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Effective Dynamics of Spherically Symmetric Static Spacetime

Etera R. Livine, Yuki Yokokura

Abstract

In general relativity, the Einstein equations provide spherically symmetric static spacetimes with dynamics defined as an evolution along the radial coordinate $r$. The geometrical sector becomes a one-dimensional mechanical system, with the Misner-Sharp mass and lapse as canonically conjugate variables, and a vanishing Hamiltonian for pure gravity. Coupling classical or quantum matter fields, or introducing (quantum) corrections to general relativity, then generate a non-vanishing effective Hamiltonian, leading to non-trivial evolutions of the mass and lapse. We illustrate this mechanism through various examples of classical matter fields and identify Hamiltonians describing the effective dynamics of gravity coupled to perfect fluids with linear barotropic equation of state. Finally, we derive effective Hamiltonians that reproduce the gravitational semi-classical dynamics coupled to renormalized quantum matter fields and discuss the conditions for which the singularity at $r=0$ is resolved. In particular, we find a singularity-free black-hole-like solution, stabilized by quantum matter, smoothly transitioning from a bulk with constant negative Ricci scalar to the standard outside Schwarzschild metric. This opens new possibilities for the modeling of both semi-classical corrections and deep quantum effects on the interior structure of self-gravitating compact objects and black holes.

Effective Dynamics of Spherically Symmetric Static Spacetime

Abstract

In general relativity, the Einstein equations provide spherically symmetric static spacetimes with dynamics defined as an evolution along the radial coordinate . The geometrical sector becomes a one-dimensional mechanical system, with the Misner-Sharp mass and lapse as canonically conjugate variables, and a vanishing Hamiltonian for pure gravity. Coupling classical or quantum matter fields, or introducing (quantum) corrections to general relativity, then generate a non-vanishing effective Hamiltonian, leading to non-trivial evolutions of the mass and lapse. We illustrate this mechanism through various examples of classical matter fields and identify Hamiltonians describing the effective dynamics of gravity coupled to perfect fluids with linear barotropic equation of state. Finally, we derive effective Hamiltonians that reproduce the gravitational semi-classical dynamics coupled to renormalized quantum matter fields and discuss the conditions for which the singularity at is resolved. In particular, we find a singularity-free black-hole-like solution, stabilized by quantum matter, smoothly transitioning from a bulk with constant negative Ricci scalar to the standard outside Schwarzschild metric. This opens new possibilities for the modeling of both semi-classical corrections and deep quantum effects on the interior structure of self-gravitating compact objects and black holes.

Paper Structure

This paper contains 29 sections, 141 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spherically Symmetric Spacetimes
  3. Pure gravity
  4. Effective Hamiltonian
  5. Classical Matter Fields
  6. Cosmological constant
  7. Massless scalar field
  8. Maxwell field
  9. Non-linear Electromagnetic Theory
  10. General structure of $H_{\mathrm{eff}}$
  11. Compatibility with Einstein equations
  12. Linear Equations of State
  13. Basic Hamiltonian Ansatz
  14. Polynomial ansatz in powers of mass
  15. Effective Hamiltonians for Fluid Matter
  16. ...and 14 more sections

Figures (4)

  • Figure 1: The dense object, gravitational condensate, represented by the KY metric \ref{['metric_KY']}. The central region beyond the semi-classical approximation \ref{['semi_Einstein']} is unclear yet, but it has been checked at least that no large singularity exists.
  • Figure 2: Mass function $a(r)$ for $\sigma=1$: \ref{['ab_KY_reg']} (blue) and \ref{['ab_KY']} (orange). Note that \ref{['ab_KY']} is originally applicable only for $r\gg l_p$, and we here extrapolate it to $r=0$ formally.
  • Figure 3: $- {\cal T}{}_t{}^t(r)= {\cal T}{}_r{}^r(r)$ in \ref{['J_reg1']}(blue), ${\cal T}{}_\theta{}^\theta(r)$ in \ref{['J_reg2']}(green), and (an extrapolation of) $\langle\Psi|-T{}_t{}^t(r)|\Psi\rangle_{KY}=\langle\Psi|T{}_r{}^r(r)|\Psi\rangle_{KY}|_{\eta=1}$ in \ref{['EMT_KY']} (orange). $8\pi G=1$, $\sigma=1$.
  • Figure 4: The interior structure of the quantum black hole described by the improved KY metric \ref{['metric_KY_reg']}.