Non existence of closed and bounded null geodesics in Kerr spacetimes

Abstract

The Kerr-star spacetime is the extension over the horizons and in the negative radial region of the slowly rotating Kerr black hole. It is known that below the inner horizon, there exist both timelike and null (lightlike) closed curves. Nevertheless, we prove that null geodesics can be neither closed nor even contained in a compact subset of the Kerr-star spacetime.

Figures (18)

• Figure 1: This picture shows a $t$-slice $\{t\}\times\mathbb{R} \times S^2$, with the radius drawn as $e^r$, so that $r=-\infty$ is at the center of the figure. The Ergoregion$\{\mathbf{g}(\partial_t,\partial_t)>0\}$ (at fixed time $t$) is the region between the purple ellipsoids in which $\partial_t$ becomes spacelike.
• Figure 2: For $Q<0$ null geodesics, $R(r)$ has either zero or two negative roots (multiplicity two is allowed).
• Figure 3: All the geodesic types which have to be studied.
• Figure 4: Plot of $R(r)$ in the case $E=0,\; L^2+Q\neq 0,\; Q>0$ with $a=3,M=6,L=2,Q=4$.
• Figure 5: Plot of $R(r)$ in the case $E=0,\; L^2+Q\neq 0,\; Q=0$ with $a=3,M=6,L=2,Q=0$.

Theorems & Definitions (87)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Lemma 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Lemma 2.7
• Definition 2.8