Wick rotations in 3D gravity: ML(H2)-spacetimes

TL;DR

This paper develops a theory of Wick rotations in 3D gravity, focusing on ML(H2)-spacetimes, to understand how different curvature spacetimes relate through rescaling guided by cosmological times.

Contribution

It introduces a consistent sector of Wick rotation rescaling in 3D gravity, specifically for ML(H2)-spacetimes, and analyzes their behavior under curvature changes and symmetry considerations.

Findings

Rescaling functions depend only on the cosmological time values.

Analysis of rays from static ML(H2)-spacetimes reveals their initial derivatives.

Cocompact G cases exhibit the tamest behavior compared to non-compact cases.

Abstract

"Ends of hyperbolic 3-manifolds should support canonical Wick Rotations, so they realize effective interactions of their ending globally hyperbolic spacetimes of constant curvature." We develop a consistent sector of WR-rescaling theory in 3D gravity, that, in particular, concretizes the above guess for many geometrically finite manifolds. ML(H2)-spacetimes are solutions of pure Lorentzian 3D gravity encoded by measured geodesic laminations of the hyperbolic plane H2, possibly invariant by any given torsion-free discrete isometry group G. The rescalings which correlate spacetimes of different curvature, as well as the conformal Wick rotations towards hyperbolic structures, are directed by the gradient of the respective canonical cosmological times, and have universal rescaling functions that only depend on their value. We get an insight into the WR-rescaling mechanism by studying rays of ML(H2)-spacetimes emanating from the static case. In particular, we determine the "derivatives" at the starting point of each ray. We point out the tamest behaviour of the cocompact G case against the different general one, even when G is of cofinite area, but non-compact. We analyze brocken T-symmetry of AdS ML(H2)-spacetimes and related earthquake failure. This helps us to figure out the main lines of development in order to achieve a complete WR rescaling theory.