Soft graviton exchange and the information paradox

Abstract

We show that there is a remarkable phase in quantum gravity where gravitational scattering amplitudes mediated by virtual gravitons can be calculated explicitly in effective field theory, when the impact parameter $b$ satisfies $L_{Pl}\ll b \lesssim R_S$, with $R_S$ being the Schwarzschild radius. This phase captures collisions with energies satisfying $\sqrt{s}\gg \gamma M_{Pl}$ (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 $M_{Pl}/M_{BH}$. 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 upon this re-summation and is agnostic of Planckian physics and any specific ultraviolet completion. In contrast to the flat space eikonal limit, the black hole eikonal phase captures collisions of extremely low energy near the horizon.