Abstract
We write down Crofton formulas — expressions that compute lengths of space- like curves in asymptotically AdS3 geometries as integrals over kinematic space — which apply when the curve and/or the background spacetime is time-dependent. Relative to their static predecessor, the time-dependent Crofton formulas display several new features, whose origin is the local null rotation symmetry of the bulk geometry. In pure AdS3 where null rotations are global symmetries, the Crofton formulas simplify and become integrals over the null planes, which intersect the bulk curve.
Highlights
Theories and subject to reasonable assumptions detailed below, the correct measure for this ‘counting’ problem has a direct information theoretic meaning on the boundary: it is the conditional mutual information of regions, which are selected by the geodesics
The covariant Crofton formula (3.4) has a number of interesting features, one of which is that there are many such formulas for a single curve! Different formulas that compute the length of the same curve are related to one another by a certain ‘gauge freedom,’ which is generated in the bulk by local null rotations
The further freedom is in choosing a continuous, λ-dependent rapidity parameter, which sets the magnitude of the local null rotation separating the null vector-aligned (NVA) geodesics from the tangent geodesics at each λ
Summary
The setup of this paper is the AdS3/CFT2 correspondence. We assume that the low energy bulk theory is Einstein gravity so that entanglement entropies of CFT intervals are computed by lengths of bulk geodesics [1,2,3,4]. The starting point is the differential entropy formula [14, 15], which expresses the length of a general spacelike bulk curve in terms of lengths of geodesics, i.e. — when applicable — in terms of entanglement entropies of CFT intervals. We review differential entropy as well as other concepts, which will be useful in the remainder of the paper.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
