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Ultraviolet Behavior of the Wheeler-DeWitt Equation in Horava-Lifshitz Gravity

Takamasa Kanai

TL;DR

This work assesses how ultraviolet modifications in Horava–Lifshitz gravity alter the quantum interiors of black holes, using the minisuperspace Wheeler–DeWitt equation in Kantowski–Sachs-type models. By solving the UV-dominated WDW equation for both the original and analytically continued HL actions, the authors show horizon regularity but action-dependent behavior near the singularity, with exponential suppression in the original action and oscillatory interior dynamics in the analytic continuation. Crucially, allowing the HL scaling parameter $\lambda$ to run away from unity suppresses the GR-like annihilation-to-nothing scenario, suggesting that singularity resolution via annihilation is not a generic feature of UV-complete gravity and depends on RG flow. The results, extended to planar and zero-cosmological-constant cases, indicate a robust qualitative distinction between HL gravity and GR in the deep interior, with implications for understanding quantum singularities in UV-complete theories.

Abstract

We investigate the quantum structure of black hole interiors in Horava-Lifshitz gravity by analyzing the Wheeler-DeWitt equation in minisuperspace. Focusing on the ultraviolet regime, where higher-order spatial curvature terms dominate, we derive analytical solutions in this UV limit for both the original Horava-Lifshitz action and its analytically continued counterpart. We study their behavior near the event horizon and the classical singularity, with particular attention to the interpretation of the wave function in terms of the annihilation-to-nothing scenario proposed in general relativity. In this paper, we have considered cases in which the two-dimensional spatial section is spherical, planar, or hyperbolic, as well as models with positive, negative, or vanishing cosmological constant. In all cases, we find that the terms dominating in the ultraviolet regime, together with the effects of the running scaling parameter, act to suppress the annihilation-to-nothing behavior. These results suggest that, at least within the range explored in this study, the characteristic annihilation-to-nothing behavior does not appear in the ultraviolet regime of Horava-Lifshitz gravity, and provide a new perspective on the understanding of singularity resolution in quantum gravity.

Ultraviolet Behavior of the Wheeler-DeWitt Equation in Horava-Lifshitz Gravity

TL;DR

This work assesses how ultraviolet modifications in Horava–Lifshitz gravity alter the quantum interiors of black holes, using the minisuperspace Wheeler–DeWitt equation in Kantowski–Sachs-type models. By solving the UV-dominated WDW equation for both the original and analytically continued HL actions, the authors show horizon regularity but action-dependent behavior near the singularity, with exponential suppression in the original action and oscillatory interior dynamics in the analytic continuation. Crucially, allowing the HL scaling parameter to run away from unity suppresses the GR-like annihilation-to-nothing scenario, suggesting that singularity resolution via annihilation is not a generic feature of UV-complete gravity and depends on RG flow. The results, extended to planar and zero-cosmological-constant cases, indicate a robust qualitative distinction between HL gravity and GR in the deep interior, with implications for understanding quantum singularities in UV-complete theories.

Abstract

We investigate the quantum structure of black hole interiors in Horava-Lifshitz gravity by analyzing the Wheeler-DeWitt equation in minisuperspace. Focusing on the ultraviolet regime, where higher-order spatial curvature terms dominate, we derive analytical solutions in this UV limit for both the original Horava-Lifshitz action and its analytically continued counterpart. We study their behavior near the event horizon and the classical singularity, with particular attention to the interpretation of the wave function in terms of the annihilation-to-nothing scenario proposed in general relativity. In this paper, we have considered cases in which the two-dimensional spatial section is spherical, planar, or hyperbolic, as well as models with positive, negative, or vanishing cosmological constant. In all cases, we find that the terms dominating in the ultraviolet regime, together with the effects of the running scaling parameter, act to suppress the annihilation-to-nothing behavior. These results suggest that, at least within the range explored in this study, the characteristic annihilation-to-nothing behavior does not appear in the ultraviolet regime of Horava-Lifshitz gravity, and provide a new perspective on the understanding of singularity resolution in quantum gravity.
Paper Structure (10 sections, 44 equations, 1 figure)

This paper contains 10 sections, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Minisuperspace Wheeler-DeWitt equation inside spherical black holes
  3. Minisuperspace WDW equation
  4. Wave-packet solutions and Yeom’s interpretation
  5. Hořava--Lifshitz gravity: basic formulation
  6. Hamiltonian constraint
  7. Spherical and Hyperbolic Black Hole Interiors
  8. Planar Black hole interiors
  9. Non cosmological constant Black hole interiors
  10. Summary and Discussion

Figures (1)

  • Figure 1: Modulus squared of the wave function \ref{['solution2']} for different values of the HL parameter $\lambda$. The left (right) panels correspond to solutions expressed in terms of modified Bessel functions $K$ (Bessel functions $J$). In all panels, we set $A=\sigma=\nu/2=1$ and integrate the wave number over $k\in[-8,8]$.