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 MPl/MBH. Such processes can generically be calculated in effective field theory in the black hole eikonal phase where scattering energies satisfy EMBH » {M}_{mathrm{Pl}}^2 . 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.
Highlights
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 find that the characteristic time spent by the particles in the scattering region is Page time, confirming the long-standing expectation
Whereas when particle production is considered (N > 1), we find the time spent in the scattering region to be approximately G2M 3, namely Page time
Summary
It was shown in [18, 19, 21] that the even and odd parity graviton modes decouple in the Regge-Wheeler gauge [22]. The interactions between the virtual gravitons and scalar legs are me√diated by the three vertex iγpμpν governed by the coupling constant γ = κR−1 where κ = 8πG. These rules focus on the dynamics of the near-horizon region. It would be interesting to incorporate effects of the classical potential barrier further away, as was done for 2 → 2 scattering in [20] These effects are not important for the = 0 modes that we concern ourselves with in this article
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