Ideal hydrodynamics inside as well as outside non-rotating black hole: Hamiltonian description in the Painlev{é}-Gullstrand coordinates

Abstract

It is demonstrated that with using Painlev{é}-Gullstrand coordinates in their quasi-Cartesian variant, the Hamiltonian functional for relativistic perfect fluid hydrodynamics near a non-rotating black hole differs from the corresponding flat-spacetime Hamiltonian just by a simple term. Moreover, the internal region of the black hole is then described uniformly together with the external region, because in Painlev{é}-Gullstrand coordinates there is no singularity at the event horizon. An exact solution is presented which describes stationary accretion of an ultra-hard matter ($\epsilon \propto n^2$) onto a moving black hole until reaching the central singularity. Equation of motion for a thin vortex filament on such accretion background is derived in the local induction approximation. The Hamiltonian for a fluid having ultra-relativistic equation of state $\epsilon \propto n^{4/3}$ is calculated in explicit form, and the problem of centrally-symmetric stationary flow of such matter is solved analytically.

revtex4, 7 pages, no figures

References in corpus (3)

• Fully General Relativistic Simulations of Black Hole-Neutron Star Mergers
• Ultra-hard fluid and scalar field in the Kerr-Newman metric
• Spherically Symmetric Black Hole Formation in Painlevé-Gullstrand Coordinates