What Exactly is the Information Paradox?

Abstract

The black hole information paradox tells us something important about the way quantum mechanics and gravity fit together. In these lectures I try to give a pedagogical review of the essential physics leading to the paradox, using mostly pictures. Hawking's argument is recast as a `theorem': if quantum gravity effects are confined to within a given length scale and the vacuum is assumed to be unique, then there will be information loss. We conclude with a brief summary of how quantum effects in string theory violate the first condition and make the interior of the hole a `fuzzball'.

Figures (17)

• Figure 1: (a) The potential characterizing a given fourier mode, and the vacuum wavefunction for this potential. (b) If the spacetime distorts, the potential changes to a new one, with its own vacuum wavefunction. (c) If the potential changes suddenly, we have the new potential but the old wavefunction, which will not be the vacuum wavefunction for this changed potential; thus we will see particles.
• Figure 2: The Penrose diagram for a black hole (without the backreaction effects of Hawking evaporation). Null rays are straight lines at $45^o$. Thus we see that the horizon is a null surface. Hawking radiation collects at future null infinity.
• Figure 3: A schematic picture of the dotted box in fig.\ref{['matfsix']}. The horizon has been rotated to be vertical. One coordinate is $r$. The other axis has been called $\tau$, but there is no canonical choice of $\tau$ (the metric will degenerate at the horizon anyway if we try to make it independent of $\tau$). We see that the null geodesics on the two sides of the horizon move away from $r=2GM$ as they evolve.
• Figure 4: Constructing a slicing of the black hole geometry. For $r>3GM$ we have the part $S_{out}$ as a $t=constant$ slice. The 'connector' part $S_{con}$ is almost the same on all slices, and has a smooth intrinsic metric as the surface crosses the horizon. The inner part of the slice $S_{in}$ is a $r=constant$ surface, with the value of $r$ kept away from the singularity at $r=0$. The coordinate $\tau$ is only schematic; it will degenerate at the horizon.
• Figure 5: The slices of fig.\ref{['matfthree']} redrawn in a different way to show the changes from one slice to the next.