $\lambda\rightarrow \infty$ , during a time evolution when choosing different initial data for the scalar field. In deed, it is seen that spacetime evolves either to a Minkowski spacetime (i.e. is connected with an infinite observer) or to a black hole spacetime (i.e. has no connection with an infinite observer) depending on the value of the initial data parameter $b$ .

Temporal behaviour of $H(u) = \lim_{\lambda\rightarrow \infty}\lambda^{-1}r(u, \lambda)$ for different values of the initial data and as a function of the proper time at the origin. Values $H=1$ correspond to an asymptotic Minkowski frame while $H=0$ indicates that the observer at large radii is hidden behind an event horizon. It also becomes apparent that for values of $b$ very close to a critical value $b^*$ , the field $H(u)$ for times $u<u^*\approx 1.5$ is the same for both the Minkowski or black hole fate of the spacetime. This time $u^*$ is the so-called critical collapse/dispersion time of the scalar field. Its value, $u^*$ , depends on the initial data, but the shape of $H(u)$ for $u\lesssim u^*$ is universally same for any(!) initial data.