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On the Null Energy Condition and Causality in Lifshitz Holography

Carlos Hoyos, Peter Koroteev

TL;DR

This work shows that in Lifshitz holography, the bulk null energy condition directly constrains causality in the boundary theory: NEC satisfaction necessitates a bulk local light speed that increases toward the boundary, which is only compatible with Lifshitz exponents $z\ge 1$. Using a WKB analysis, shock-wave propagation, and holographic scalar correlators, the authors demonstrate that NEC violation for $z<1$ leads to superluminal signal propagation and causality breakdown, thereby ruling out $z<1$ Lifshitz holographic duals. The paper also analyzes scalar two-point functions across different $z$ and examines higher-derivative (curvature-squared) gravity corrections to map regions of parameter space where NEC can be violated. Together these results connect bulk energy conditions, bulk light-speed profiles, and boundary causality, providing NEC-based constraints for holographic model-building and c-theorem considerations in nonrelativistic holography.

Abstract

We use a WKB approximation to establish a relation between the wavefront velocity in a strongly coupled theory and the local speed of light in a holographic dual, with our main focus put on systems with Lifshitz scaling with dynamical exponent z. We then use Einstein equations to relate the behavior of the local speed of light in the bulk with the null energy condition (NEC) for bulk matter, and we show that it is violated for Lifshitz backgrounds with z<1. We study signal propagation in the gravity dual and show that violations of the NEC are incompatible with causality in the strongly coupled theory, ruling out as holographic models Lifshitz backgrounds with z<1. We argue that causality violations in z<1 theories will show up in correlators as superluminal modes and confirm this for a particular example with z=1/2. Finally, as an application, we use z<1 solutions to uncover regions of the parameter space of curvature squared corrections to gravity where the NEC can be violated.

On the Null Energy Condition and Causality in Lifshitz Holography

TL;DR

This work shows that in Lifshitz holography, the bulk null energy condition directly constrains causality in the boundary theory: NEC satisfaction necessitates a bulk local light speed that increases toward the boundary, which is only compatible with Lifshitz exponents . Using a WKB analysis, shock-wave propagation, and holographic scalar correlators, the authors demonstrate that NEC violation for leads to superluminal signal propagation and causality breakdown, thereby ruling out Lifshitz holographic duals. The paper also analyzes scalar two-point functions across different and examines higher-derivative (curvature-squared) gravity corrections to map regions of parameter space where NEC can be violated. Together these results connect bulk energy conditions, bulk light-speed profiles, and boundary causality, providing NEC-based constraints for holographic model-building and c-theorem considerations in nonrelativistic holography.

Abstract

We use a WKB approximation to establish a relation between the wavefront velocity in a strongly coupled theory and the local speed of light in a holographic dual, with our main focus put on systems with Lifshitz scaling with dynamical exponent z. We then use Einstein equations to relate the behavior of the local speed of light in the bulk with the null energy condition (NEC) for bulk matter, and we show that it is violated for Lifshitz backgrounds with z<1. We study signal propagation in the gravity dual and show that violations of the NEC are incompatible with causality in the strongly coupled theory, ruling out as holographic models Lifshitz backgrounds with z<1. We argue that causality violations in z<1 theories will show up in correlators as superluminal modes and confirm this for a particular example with z=1/2. Finally, as an application, we use z<1 solutions to uncover regions of the parameter space of curvature squared corrections to gravity where the NEC can be violated.

Paper Structure

This paper contains 7 sections, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Null energy condition and causality
  3. Phase velocity from the WKB approximation.
  4. Growing versus decreasing speed of light.
  5. Causality from shock waves.
  6. Speed of light and the null energy condition.
  7. Scalar two-point correlation functions.

Figures (2)

  • Figure 1: Quantum mechanical potentials for $z=0.2, 0.4, 0.6, 0.8$ as functions of the radial coordinate $\rho$. The plots are ordered by their slopes at large $\rho$, the steepest curve corresponds to $z=0.8$. Spatial momentum $k$ is set to unity together with the bulk curvature scale $L$.
  • Figure 2: A source at the boundary produces a shock wave propagating through the bulk. When the shock wave crosses a static probe a pulse of radiation is sent to the boundary. In the holographic dual the front of radiation produced by the source is determined by the the time and position of the emitted bulk radiation when it reaches the boundary.