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On Modelling the Surfaces of Celestial Bodies in Quantum Gravity

Xavier Calmet, Marco Sebastianutti

TL;DR

This work addresses divergences that arise when modelling stellar surfaces in quantum gravity using discontinuous density profiles. It employs the Vilkovisky--DeWitt unique effective action to derive universal exterior quantum corrections at order $\ell_{\rm P}^2$, solving perturbatively in the compactness $C$ and smoothing the interior density via a modified Tolman VII profile controlled by $\lambda$. The analysis shows that a density profile with $\lambda>1$ yields finite perturbations at the surface, while $\lambda>2$ also regularizes curvature invariants, enabling the exterior solution to extend to $r\ge R_{\rm s}$; the interior-dependence appears as a quantum hair at order $r^{-5}$ in the far-field, and the observable deflection-angle correction is highly suppressed, indicating practical invisibility for stellar objects. The results demonstrate a model-independent, principled way to incorporate quantum gravity corrections in astrophysical contexts by ensuring boundary regularity through smoother density profiles.

Abstract

We discuss how to model the surface of celestial bodies (such as stars) in quantum gravity to ensure the regularity of quantum corrections to classical solutions of general relativity at the surface of such bodies. Specifically, we use the Vilkovisky--DeWitt unique effective action to calculate universal quantum corrections to the exterior metric for a class of stellar models. Previous descriptions, obtained via a Heaviside density profile, are ``pathological'' at the surface of the star due to the divergence of the metric functions and associated curvature invariants. Introducing a modified version of the Tolman VII density profile, we determine the minimal degree of differentiability required for this function to generate regular quantum corrections at the star's surface.

On Modelling the Surfaces of Celestial Bodies in Quantum Gravity

TL;DR

This work addresses divergences that arise when modelling stellar surfaces in quantum gravity using discontinuous density profiles. It employs the Vilkovisky--DeWitt unique effective action to derive universal exterior quantum corrections at order , solving perturbatively in the compactness and smoothing the interior density via a modified Tolman VII profile controlled by . The analysis shows that a density profile with yields finite perturbations at the surface, while also regularizes curvature invariants, enabling the exterior solution to extend to ; the interior-dependence appears as a quantum hair at order in the far-field, and the observable deflection-angle correction is highly suppressed, indicating practical invisibility for stellar objects. The results demonstrate a model-independent, principled way to incorporate quantum gravity corrections in astrophysical contexts by ensuring boundary regularity through smoother density profiles.

Abstract

We discuss how to model the surface of celestial bodies (such as stars) in quantum gravity to ensure the regularity of quantum corrections to classical solutions of general relativity at the surface of such bodies. Specifically, we use the Vilkovisky--DeWitt unique effective action to calculate universal quantum corrections to the exterior metric for a class of stellar models. Previous descriptions, obtained via a Heaviside density profile, are ``pathological'' at the surface of the star due to the divergence of the metric functions and associated curvature invariants. Introducing a modified version of the Tolman VII density profile, we determine the minimal degree of differentiability required for this function to generate regular quantum corrections at the star's surface.

Paper Structure

This paper contains 7 sections, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Unique effective action framework
  3. Modified Tolman VII density profile
  4. Exterior perturbative solutions
  5. Quantum corrections at the surface
  6. Quantum corrections far from the surface
  7. Conclusions

Figures (1)

  • Figure 1: Modified Tolman VII density profile (in units of $\rho_0$) for different values of $\lambda$. Any $\lambda>0$ produces a continuos density profile at $r={R_{\rm s}}$; $\lambda=1$ corresponds to the standard Tolman VII density profile, any $\lambda>2$ corresponds to a $\rho(r)$ of class at least $C^2$.