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Consistent Gauge Conditions for Dust-Shell Dynamics in Effective Quantum Gravity

Dongxue Qu, Cong Zhang

Abstract

Previous analyses of shocks generated by shell-crossing singularities are affected by inappropriate gauge choices, and no systematic method is available for selecting a consistent gauge. In this work, we develop such a method for effective quantum gravity coupled to a dust shell. We illustrate it in classical general relativity and verify it numerically, with results in agreement with the Israel junction condition. We also show that gauges such as the Painlevé-Gullstrand and Schwarzschild ones are incompatible with the presence of a dust shell when imposed on the whole spatial slice. This explains the difficulties in previous treatments. The framework developed here provides a basis for studying shell-crossing singularities and shock dynamics in generally covariant effective black-hole models.

Consistent Gauge Conditions for Dust-Shell Dynamics in Effective Quantum Gravity

Abstract

Previous analyses of shocks generated by shell-crossing singularities are affected by inappropriate gauge choices, and no systematic method is available for selecting a consistent gauge. In this work, we develop such a method for effective quantum gravity coupled to a dust shell. We illustrate it in classical general relativity and verify it numerically, with results in agreement with the Israel junction condition. We also show that gauges such as the Painlevé-Gullstrand and Schwarzschild ones are incompatible with the presence of a dust shell when imposed on the whole spatial slice. This explains the difficulties in previous treatments. The framework developed here provides a basis for studying shell-crossing singularities and shock dynamics in generally covariant effective black-hole models.
Paper Structure (14 sections, 63 equations, 3 figures)

This paper contains 14 sections, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Phase space structure for vacuum theory
  4. Minimal assumptions on dust shell
  5. The equation constraining the choice of gauge
  6. Application of the method to classical GR
  7. How the jump condition determines the trajectory of the dust shell
  8. Solving the dynamics
  9. Rewriting the equations of motion for numerical investigation
  10. Numerical results
  11. Initial condition
  12. Comparison with the classical Israel junction condition
  13. New Approach Without Separating the Interior and Exterior Regions
  14. Conclusion and outlook

Figures (3)

  • Figure 1: Initial (solid blue) and evolved (dashed red) profiles of $E^I$, $K_I$, $N$, and $N^x$ for $m_i=1/2$ and $m_e=1$, shown at $t=0.4$.
  • Figure 2: Numerical results obtained using the new approach, with the same parameters and initial data as in Fig. \ref{['fig:num1']}, shown at $t=0.4$. The results agree with those obtained from the previous approach up to numerical errors.
  • Figure :