Holographic stress tensor of colored Lifshitz spacetimes and hairy black holes

TL;DR

This paper calculates the holographic stress tensor for colored Lifshitz spacetimes, constructs a finite action for Einstein-Yang-Mills models, and verifies thermodynamic relations for hairy black holes with various dynamical exponents.

Contribution

It provides a finite on-shell action and stress tensor for Einstein-Yang-Mills Lifshitz spacetimes, and demonstrates the thermodynamic consistency of hairy black holes without global Yang-Mills charge.

Findings

Finite, conserved stress tensor obeying Ward identities.

Verification of the first law of thermodynamics for hairy black holes.

Confirmation that Lifshitz black holes are hairy without global Yang-Mills charge.

Abstract

We compute the holographic stress tensor of colored Lifshitz spacetimes following the proposal by Ross-Saremi for gravity duals of non-relativistic theories. For a well-defined variational principle, we first construct a finite on-shell action for the Einstein-Yang-Mills model in four dimensions with Lifshitz spacetime as a solution. We then solve the linearised equations of motion and identify the modes that preserve the asymptotically Lifshitz condition. Employing these modes, we also show that the stress tensor is finite, obeying the scaling and the diffeomorphism Ward identities, i.e., conservations laws. As a final application, we evaluate the energy density and the spatial stress tensor of the previously found numerical black hole solutions with various dynamical exponents $z$. The alternative Smarr relation that has been used in Lifshitz black holes and the first law of thermodynamics are shown to hold without a global Yang-Mills charge, indicating the black holes in question are hairy.