Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lorentzian-Euclidean singularity-free solutions to gravitational collapse

Sune Rastad Bahn, Michael Cramer Andersen

Abstract

This study explores singularity-free solutions to the static, spherical symmetric Einstein equations with the standard Schwarzschild solution as a boundary condition. Imposing the absence of curvature singularities and requiring differentiability of the time component of the metric leads to a sign change across the horizon, violating the Principle of Equivalence locally. We find a solution within the event horizon with a simple ``cosmological constant'' stress-energy tensor. Considering the impact of sign change to a compact stellar remnant, modeled by an incompressible perfect fluid obeying the Tolman-Oppenheimer-Volkoff equation, we rediscover the same geometry, indicating both mathematical and physical feasibility of the model. We also find a new theoretical limit M/R=3/8, which is lower than the Buchdahl limit of M/R=4/9 for the density of a perfect fluid that will recede behind an event horizon. The equation of state is discussed, and we propose that the final state is described by a Higgs-like free scalar field.

Lorentzian-Euclidean singularity-free solutions to gravitational collapse

Abstract

This study explores singularity-free solutions to the static, spherical symmetric Einstein equations with the standard Schwarzschild solution as a boundary condition. Imposing the absence of curvature singularities and requiring differentiability of the time component of the metric leads to a sign change across the horizon, violating the Principle of Equivalence locally. We find a solution within the event horizon with a simple ``cosmological constant'' stress-energy tensor. Considering the impact of sign change to a compact stellar remnant, modeled by an incompressible perfect fluid obeying the Tolman-Oppenheimer-Volkoff equation, we rediscover the same geometry, indicating both mathematical and physical feasibility of the model. We also find a new theoretical limit M/R=3/8, which is lower than the Buchdahl limit of M/R=4/9 for the density of a perfect fluid that will recede behind an event horizon. The equation of state is discussed, and we propose that the final state is described by a Higgs-like free scalar field.
Paper Structure (12 sections, 15 equations, 1 figure)

This paper contains 12 sections, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Method of the Study
  3. Solving the Einstein Equations
  4. The Schwarzschild solution
  5. A singularity-free solution
  6. Final State of Gravitational Collapse
  7. Incompressible perfect fluid solution
  8. Singularity-free dark compact object solution
  9. Matching the metric across the event horizon
  10. Relation to Other Dark Compact Object Models
  11. Comparison with Astrophysical Observations
  12. Summary and Outlook

Figures (1)

  • Figure 1: Components of the metric in equation (\ref{['lineelement']}) shown with solid curves. The singular inner parts of the Schwarzschild metric are shown with thin dashed curves for comparison.