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 if the black hole is rotating. 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.
Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A ...prominent example is the binary black-hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity.
The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein’s equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.
This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (
M
,
g
) of arbitrary dimension. ...Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focused on the tangent bundle formulation, here we work on the cotangent bundle associated with (
M
,
g
) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle
T
∗
M
, starting with the natural symplectic form on
T
∗
M
, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on
T
∗
M
and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov–Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (
M
,
g
). To this end, we introduce the collision manifold, describing the set of all possible binary elastic collisions and discuss its most important geometric properties, including the metric and volume form it is equipped with and its symmetries. We show how imposing full Lorentz symmetry leads to microscopic reversibility and the relativistic
H
theorem. The meaning of global and local equilibrium and the stringent restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs.
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