Geodesics, holography and black holes in three dimensions

I have a long-standing interest in understanding the description of black holes in the AdS/CFT correspondence. Black hole spacetimes have non-trivial causal structures, so it is difficult to understand how to extend the usual UV/IR relation to them. In the context of 2+1-dimensional gravity, it is possible to give a complete description of an idealised process of black hole formation, when two point particles collide. We can use the AdS/CFT correspondance to learn how such a process might be described in the field theory.

In [1], Vijay Balasubramanian and I explored the description of these solutions in the dual field theory, by considering the predictions of the AdS/CFT correspondence for the behaviour of observables in the field theory state dual to point particle spacetimes. We found that the predictions for one-point functions in the field theory did not depend on the locations of the particles in the bulk, but the predictions for two-point correlators did. These two-point correlators were calculated in an approximation using the geodesics of the bulk spacetime. In the case which describes black hole formation, we found that the prediction for the two-point correlator involved geodesics passing inside the black hole event horizon.

This result is somewhat surprising, as one would not expect correlators near the boundary to depend on the causally disconnected region behind the horizon. In [2], Jorma Louko, Don Marolf and I examined the approximation employed in [1], and discovered that there could be problems with this approximation in the context of dynamical spacetimes such as that describing black hole formation. We applied the approximation to static black hole spacetimes, obtaining answers in agreement with our expectations. This can be regarded as a validation of the method in the case of static spacetimes. We also found in one of the static black hole spacetimes that the correlators could contain information about the part of the spacetime behind the horizon. However, understanding to what extent our earlier results for the more interesting dynamical cases are valid requires further work.

[1] V. Balasubramanian and S. F. Ross, "Holographic particle detection", Phys. Rev. D 61:044007 (2000), hep-th/9906226.

[2] J. Louko, D. Marolf and S.F. Ross, "On geodesic propagators and black hole holography", hep-th/0002111.