论文标题
Birman-Schwinger原理总体相对论：黑洞周围无碰撞物质的线性稳定壳
A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole
论文作者
论文摘要
我们开发了一个Birman-Schinginger原理，用于球形对称，渐近平坦的Einstein-Vlasov系统。它表征了稳态的稳定性特性，例如安东尼诺夫型操作员的积极确定性或Hilbert-Schmidt操作员的一维变异问题而言，存在指数增长的模式。这需要对使用动作角度变量线性化的系统产生的运算符进行完善的分析。对于后者，需要具有有效电位的单孔结构的稳态粒子流动。可以为广泛的无奇异稳态验证这种自然特性。作为应用Birman-Schinginger原则的一个特殊示例，我们考虑了Schwarzschild黑洞被Vlasov物质包围的稳态状态。如果与黑洞的质量相比，我们证明存在这种稳态的存在并得出线性稳定性。
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.
