Quantization in black hole backgrounds

Quantum field theory in a semiclassical background can be derived as an approximation to quantum gravity from a weak-coupling expansion in the inverse Planck mass. Such an expansion is studied for evolution on “nice slices” in the spacetime describing a black hole of mass $M$. Arguments for a breakdown of this expansion are presented, due to significant gravitational coupling between fluctuations, which is consistent with the statement that existing calculations of information loss in black holes are not reliable. For a given fluctuation, the coupling to subsequent fluctuations becomes of order unity by a time of order $M^3$. Lack of a systematic derivation of the weakly coupled/semiclassical approximation would indicate a role for the nonperturbative dynamics of gravity, and possibly for the proposal that such dynamics has an essentially nonlocal quality.

Figure 1

A sketch of two nice slices in the Eddington-Finkelstein representation of the black hole geometry. The dotted line is the horizon. The stretching effect is easily seen, as the distance from point $P$ to radius $r_o$ on subsequent slices increases linearly with $T$.

Figure 2

An illustration of the comparison of the state on the internal part of a nice slice with and without the effects of the perturbations ${\mathcal{H}}_n$, $n\ge 3$. Vertical marks represent the centers of corresponding wave packets. Initially the mismatch is small, but it increases along the slice. A similar mismatch develops on part of the slice outside the black hole.

Figure 3

A Feynman diagram representation of the coupling between fluctuations at different times. The wavy line is the graviton propagator.

Figure 4

An extended Penrose diagram for de Sitter space. Horizontal lines correspond to $D-1$ spheres which are thought of as nice slices; $N$ and $S$ label the north and south poles, respectively. A fluctuation that is $s$ wave with respect to the observer at $N$ produces a pair of particles (shown as dashed lines) on either side of the horizon. Regions I and III are thus de Sitter space, but region II is described as Schwarzschild-de Sitter. The observer at $O$ is as a result expected to see a significant late-time phase shift in the nice-slice state, relative to the corresponding state without backreaction included.