A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole

Abstract

We develop a Birman-Schwinger principle for the spherically symmetric, asymptotically flat Einstein-Vlasov system. It characterizes stability properties of steady states such as the positive definiteness of an Antonov-type operator or the existence of exponentially growing modes in terms of a one-dimensional variational problem for a Hilbert-Schmidt operator. This requires a refined analysis of the operators arising from linearizing the system, which uses action-angle type variables. For the latter, a single-well structure of the effective potential for the particle flow of the steady state is required. This natural property can be verified for a broad class of singularity-free steady states. As a particular example for the application of our Birman-Schwinger principle we consider steady states where a Schwarzschild black hole is surrounded by a shell of Vlasov matter. We prove the existence of such steady states and derive linear stability if the mass of the Vlasov shell is small compared to the mass of the black hole.

Stability and instability results for equilibria of a (relativistic) self-gravitating collisionless gas—a review

“…L 0 > 0, then the resulting steady states have a vacuum region at the center, if L 0 = 0 they do not. The static shell solutions with L 0 > 0 look somewhat artificial, but they become more interesting if one places a Schwarzschild black hole (or a point mass in the (VP) case) into the vacuum region, which is then surrounded by a static shell of Vlasov matter, see [37,59,93,96].…”

“…and the resulting operator T : D(T ) → H is the transport operator. In view of (5.19) we also define B : D(T ) → H by That the transport operator is symmetric with respect to the scalar product on the Hilbert space H is easy to see; for the details of the above results we refer to [37] or [47]. We use these operators to put the linearized (EV) system, i.e.…”

“…An important step in the previous two sections was to relate the generator(s) of the linearized (EV) dynamics to some operator(s) defined on functions which depend only on the radial variable r. An alternative way to do this is the Birman-Schwinger principle, which for (VP) we discussed in section 4.3. In [37] a Birman-Schwinger type principle was developed for (EV), and we now discuss the main features of this approach.…”

“…For a detailed proof we refer to [37]. Here we want to discuss an application of these techniques yielding a result which was not obtained by the methods in the previous sections, namely, we want to consider the stability of a shell of Vlasov matter surrounding a Schwarzschild black hole.…”

Accretion of the relativistic Vlasov gas in the equatorial plane of the Kerr black hole

“…Some aspects of the dynamics of the massless Vlasov gas in a slowly rotating Kerr spacetime are analysed in [24]. We would also like to mention recent studies of self-gravitating static Vlasov configurations around black holes [25,26].…”