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Singularity and differentiability at the origin of static and spherically symmetric black holes

Tommaso Antonelli, Marco Sebastianutti

Abstract

The divergence of curvature invariants at a given point signals the impossibility of extending the spacetime to that point, with the derivative order of these diverging invariants determining the differentiability class of the considered spacetime. We hereby focus on a general static and spherically symmetric geometry and determine, in the full non-linear regime and in a model-independent way, the conditions that the metric functions must satisfy in order to achieve finiteness of all curvature invariants at the origin. Our findings have direct implications regarding the extendibility of such spacetimes, which we illustrate by making explicit examples of various black hole geometries. This work is structured around a central theorem, which relates the finiteness of curvature invariants at the origin to the leading order behavior and parity properties of the metric functions. The detailed proof of this theorem constitutes the main result of the paper.

Singularity and differentiability at the origin of static and spherically symmetric black holes

Abstract

The divergence of curvature invariants at a given point signals the impossibility of extending the spacetime to that point, with the derivative order of these diverging invariants determining the differentiability class of the considered spacetime. We hereby focus on a general static and spherically symmetric geometry and determine, in the full non-linear regime and in a model-independent way, the conditions that the metric functions must satisfy in order to achieve finiteness of all curvature invariants at the origin. Our findings have direct implications regarding the extendibility of such spacetimes, which we illustrate by making explicit examples of various black hole geometries. This work is structured around a central theorem, which relates the finiteness of curvature invariants at the origin to the leading order behavior and parity properties of the metric functions. The detailed proof of this theorem constitutes the main result of the paper.

Paper Structure

This paper contains 8 sections, 5 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Formulation of the theorem
  3. Proof of the backward direction
  4. Proof of the forward direction
  5. A remark on the action of box on rank-two tensors
  6. Generalizations of the main theorem
  7. Applications and conclusions
  8. Properties of d-even and d-odd functions

Key Result

Theorem 1

Given the metric in Eq. eq:metric, where $A(r)$ and $B(r)$ satisfy the assumptions in Eq. eq:ansatz_metric_functions, all curvature invariants are finite at $r=0$ if and only if $m=n=0$, $b_0=1$, and $\tilde{A}(r)$, $\tilde{B}(r)$ are d-even functions of $r$.

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4