$2 \rightarrow 2N$ scattering: Eikonalisation and the Page curve

TL;DR

This paper calculates the scattering amplitude for particles interacting with a black hole, confirming that the time delay matches the Page time, and shows how information re-emerges after this period.

Contribution

It provides an all-orders calculation of black hole scattering amplitudes in the eikonal approximation, linking the time delay to the Page time and information recovery.

Findings

Time delay equals Page time, confirming long-standing expectations.

Scattering processes involve fluctuating metrics, not fixed geometries.

Information begins to re-emerge after Page time.

Abstract

In the spirit of studying the information paradox as a scattering problem, we pose and answer the following questions: i) What is the scattering amplitude for $N$ particles to emerge from a large black hole when two energetic particles are thrown into it? ii) How long would we have to wait to recover the information sent in? The answer to the second question has long been expected to be Page time, a quantity associated with the lifetime of the black hole. We answer the first by evaluating an infinite number of `ladder of ladders' Feynman diagrams to all orders in $M_{Pl}/M_{BH}$. Such processes can generically be calculated in effective field theory in the black hole eikonal phase where scattering energies satisfy $E M_{BH} \gg M^{2}_{Pl}$. Importantly, interactions are mediated by a fluctuating metric; a fixed geometry is insufficient to capture these effects. We find that the characteristic time spent by the particles in the scattering region (the so-called Eisenbud-Wigner time delay) is indeed Page time, confirming the long-standing expectation. This implies that the entropy of radiation continues to increase, after the particles are thrown in, until after Page time, when information begins to re-emerge.