Unitarity and the Hilbert space of quantum gravity

Abstract

Under the premises that physics is unitary and black hole evaporation is complete (no remnants, no topology change), there must exist a one-to-one correspondence between states on future null and timelike infinity and on any earlier spacelike Cauchy surface (e.g., slices preceding the formation of the hole). We show that these requirements exclude a large set of semiclassical spacetime configurations from the Hilbert space of quantum gravity. In particular, the highest entropy configurations, which account for almost all of the volume of semiclassical phase space, would not have quantum counterparts, i.e. would not correspond to allowed states in a quantum theory of gravity.

Figures (3)

• Figure 1: Penrose diagram of a collapsing spherically symmetric object of mass $M$: as it contracts, a horizon forms behind which eventually all matter falls into the black hole singularity $r=0$. Initially (event A), an outside observer sees a shrinking star; after a while (B), her observations are consistent with the existence of a black hole, i.e. dim red-shifted light from the frozen surface and outgoing thermal Hawking radiation, which causes decrease in the central mass; finally (C), all Hawking radiation has passed the observer and the black hole has evaporated completely, leaving behind empty space with all information in the radiation at $\mathscr{I}^+$ and $i^+$. Also shown is a spacelike Cauchy slice $\Sigma_0$ preceding the black hole. Under our assumptions of unitarity and complete evaporation there exists a one-to-one correspondence between states $\Sigma_0$ (matter+gravity) and states on future infinity $\mathscr{I}^+ \cup i^+$.
• Figure 2: (a) Embedding of the monster configuration $\Sigma_0$ into flat space with one angular dimension suppressed. The "neck" has proper length much bigger than $\left(R-r_0\right)$, due to the huge factor $\epsilon(r)^{-1/2}$, and contains all of the initial entropy $S_{\Sigma_0}$. For $r>R$ the geometry is just that of a Schwarzschild slice with mass $M = M_{ADM}$. (b) The monster's future time evolution is similar to ordinary gravitational collapse (cf. Figure 1): (almost) all matter and entropy, if it was not already initially, will fall behind a horizon (infall of outer layers soon creates trapped surfaces) and form a black hole which then evaporates, radiating away entropy $S_+ \sim M^2 < S_{\Sigma_0}$ past the external observer to future infinity $\mathscr{I}^+ \cup i^+$.
• Figure 3: (a) Embedding of glued Kruskal-FRW initial slice $\Sigma_0$ into flat space with one angular dimension suppressed. $R$ is the proper radial distance from the innermost point and $r=r(R)$ gives the radius of the 2-sphere labeled $R$. Additional or larger closed FRW pieces could be adjoined, and there could also be a second asymptotic Kruskal piece (even with mass parameter different from $M_1$) if the far left were not closed off with a 3-sphere. (b) By considering the rightmost Einstein-Rosen bridge, standard energy conditions suffice to show that a singularity will form and that the external observer will see a black hole of mass $M_1$ whose Hawking radiation then contains potentially much less entropy than was present on $\Sigma_0$. In the case of pressureless dust, the time evolved spacetime can be given analytically as Kruskal spacetimes and FRW universes appropriately sewn together (Oppenheimer-Snyder collapse RelativistsToolkitMTW).