标题： Kerr黑洞外部的Vlasov气体相空间混合 显示英文标题

标题： Phase space mixing of a Vlasov gas in the exterior of a Kerr black hole

摘要： 我们研究了在外子超极限克尔黑洞时空外部遵循未来指向类时且空间有界的测地线的碰撞less动能气体的动力学。 基于广义的作用-角度变量的使用，我们分析了与该气体相关的宏观可观测量的大时间渐近行为。 我们证明了，只要系统的本征频率满足一个合适的非退化条件，这些宏观可观测量就会随时间收敛到由平均分布函数确定的相应可观测量。 特别是，这意味着最终状态由一个分布函数表征，该分布函数对于系统的整个对称群是不变的，即它是定态的、轴对称的，并且与与卡特常数相关的运动积分具有泊松对易性。 作为我们的结果的一个推论，我们在设定中证明了强让斯定理的有效性，指出属于定态的分布函数必须是独立于广义角度变量的函数。 在假设定态被卡特流不变性要求取代的情况下，也证明了一个类似的定理。 最后，我们证明了上述非退化条件成立。 这是通过提供能量和卡特常数关于轨道具有足够大半径的动作变量的合适渐近展开式实现的，并利用了基本频率对运动积分的解析依赖性。 显示英文摘要

摘要： We study the dynamics of a collisionless kinetic gas whose particles follow future-directed timelike and spatially bound geodesics in the exterior of a sub-extremal Kerr black hole spacetime. Based on the use of generalized action-angle variables, we analyze the large time asymptotic behavior of macroscopic observables associated with the gas. We show that, as long as the fundamental frequencies of the system satisfy a suitable non-degeneracy condition, these macroscopic observables converge in time to the corresponding observables determined from an averaged distribution function. In particular, this implies that the final state is characterized by a distribution function which is invariant with respect to the full symmetry group of the system, that is, it is stationary, axisymmetric and Poisson-commutes with the integral of motion associated with the Carter constant. As a corollary of our result, we demonstrate the validity of the strong Jeans theorem in our setting, stating that the distribution function belonging to a stationary state must be a function which is independent of the generalized angle variables. An analogous theorem in which the assumption of stationarity is replaced with the requirement of invariance with respect to the Carter flow is also proven. Finally, we prove that the aforementioned non-degeneracy condition holds. This is achieved by providing suitable asymptotic expansions for the energy and Carter constant in terms of action variables for orbits having sufficiently large radii, and by exploiting the analytic dependency of the fundamental frequencies on the integrals of motion.