73 research outputs found

    Tangent bundle formulation of a charged gas

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    We discuss the relativistic kinetic theory for a simple, collisionless, charged gas propagating on an arbitrary curved spacetime geometry. Our general relativistic treatment is formulated on the tangent bundle of the spacetime manifold and takes advantage of its rich geometric structure. In particular, we point out the existence of a natural metric on the tangent bundle and illustrate its role for the development of the relativistic kinetic theory. This metric, combined with the electromagnetic field of the spacetime, yields an appropriate symplectic form on the tangent bundle. The Liouville vector field arises as the Hamiltonian vector field of a natural Hamiltonian. The latter also defines natural energy surfaces, called mass shells, which turn out to be smooth Lorentzian submanifolds. A simple, collisionless, charged gas is described by a distribution function which is defined on the mass shell and satisfies the Liouville equation. Suitable fibre integrals of the distribution function define observable fields on the spacetime manifold, such as the current density and stress-energy tensor. Finally, the geometric setting of this work allows us to discuss the relationship between the symmetries of the electromagnetic field, those of the spacetime metric, and the symmetries of the distribution function. Taking advantage of these symmetries, we construct the most general solution of the Liouville equation an a Kerr-Newman black hole background.Comment: 16 pages, 2 figures, prepared for the proceedings of the Fifth Leopoldo Garc\'ia-Col\'in Mexican Meeting on Mathematical and Experimental Physics, Mexico, September 201

    Outer boundary conditions for Einstein's field equations in harmonic coordinates

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    We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions, which is constraint-preserving and sufficiently general to include recent proposals for reducing the amount of spurious reflections of gravitational radiation. In particular, our class comprises the boundary conditions recently proposed by Kreiss and Winicour, a geometric modification thereof, the freezing-Ψ0 boundary condition and the hierarchy of absorbing boundary conditions introduced by Buchman and Sarbach. Using the recent technique developed by Kreiss and Winicour based on an appropriate reduction to a pseudo-differential first-order system, we prove well posedness of the resulting initial-boundary value problem in the frozen coefficient approximation. In view of the theory of pseudo-differential operators, it is expected that the full nonlinear problem is also well posed. Furthermore, we implement some of our boundary conditions numerically and study their effectiveness in a test problem consisting of a perturbed Schwarzschild black hole

    Michel accretion of a polytropic fluid with adiabatic index gamma > 5/3: Global flows versus homoclinic orbits

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    We analyze the properties of a polytropic fluid which is radially accreted into a Schwarzschild black hole. The case where the adiabatic index gamma lies in the range 1 < gamma <= 5/3 has been treated in previous work. In this article we analyze the complementary range 5/3 < gamma <= 2. To this purpose, the problem is cast into an appropriate Hamiltonian dynamical system whose phase flow is analyzed. While for 1 < gamma <= 5/3 the solutions are always characterized by the presence of a unique critical saddle point, we show that when 5/3 < gamma <= 2, an additional critical point might appear which is a center point. For the parametrization used in this paper we prove that whenever this additional critical point appears, there is a homoclinic orbit.Comment: 13 pages, 3 figure
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