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Regularity of a double null coordinate system for Kerr-Newman-de Sitter spacetimes

Anne T. Franzen, Pedro M. Girão

Abstract

We construct a double null coordinate system $(u,v,θ_\star,φ_\star)$ for Kerr-Newman-de Sitter spacetimes and prove that the two-spheres given by the intersection of the hypersurfaces $u=\mbox{constant}$ and $v=\mbox{constant}$ are $C^\infty$ in Boyer-Lindquist coordinates (including at the "poles"). The null coordinates allow one to immediately extend some results previously proven for Kerr. As an example, we illustrate how Sbierski's result, for the wave equation on the black hole interior, for Reissner-Nordström and Kerr spacetimes, applies to Kerr-Newman-de Sitter spacetimes.

Regularity of a double null coordinate system for Kerr-Newman-de Sitter spacetimes

Abstract

We construct a double null coordinate system for Kerr-Newman-de Sitter spacetimes and prove that the two-spheres given by the intersection of the hypersurfaces and are in Boyer-Lindquist coordinates (including at the "poles"). The null coordinates allow one to immediately extend some results previously proven for Kerr. As an example, we illustrate how Sbierski's result, for the wave equation on the black hole interior, for Reissner-Nordström and Kerr spacetimes, applies to Kerr-Newman-de Sitter spacetimes.

Paper Structure

This paper contains 40 sections, 20 theorems, 401 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A double null coordinate system
  3. Construction of the double null coordinate system
  4. The coordinates $r_\star$ and $\theta_{\star}$
  5. The metric in $(t,r_\star,\theta_{\star},\phi)$ coordinates
  6. Definition of $\phi_{\star}$ and the metric in double null coordinates $(u,v,\theta_{\star},\phi_{\star})$
  7. Normals to hypersurfaces and volume elements
  8. Regularity of the change of coordinates
  9. The coordinate $\theta_{\star}$ is well defined and continuous
  10. The derivative of the function defining $\theta_\star$
  11. The coordinates $r$ and $\theta$ as functions of $r_\star$ and $\theta_\star$
  12. First partial derivatives
  13. Higher order derivatives
  14. Regularity at $\theta_\star=0$ and $\theta_\star=\pi$
  15. Regularity of the metric.
  16. ...and 25 more sections

Key Result

Lemma 2.4

$\theta_{\star}$ is well defined.

Figures (7)

  • Figure 1: Behavior of the coordinates $u$ and $v$.
  • Figure 2: The graph of $l$ bounds the region where the parameters $\alpha=\frac{r_+}{r_-}$, $\epsilon=\Lambda a^2$ and $\gamma=\Lambda e^2$ can vary.
  • Figure 3: Sketch of the graphs of $l(\,\cdot\,,0)$, $l(\,\cdot\,,0.05)$, $l(\,\cdot\,,0.1)$, $l(\,\cdot\,,0.15)$, $l(\,\cdot\,,0.20)$, $l(\,\cdot\,,0.24)$.
  • Figure 4: The graph of $l(\,\cdot\,,1/8)$.
  • Figure 5: Sketch the graphs of $r_-$, $r_+$ and $r_c$, for $\Lambda e^2=\gamma=1/8$ and $\Lambda a^2=\epsilon=(21-6\sqrt{12+\gamma})/2$.
  • ...and 2 more figures

Theorems & Definitions (44)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 34 more