Soft graviton exchange and the information paradox

Abstract

We show that there is a remarkable soft limit in quantum gravity where the information paradox is readily resolved due to virtual soft graviton exchange on the black hole horizon. This regime is where collision energies satisfy $\sqrt{s}\gg \gamma M_{Pl}$ (with $\gamma \sim M_{Pl}/M_{BH}$) near the horizon. We call this the black hole eikonal phase, in contrast to its flat space analogue where collisions are trans-Planckian. Hawking's geometric optics approximation neglects gravitational interactions near the horizon, and results in thermal occupation numbers in the Bogoliubov coefficients. We show that these interactions are mediated by graviton exchange in $2 \rightarrow 2$ scattering near the horizon, and explicitly calculate the S-matrix non-perturbatively in $M_{Pl}/M_{BH}$ and $\hbar$. This involves a re-summation of infinitely many ladder diagrams near the horizon, all mediated by virtual soft gravitons. The S-matrix turns out to be a pure phase \textit{only} upon this re-summation. The impact parameter $b$ satisfies $L_{Pl}\ll b \lesssim R_S$, where $R_S$ is the Schwarzschild radius; therefore, our results are agnostic of Planckian physics. Our calculation shows that non-renormalisability of gravity is irrelevant for a resolution of the information problem, and is agnostic of any specific ultraviolet completion. In contrast to the flat space eikonal limit, the black hole eikonal phase involves collisions of extremely low energy near the horizon, thereby avoiding firewalls for black holes much larger than Planck size.