Chaos in matrix models and black hole evaporation

Is the evaporation of a black hole described by a unitary theory? In order to shed light on this question—especially aspects of this question such as a black hole’s negative specific heat—we consider the real-time dynamics of a solitonic object in matrix quantum mechanics, which can be interpreted as a black hole (black zero-brane) via holography. We point out that the chaotic nature of the system combined with the flat directions of its potential naturally leads to the emission of D0-branes from the black brane, which is suppressed in the large $N$ limit. Simple arguments show that the black zero-brane, like the Schwarzschild black hole, has negative specific heat, in the sense that the temperature goes up when it evaporates by emitting D0-branes. While the largest Lyapunov exponent grows during the evaporation, the Kolmogorov-Sinai entropy decreases. These are consequences of the generic properties of matrix models and gauge theory. Based on these results, we give a possible geometric interpretation of the eigenvalue distribution of matrices in terms of gravity. Applying the same argument in the M-theory parameter region, we provide a scenario to derive the Hawking radiation of massless particles from the Schwarzschild black hole. Finally, we suggest that by adding a fraction of the quantum effects to the classical theory, we can obtain a matrix model whose classical time evolution mimics the entire life of the black brane, from its formation to the evaporation.

Figure 1

A possible geometric interpretation of the eigenvalue distribution.

Figure 2

A phenomenological model of the black zero-brane/black hole.

Figure 3

The observed numerical value of $c_{N,d}/(2N-3)$ as a function of $N$, for several values of $N$ and $d$. Thin black lines indicate where $c_{N,d}=(d-1)(2N-3)$ as in (b6). We fit a line to every span of points to the right of the maximum of the corresponding numerical distribution and constructed a weighted histogram according to goodness of fit. The best values correspond to the maximum of that histogram, while the error bars were determined by finding where that histogram fell by a factor of $e$ to either side.

Figure 4

The observed numerical distribution as a function of $r$ for $N=2$, $d=3$, 4, 6, 9. The powers $c_{2,d}$ take the same values within error. We divide by $d$ simply to visually offset the different distributions from one another.