Gravitational collapse in 2+1 dimensional AdS spacetime

TL;DR

This work analyzes gravitational collapse of a massless scalar field in 2+1D AdS spacetime, revealing BTZ-like exterior spacetimes with no scalar hair and a distinct, spacelike interior singularity. Using a detailed Einstein–Klein–Gordon formulation with a tailored metric and singularity excision, the authors identify a continuously self-similar critical solution at the threshold of black hole formation, and quantify a universal scaling exponent $\gamma \approx 1.2$ via subcritical curvature growth. The critical solution remains universal across initial data families, up to a phase shift caused by an angle deficit from a central point particle, indicating robust CSS behavior despite the AdS boundary conditions that prevent simple outgoing radiation. These results illuminate critical phenomena in AdS gravity, echoing BTZ physics while highlighting interior dynamics and potential connections to AdS/CFT, and open avenues for exploring more general couplings and rotational effects.

Abstract

We present results of numerical simulations of the formation of black holes from the gravitational collapse of a massless, minimally-coupled scalar field in 2+1 dimensional, axially-symmetric, anti de-Sitter (AdS) spacetime. The geometry exterior to the event horizon approaches the BTZ solution, showing no evidence of scalar `hair'. To study the interior structure we implement a variant of black-hole excision, which we call singularity excision. We find that interior to the event horizon a strong, spacelike curvature singularity develops. We study the critical behavior at the threshold of black hole formation, and find a continuously self-similar solution and corresponding mass-scaling exponent of approximately 1.2. The critical solution is universal to within a phase that is related to the angle deficit of the spacetime.

Figures (20)

• Figure 1: $\Phi(r,0) = \phi_{,r}(r,0)$ for each family of initial data studied. The three compact families are initially ingoing, thus $\Pi(r,0)=\Phi(r,0)$, while the harmonic function is time-symmetric with $\Pi(r,0)=0$ ($\ell=2/\pi$, so $\mathcal{I}$ is at $r=1$).
• Figure 2: $A(r,0)$ (left-most figure) for a gaussian with $P=0.133051$, as obtained by solving the Hamiltonian constraint with $B(r,0)=0$. This amplitude is used as an example in section \ref{['sing']} when we discuss the singularity structure, so for reference we also show $A(r,0.6)$ and $B(r,0.6)$. Notice in particular how large and negative $B$ is towards the origin, indicating that in this region of the grid we are looking at very small scales in the problem (the proper circumference element is $\bar{r}=\ell\tan(r/\ell) e^B$).
• Figure 3: A plot of $\Phi(r,t)$, the spatial gradient of the scalar field, for sample gaussian initial data with $P=0.1302$. In this case, a black hole is not formed. This plot clearly demonstrates the nature of the Dirichlet boundary conditions on $\phi$ at $\mathcal{I}$ ($r=1$ in these coordinates). Even though a black hole does not form, back reaction is significant here---notice the non-linear interaction between ingoing and outgoing components of the field: when the ingoing and outgoing pulses cross, the ingoing component is amplified, while at the same time the outgoing component is surpressed. The effect is most apparent on this plot at around $t=3$ near the outer boundary; and note that the initial outgoing component of the field is quite small and not visible in the picture.
• Figure 4: Asymptotic mass as a function of pulse amplitude for an initially ingoing gaussian (\ref{['gauss']}) of width $0.05$, centered at $r=0.2$ in a cosmology with $\ell=2/\pi$. For the amplitudes that formed an apparent horizon within the simulation time of $t=2$, the mass estimate at time of AH formation is also shown (it is not clear in the figure but this curve does not touch the asymptotic mass curve). The dashed vertical line, labeled by $P^\star$, is the critical amplitude---see sec. \ref{['sec_crit']}.
• Figure 5: Asymptotic mass as a function of pulse amplitude for the time-symmetric $n=1$ harmonic function (\ref{['harm']}). For the amplitudes that formed an apparent horizon within the simulation time of $t=50$, the mass estimate at time of AH formation is also shown. The dashed vertical line, labeled by $P^\star$, is the critical amplitude as discussed in sec. \ref{['sec_crit']}. Notice the discontinuity of the initial mass estimate curve just to the right of $P^\star$, and compare the gaussian case in Fig. \ref{['gmass']}. The reason for the sudden jump, and difference from the gaussian case, is that around $t=1$ for those amplitudes near $P^\star$ an apparent horizon is close to forming in two locations; to the left of the discontinuity it first forms at larger radii, to the right at smaller (see Fig. \ref{['cr0t0']}).