# Holographic integral geometry with time dependence

## TL;DR

This paper extends Crofton formulas to time-dependent asymptotically AdS$_3$ geometries, enabling the calculation of spacelike curve lengths via integrals over kinematic space, with new features arising from local null rotation symmetry.

## Contribution

It introduces time-dependent Crofton formulas for asymptotically AdS$_3$ geometries, revealing new features linked to null rotation symmetry.

## Key findings

- Derived Crofton formulas for time-dependent backgrounds
- Identified simplifications in pure AdS$_3$ due to null rotation symmetry
- Formulated integrals over null planes for bulk curve lengths

## Abstract

We write down Crofton formulas--expressions that compute lengths of spacelike curves in asymptotically AdS$_3$ 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 AdS$_3$ where null rotations are global symmetries, the Crofton formulas simplify and become integrals over the null planes, which intersect the bulk curve.