Emergence of the fuzzy horizon through gravitational collapse

For a large enough Schwarzschild black hole, the horizon is a region of space where gravitational forces are weak; yet it is also a region leading to numerous puzzles connected to stringy physics. In this work, we analyze the process of gravitational collapse and black hole formation in the context of light-cone M-theory. We find that, as a shell of matter contracts and is about to reveal a black hole horizon, it undergoes a thermodynamic phase transition. This involves the binding of D0 branes into D2’s, and the new phase leads to large membranes of the size of the horizon. These in turn can sustain their large size through back-reaction and the dielectric Myers effect—realizing the fuzzball proposal of Mathur and the Matrix black hole of M(atrix) theory. The physics responsible for this phenomenon lies in strongly coupled $2+1$ dimensional noncommutative dynamics. The phenomenon has a universal character and appears generic.

Figure 1

The collapse of a thin shell of D0 brane matter as a function of time from (a) to (c); In (b) the radius of the shell $R(t)$ equals that of its horizon at $r_0$ and a D0 black hole emerges.

Figure 2

Cartoons of various geometrical phase transitions: (a) A black hole of horizon size $r_0$ is on a circle of size $R$; the dotted circles denote the images arising from the periodic boundary conditions; as the box size decreases to $R\simeq r_0$, a Gregory-LaFlamme transition to a phase more uniform along the circle is expected. (b) A black string is wrapped on the circle; as the circle shrinks in size to $r_0\gg l_s$, no phase transition is expected; the wrapped string charge is conserved. (c) A ring of black holes shrinks in size; when $R\simeq r_0$, the horizons touch (much like in (a)) and a phase transition is again expected; in this case brane charge of multipole order need not be conserved.

Figure 3

The phase diagram describing the matter making up the collapsing shell in the rest frame of shell. The faint lines on the diagram correspond to various duality transformations explained in Appendix .

Figure 4

The phase diagram of $2+1$ dimensional NCSYM on a 2-sphere of radius $\sqrt{V_2}=\sqrt{v_2}{\alpha }^{\prime }$. The labels on the various curves are referenced in the main text. Dotted lines denote duality transformations, whereas solid lines are expected to describe phase transitions. Curve C is the main black hole formation transition of interest in this work.