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Black holes and covariance in effective quantum gravity: A solution without Cauchy horizons

Cong Zhang, Jerzy Lewandowski, Yongge Ma, Jinsong Yang

Abstract

The issue of general covariance in effective quantum gravity models within the Hamiltonian framework is addressed. The previously proposed equations for the covariance condition in spherically symmetric models are explicitly derived. By solving this equation, a new effective Hamiltonian constraint is obtained, incorporating free functions that can account for quantum gravity effects. The resulting spacetime structure is analyzed by specifying the free functions. Remarkably, in this model, the classical singularity is replaced by a region where the metric asymptotically approaches a Schwarzschild-de Sitter one with negative mass. Thus, this new quantum-corrected black hole model avoids the Cauchy horizons presented typically in previously studied models. The covariant approach is also applicable to matter coupling in the models.

Black holes and covariance in effective quantum gravity: A solution without Cauchy horizons

Abstract

The issue of general covariance in effective quantum gravity models within the Hamiltonian framework is addressed. The previously proposed equations for the covariance condition in spherically symmetric models are explicitly derived. By solving this equation, a new effective Hamiltonian constraint is obtained, incorporating free functions that can account for quantum gravity effects. The resulting spacetime structure is analyzed by specifying the free functions. Remarkably, in this model, the classical singularity is replaced by a region where the metric asymptotically approaches a Schwarzschild-de Sitter one with negative mass. Thus, this new quantum-corrected black hole model avoids the Cauchy horizons presented typically in previously studied models. The covariant approach is also applicable to matter coupling in the models.

Paper Structure

This paper contains 17 sections, 1 theorem, 163 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General covariance in classical theory
  3. Constructing spacetimes from physical states
  4. Covariance in Hamiltonian formulation
  5. Conditions of general covariance for effective models
  6. Covariance equations
  7. Solutions to the covariance equations
  8. The spacetime metric from $H_{\rm eff}^{(3)}$ with $\mathcal{R}=0$
  9. Spacetime structure of $\dd s_{(3)}^2$ in a concrete example
  10. Spacetime structure for $GM<m_o$
  11. Spacetime structure for $GM=m_o$
  12. Spacetime structure for $GM>m_o$
  13. Matter coupling
  14. Summary
  15. The metrics associated with $M_{\rm eff}^{(1)}$ and $M_{\rm eff}^{(2)}$
  16. ...and 2 more sections

Key Result

Proposition 1

The totally constrained Hamiltonian theory with the constraint algebra eq:effconstraintalgebra is covariant with respect to the metric $g_{\rho\sigma}^{(\mu)}$ given in eq:effg, namely equation holds for all smeared functions $\alpha$ and smeared vector fields $\beta^x\partial_x$ if and only if

Figures (1)

  • Figure :

Theorems & Definitions (1)

  • Proposition 1