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A boundary stress tensor for higher-derivative gravity in AdS and Lifshitz backgrounds

Olaf Hohm, Erik Tonni

TL;DR

This work develops a universal method to define Brown-York boundary stress tensors for curvature-squared gravity by introducing auxiliary tensor fields to reduce derivative orders, enabling a well-posed variational principle and covariant boundary terms. It validates the construction in 3D new massive gravity by computing central charges and BTZ black hole masses, and extends to Lifshitz backgrounds with a covariant counterterm, highlighting both successes (correct central charges, chiral point behavior) and ambiguities (Lifshitz counterterms not uniquely fixed). It provides explicit formulas for the boundary stress tensor and demonstrates finite results after appropriate counterterms, with implications for AdS/CFT and non-relativistic holography. The results clarify the trade-offs between bulk unitarity and boundary energy in higher-derivative gravity and lay groundwork for extending the approach to higher dimensions and additional Lifshitz solutions.

Abstract

We investigate the Brown-York stress tensor for curvature-squared theories. This requires a generalized Gibbons-Hawking term in order to establish a well-posed variational principle, which is achieved in a universal way by reducing the number of derivatives through the introduction of an auxiliary tensor field. We examine the boundary stress tensor thus defined for the special case of `massive gravity' in three dimensions, which augments the Einstein-Hilbert term by a particular curvature-squared term. It is shown that one obtains finite results for physical parameters on AdS upon adding a `boundary cosmological constant' as a counterterm, which vanishes at the so-called chiral point. We derive known and new results, like the value of the central charges or the mass of black hole solutions, thereby confirming our prescription for the computation of the stress tensor. Finally, we inspect recently constructed Lifshitz vacua and a new black hole solution that is asymptotically Lifshitz, and we propose a novel and covariant counterterm for this case.

A boundary stress tensor for higher-derivative gravity in AdS and Lifshitz backgrounds

TL;DR

This work develops a universal method to define Brown-York boundary stress tensors for curvature-squared gravity by introducing auxiliary tensor fields to reduce derivative orders, enabling a well-posed variational principle and covariant boundary terms. It validates the construction in 3D new massive gravity by computing central charges and BTZ black hole masses, and extends to Lifshitz backgrounds with a covariant counterterm, highlighting both successes (correct central charges, chiral point behavior) and ambiguities (Lifshitz counterterms not uniquely fixed). It provides explicit formulas for the boundary stress tensor and demonstrates finite results after appropriate counterterms, with implications for AdS/CFT and non-relativistic holography. The results clarify the trade-offs between bulk unitarity and boundary energy in higher-derivative gravity and lay groundwork for extending the approach to higher dimensions and additional Lifshitz solutions.

Abstract

We investigate the Brown-York stress tensor for curvature-squared theories. This requires a generalized Gibbons-Hawking term in order to establish a well-posed variational principle, which is achieved in a universal way by reducing the number of derivatives through the introduction of an auxiliary tensor field. We examine the boundary stress tensor thus defined for the special case of `massive gravity' in three dimensions, which augments the Einstein-Hilbert term by a particular curvature-squared term. It is shown that one obtains finite results for physical parameters on AdS upon adding a `boundary cosmological constant' as a counterterm, which vanishes at the so-called chiral point. We derive known and new results, like the value of the central charges or the mass of black hole solutions, thereby confirming our prescription for the computation of the stress tensor. Finally, we inspect recently constructed Lifshitz vacua and a new black hole solution that is asymptotically Lifshitz, and we propose a novel and covariant counterterm for this case.

Paper Structure

This paper contains 15 sections, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The generalized Gibbons-Hawking term
  3. Massive gravity in $D=3$
  4. Curvature-squared theories in general dimensions
  5. The boundary stress tensor and its counterterms
  6. Dual CFT and black hole parameters
  7. Central charges
  8. BTZ black holes
  9. New black holes at the chiral point
  10. Lifshitz backgrounds and black holes
  11. Lifshitz vacua and their stress tensor
  12. Asymptotically Lifshitz black holes
  13. Conclusions and discussion
  14. ADM variables and boundary diffeomorphisms
  15. Auxiliary fields in arbitrary dimensions

Figures (1)

  • Figure 1: Lifshitz vacua corresponding to $z_{+}(\lambda)$, indicated by a solid line, and $z_{-}(\lambda)$, indicated by a dashed line, see (\ref{['zsol']}). $P_1$ and $P_2$ are the special points mentioned in the text.