Existence of steady states of the massless Einstein-Vlasov system surrounding a Schwarzschild black hole

Abstract

We show that there exist steady states of the massless Einstein-Vlasov system which surround a Schwarzschild black hole. The steady states are (thick) shells with finite mass and compact support. Furthermore we prove that an arbitrary number of shells, necessarily well separated, can surround the black hole. To our knowledge this is the first result of static self-gravitating solutions to any massless Einstein-matter system which surround a black hole. We also include a numerical investigation about the properties of the shells.

Figures (13)

• Figure 1: Not a proper shell ($k=0, l=1/2$).
• Figure 2: A proper shell ($k=0, l=1/2$). $L_0=1.86, \Gamma=0.74$ and $r_*=0.75$.
• Figure 3: Not a proper shell ($k=1, l=1/2$).
• Figure 4: A proper shell ($k=1, l=1/2$). $L_0=5.5, \Gamma=0.8$ and $r_*=0.75$.
• Figure 5: A proper shell ($k=0, l=1/2$). $L_0=108.1, \Gamma=0.68$ and $r_*=6.0$.

Theorems & Definitions (16)

• Remark 2.1
• Remark 3.1
• Lemma 3.2
• Lemma 3.3
• Remark 3.4
• Theorem 3.5
• Remark 3.6
• Remark 3.7
• Remark 3.8
• Remark 3.9