Abstract
We consider the eikonal phase associated with the gravitational scattering of a highly energetic light particle off a very heavy object in AdS spacetime. A simple expression for this phase follows from the WKB approximation to the scattering amplitude and has been computed to all orders in the ratio of the impact parameter to the Schwarzschild radius of the heavy particle. The eikonal phase is related to the deflection angle by the usual stationary phase relation. We consider the flat space limit and observe that for sufficiently small impact parameters (or angular momenta) the eikonal phase develops a large imaginary part; the inelastic cross-section is exactly the classical absorption cross-section of the black hole. We also consider a double scaling limit where the momentum becomes null simultaneously with the asymptotically AdS black hole becoming very large. In the dual CFT this limit retains contributions from all leading twist multi stress tensor operators, which are universal with respect to the addition of higher derivative terms to the gravitational lagrangian. We compute the eikonal phase and the associated Lyapunov exponent in the double scaling limit.
Highlights
The two-point function is determined by the length of spacelike geodesics and is peaked around points connected by null geodesics
There is a null geodesic which gives the dominant contribution. The parameters of this geodesic can be determined from the stationary phase condition — it is precisely the null geodesic whose energy and angular momentum are equal to the pt and pφ parameters of the Fourier transform
We study the propagation of null geodesics in the effective metric derived in [60] and use eq (1.1) to compute the phase shift
Summary
Where ∆t and ∆φ are the time and angular displacements of the null geodesic with the energy pt and angular momentum pφ As we describe, this is a consequence of the WKB approximation to the differential equation which determines the holographic correlator. There is a null geodesic which gives the dominant contribution The parameters of this geodesic can be determined from the stationary phase condition — it is precisely the null geodesic whose energy and angular momentum are equal to the pt and pφ parameters of the Fourier transform. We take a double scaling limit where the impact parameter becomes large (the momentum approaches the lightcone) and at the same time the Schwarzschild radius becomes large in AdS units. We discuss our results in section 5. appendices contain some technical details used in the main text
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