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Fake Schwarzschild and Kerr black holes

Hideki Maeda

TL;DR

The paper addresses whether black holes can have interior structures that are qualitatively different from the standard Schwarzschild or Kerr solutions while remaining observationally indistinguishable from them. It develops exact constructions by attaching a dynamical interior region with a carefully chosen matter content to Schwarzschild or Gürses–Gürsey Kerr exteriors at Killing horizons, enforcing regular $C^{1,1}$ matching without lightlike thin shells. The main contributions include explicit fake Schwarzschild interiors using Semiz ($\chi=-\tfrac{1}{5}$) and Whittaker ($\chi=-\tfrac{1}{3}$) solutions, plus rotating fake Kerr models via Gürses–Gürsey with three mass functions, and a detailed analysis of energy conditions, horizon regularity, and causal structure. The results show that NEC/SEC can be satisfied while WEC/DEC are violated in many interior configurations, and in the rotating case DEC violations clash with conservation theorems, raising questions about physical realizability and stability of such interiors. Overall, exterior observables and thermodynamics remain identical to the standard black holes, highlighting intriguing possibilities for interior diversity within general relativity and motivating further study of stability and physical viability.

Abstract

We present exact solutions describing a fake Schwarzschild black hole that cannot be distinguished from the Schwarzschild black hole by observations. They are constructed by attaching a spherically symmetric dynamical interior solution with a matter field to the Schwarzschild exterior solution at the event horizon without a lightlike thin shell. The dynamical region inside a Killing horizon of a static spherically symmetric perfect-fluid solution obeying an equation of state $p=χρ$ for $χ\in[-1/3,0)$ can be the interior of a fake Schwarzschild black hole. The matter field inside such a black hole is an anisotropic fluid that violates at least the weak energy condition and can be interpreted as a spacelike (tachyonic) perfect fluid. While the author constructed the first model of fake Schwarzschild black holes using Semiz's solution for $χ=-1/5$, we present another one using Whittaker's solution for $χ=-1/3$ in this paper. We also present a model of fake Kerr black holes whose interior is filled with a different matter field violating only the dominant energy condition near the event horizon. Because it contradicts the conservation theorem, this configuration of black holes is, in fact, precluded by the dominant energy condition.

Fake Schwarzschild and Kerr black holes

TL;DR

The paper addresses whether black holes can have interior structures that are qualitatively different from the standard Schwarzschild or Kerr solutions while remaining observationally indistinguishable from them. It develops exact constructions by attaching a dynamical interior region with a carefully chosen matter content to Schwarzschild or Gürses–Gürsey Kerr exteriors at Killing horizons, enforcing regular matching without lightlike thin shells. The main contributions include explicit fake Schwarzschild interiors using Semiz () and Whittaker () solutions, plus rotating fake Kerr models via Gürses–Gürsey with three mass functions, and a detailed analysis of energy conditions, horizon regularity, and causal structure. The results show that NEC/SEC can be satisfied while WEC/DEC are violated in many interior configurations, and in the rotating case DEC violations clash with conservation theorems, raising questions about physical realizability and stability of such interiors. Overall, exterior observables and thermodynamics remain identical to the standard black holes, highlighting intriguing possibilities for interior diversity within general relativity and motivating further study of stability and physical viability.

Abstract

We present exact solutions describing a fake Schwarzschild black hole that cannot be distinguished from the Schwarzschild black hole by observations. They are constructed by attaching a spherically symmetric dynamical interior solution with a matter field to the Schwarzschild exterior solution at the event horizon without a lightlike thin shell. The dynamical region inside a Killing horizon of a static spherically symmetric perfect-fluid solution obeying an equation of state for can be the interior of a fake Schwarzschild black hole. The matter field inside such a black hole is an anisotropic fluid that violates at least the weak energy condition and can be interpreted as a spacelike (tachyonic) perfect fluid. While the author constructed the first model of fake Schwarzschild black holes using Semiz's solution for , we present another one using Whittaker's solution for in this paper. We also present a model of fake Kerr black holes whose interior is filled with a different matter field violating only the dominant energy condition near the event horizon. Because it contradicts the conservation theorem, this configuration of black holes is, in fact, precluded by the dominant energy condition.

Paper Structure

This paper contains 19 sections, 3 theorems, 89 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Static spherically symmetric solutions with Killing horizons
  4. Matter fields and energy conditions
  5. Whittaker perfect-fluid solution for $p=-\rho/3$
  6. Comoving quasi-global coordinates
  7. Global structure
  8. Fake Schwarzschild black holes
  9. Semiz interior for $\chi=-1/5$
  10. Whittaker interior for $\chi=-1/3$
  11. Fake Kerr black holes
  12. Gürses-Gürsey metric
  13. Violation of the dominant energy condition
  14. Three models with different mass functions
  15. Non-rotating case
  16. ...and 4 more sections

Key Result

Proposition 1

Consider a non-vacuum solution described by the metric (metric-Buchdahl-v) to the Einstein equations with a perfect fluid (EFE) obeying a linear equation of state $p=\chi\rho$ with $\chi= -1/(1+2N)$ where $N\in\mathbb{N}$ and suppose that the metric is $C^\infty$ at a non-degenerate Killing horizon

Figures (5)

  • Figure 1: Penrose diagrams of the maximally extended Whittaker spacetime with $M>0$ for (a) $\alpha>0$ and (b) $-1/(4M^2)<\alpha<0$ in the case where the values of $\alpha$ and $M$ in the regions I, II, III, and IV are the same. A zig-zag line is the curvature singularity at $r=0$. A dashed line in (a) corresponds to $r=r_{\rm b}$. The symbol $\Im^{+(-)}$ in (b) stands for a future (past) null infinity. Note that the Killing horizon $r=r_{\rm h}$ in (a) is not an event horizon due to the absence of null infinity.
  • Figure 2: A Penrose diagram of a fake Schwarzschild black hole with a Semiz dynamical interior (shaded). The metric is $C^{2,1}$ on the Killing horizon $x=x_{\rm h}(=2M_+)$.
  • Figure 3: A Penrose diagram of a fake Schwarzschild black hole with a Semiz dynamical interior only in the region II (shaded). A curve in the region II is the orbit of a fluid element of a spacelike perfect fluid (\ref{['T-spacelike']}).
  • Figure 4: Penrose diagrams of a fake Schwarzschild black hole with a Whittaker dynamical interior (shaded) (a) in the regions II and IV and (b) only in the region II. The metric in the single-null coordinates (\ref{['metric-Buchdahl-v+']}) is $C^{1,1}$ on the Killing horizon between the Schwarzschild and Whittaker regions.
  • Figure 5: Penrose diagrams of a Schwarzschild interior spacetime attached on the Killing horizon to a Whittaker exterior spacetime (shaded) in the case where the values of $\alpha$ and $M$ are the same in the regions I and III and satisfy (a) $\alpha>0$ and (b) $-1/(4M^2)<\alpha<0$. While the Killing horizon is an event horizon in (b), it is not in (a) due to the absence of null infinity.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3