String Theory and Black Hole Complementarity

Is string theory relevant to the black hole information problem? This is an attempt to clarify some of the issues involved. In spite of the great effort that the black hole information problem has inspired, the situation has in some ways changed little since the original work of Hawking. The three principal alternatives (that information is lost, stored in a remnant, or emitted with the Hawking radiation) remain, none having been convincingly ruled out or shown to be consistent. The issues have been sharpened, but there is no consensus. Perhaps the most novel proposal is the principle of black hole complementarity as realized in string theory. This talk is about an attempt to understand these ideas, and is based on work in collaboration with Lowe, Susskind, Thorlacius, and Uglum. 1. The Nice Slice Argument Before we turn to string theory itself let us ask, do we expect the Hawking radiation to depend on short distance (Planck scale) physics? Even on this basic question there are vociferous differences of opinion, and it is easy to see why. On the one hand, the horizon of a macroscopic black hole is a very smooth place. The tidal forces need be no larger than in this room, and we would expect therefore to be able to use low energy effective field theory. On the other hand, many (some would claim all) derivations of the Hawking radiation make explicit reference to ridiculously large energies, greater than the mass of the universe, and assume for example that free field theory is valid at these energies. So, can we derive the Hawking radiation in a way that takes advantage of the smoothness of the geometry? It seems that the right way to do this is in a Hamiltonian framework, pushing forward the state of the system on a series of spacelike surfaces. In order to use low energy field theory everywhere, the slices need to be smooth, without large curvatures or accelerations, and any matter (the asymptotic observer, the infalling body) must be moving with modest velocity in the local frame defined by the slice. We will refer to these as “nice slices.” To construct one family of nice slices, let us describe the Schwarzschild black hole in Kruskal-Szekeres coordinates, which we will call (x, x) rather than the usual (U, V ). In these coordinates the singularity is at xx = 16GM, and the event horizon is the surface x = 0 (x increases to the upper left). We construct a spacelike surface composed of two pieces. The first piece is the left half (x < x) of the hyperbola xx = R. This is chosen to be inside the horizon but far from the singularity, so the geometry is still smooth; for example let R = 4GM. The second piece of the nice slice is the half-line x + x = 2R for x > x. The slice is shown in fig. 1. At large distance this slice is asymptotic to the constant time surface t = 0. The slice can be pushed forward and backward in time by using the Killing symmetry of the black hole geometry, x → xe x− → xe . (1) Since the nice slices are asymptotic to surfaces of constant Schwarzschild time, they can be parametrized by t. The full set of slices can then be written xx = R , x− < ex , ex + ex = 2R , x− > ex . (2) The join between the line segment and the hyperbola on each slice should be smoothed to avoid a large extrinsic curvature gradient there. It is not hard to check that the velocity (and energy) of an infalling particle, as measured locally in terms of the time coordinate orthogonal to the nice slice, remain small even as it passes through the horizon. For a black hole formed by collapse, one can join the nice slices smoothly onto a set of smooth slices in the interior of the collapsing body. Also, as the black hole evaporates the background geometry changes. The nice slices can be adjusted along with the change in the geometry until very late in the evaporation when the curvature becomes large. Starting with the initial diffuse matter from which the black hole formed, one can evolve the state of the system forward on such nice slices until the evaporation is nearly complete and the curvature becomes large. For a large black hole, almost all of the original mass will have been converted to Hawking radiation, which will be outgoing on the exterior part of the nice slice. By construction the geometry changes smoothly from slice to slice, so the adiabatic theorem implies that only very low-energy degrees of freedom (E ∼ 1/GM) are excited from their ground state in the Hawking emission process. Thus, the state on the last nice slice is obtained from the initial diffuse state using only low energy field theory. Sin gul arit y Ho riz on , x = 0