Mixed Gauge-Global Symmetries, Elliptic Modes, and Black Hole Thermodynamics in Ho\v{r}ava-Lifshitz Gravity

TL;DR

This paper investigates the unique mixed gauge-global symmetry in Hořava-Lifshitz gravity, demonstrating its role in resolving thermodynamic inconsistencies of black holes and discussing implications for holography and quantum gravity.

Contribution

It introduces and analyzes the mixed symmetry and charge in Hořava-Lifshitz gravity, showing their importance in black hole thermodynamics and holographic arguments.

Findings

Resolved the first law failure in asymptotically AdS solutions.

Identified the role of the mixed symmetry charge in black hole thermodynamics.

Discussed how elliptic equations affect holographic gauge field assumptions.

Abstract

In Ho\v{r}ava-Lifshitz gravity, a putative consistent theory of quantum gravity for which there is evidence for both black hole thermodynamics and a holographic construction, spacetime is endowed with a preferred dynamical spacelike foliation. The theory has a leaf reparameterization symmetry that is neither global nor local gauge, hyperbolic and elliptic equations of motion, a lack of splittability, and universal horizon black hole solutions. The reparameterization symmetry is ``mixed'': it is a local symmetry in one coordinate yet global on each leaf. More broadly it is an example of both unfree and projectable gauge symmetries. The mixed symmetry and associated charge has not yet been accounted for in calculations of universal horizon thermodynamics in Ho\v{r}ava-Lifshitz gravity. This has led to problems, in particular the failure of the first law in a class of asymptotically AdS solutions where the normal to the leaves of the foliation is not aligned with the time translation Killing vector at infinity. We show how the dynamics of the charge corresponding to this symmetry coupled with the other features above resolves this issue. We then briefly comment how this mixed symmetry, the corresponding charge, and the elliptic equations of motion also conspire to evade recent holographic arguments for only local gauge fields in consistent theories of quantum gravity due to the lack of splittability of the elliptic equation and associated mode.