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Covariant constraints on hole-ograhpy
Netta Engelhardt and Sebastian Fischetti 2015 Class. Quantum Grav. 32 195021
Spacetime reconstruction in holography is limited in the presence of strong gravity.
In recent years, it has become clear that there is a deep connection between quantum entanglement and geometry. This mysterious connection has the potential to provide profound insights into the inner workings of a complete theory of quantum gravity. Many concrete hints for how geometry and entanglement are related come from the so-called AdS/CFT duality conjectured by J.Maldacena, which relates certain types of quantum field theories (the “boundary”) to string theory on a negatively-curved spacetime called anti-de Sitter (AdS) space (the “bulk”) of one higher dimension. In a certain limit, the string theory is well-approximated by Einstein’s theory of General Relativity and the dual field theory is strongly interacting. It turns out that in this limit, many field theoretic observables are dual to simple geometric objects in the AdS space.
If the goal is to use this connection to probe quantum gravity, an important issue to understand more deeply is that of bulk reconstruction. Namely: given some field theoretic data, how much of the bulk (gravitational) data can be reconstructed, and how is this reconstruction performed?
At the level of classical Einstein gravity, the most fundamental geometric object to reconstruct is, of course, the geometry itself. A promising approach, dubbed hole-ography, tries to reconstruct geometry from the entropy of regions in the dual field theory. In the specific case of a three-dimensional bulk (dual to a two-dimensional field theory), the basic idea is to reconstruct curves — “holes” — in the spacetime out of geodesics tangent to them, as shown below. By the Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi proposals, the lengths of the these geodesics give the entanglement entropies of subregions of the field theory. These entropies encode entanglement between different spatial regions of the field theory, as their name suggests.
Figure 1: A closed curve, or “hole” on a time slice of three-dimensional AdS. The geodesics tangent to are dual to entanglement entropies in the field theory. These entanglement entropies can be used to reconstruct the length of .
By shrinking the hole to a point or to a convex curve encircling two points, it is possible in certain contexts to use this hole-ographic approach to reconstruct distances in the bulk spacetime from entanglement entropies in the field theory: this is precisely geometry from entanglement.
In what contexts does this reconstruction actually succeed? This is the question we address in our recent CQG paper: we prove no-go theorems that specify regions of generic three-dimensional spacetimes that cannot be reconstructed from this approach, and prove similar theorems about a class of higher-dimensional geometries.
An obvious way in which this hole-ographic reconstruction can fail is if there are regions of the spacetime that geodesics simply don’t reach. In these regions, the geometry clearly can’t be reconstructed from the entanglement entropy of the field theory. In fact, it turns out that the interiors of stationary black holes are such regions: the geometry inside the event horizon can’t be probed from the entanglement entropy of a single dual field theory. However, the story is more complex in the more generic case of dynamical black holes, where geodesics can reach arbitrarily far into the event horizon. Even in this dynamical context, where it may seem that we should be able to carry out a full, hole-ographic reconstruction, our theorems show that hole-ography must fail inside of so-called holographic screens, which are a locally-defined analogue of event horizons.
Why does hole-ography fail inside holographic screens? The reason is subtle. Geodesics can enter holographic screens, but we show that when they do, they cannot “turn around” in a precise sense. In order to reconstruct any smooth curve inside the screen, there must exist geodesics tangent to the curve everywhere. Our theorems show that at least some of these geodesics must crash into a singularity, and therefore do not correspond to the entanglement entropy of any region of the field theory. This implies that the hole-ographic approach as it is currently formulated is insufficient to reconstruct closed curves in the interior of holographic screens, and therefore is insufficient to reconstruct the geometry therein. At best, it can reconstruct only portions of curves, as shown in the accompanying figure.
Figure 2: The plane of the figure is a time slice containing a “hole” (solid black line) inside of a holographic screen . Left: the curve will always have at least two points to which geodesics cannot be tangent; in this particular case, there are four such points, as marked. Right: by our theorems, portions of in a neighborhood of these points cannot be reconstructed from geodesics that reach the boundary on both ends.
While this is a negative result, we speculate on some potentially interesting consequences. For instance, the partial reconstruction shown in the accompanying figure can be thought of as a form of coarse-graining: hole-ography can probe certain features of the interior, but not all. It is a fascinating coincidence that holographic screens, besides their interpretation as local versions of event horizons, also (as their name suggests) have a holographic interpretation via the so-called Bousso bound: the area of a slice of a holographic screen places an upper bound on the total entropy of a region of the spacetime. Since they play a role in constraining how much of the bulk geometry can be reconstructed from field theory entanglement entropy, is there a sense in which the information lost in the coarse-graining can be associated to the holographic screen?
In a different direction, we also comment on possible ways to generalize our results to the perturbatively quantum regime: that is, ways to include quantum mechanical corrections in the bulk gravitational theory. The ultimate goal is to figure out how to reconstruct the bulk gravitational theory from field theory data, and it so happens that our theorems, while formulated for classical gravity, make use of approaches that can be extended to the perturbatively quantum regime. We therefore expect that constraints like the ones we prove and discuss should hold for perturbatively quantum extensions of the hole-ographic reconstruction.
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