Holographic integral geometry with time dependence. (English) Zbl 1457.83056
Summary: We write down Crofton formulas – expressions that compute lengths of space-like curves in asymptotically \(\mathrm{AdS}_3\) 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 \(\mathrm{AdS}_3\) where null rotations are global symmetries, the Crofton formulas simplify and become integrals over the null planes, which intersect the bulk curve.
MSC:
| 83E05 | Geometrodynamics and the holographic principle |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
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