Acceleration, streamlines and potential flows in general relativity: Analytical and numerical results (Q2752503)

This paper deals with analytical and numerical solutions for the integral curves of the velocity field (streamlines) of a steady-state flow of an ideal fluid. The starting point of the paper is the ideal fluid whose energy-momentum tensor is given by \(T_{\mu\nu }=(\rho +p)U_{\mu }U_{\nu }+pg_{\mu\nu }\), where \(p\) is the pressure, \(\rho \) is the total energy density and \(U_{\mu }\) is \(4-\)velocity. The conservation laws are given by the equations \(T_{;\nu }^{\mu\nu }=0\) which are reduced to the equations \((\rho +p)U_{;\mu }^{\mu }+\rho_{,\mu }U^{\mu }=0\) and \((\rho +p)U^{\nu }U_{\mu;\nu }+p_{,\mu }+p_{,\nu }U^{\nu }U_{\mu }=0\), the mass conservation law and Euler equation, respectively. The streamlines associated with an accelerate black hole and a rigid sphere are studied in some detail. The velocity fields of a black hole and a rigid sphere in an external dipolar field (constant acceleration field) are investigated as well. The dipole field is produced by an axially symmetric halo or shell of matter. The fluid density is studied using contour lines. Another result is that the presence of acceleration is detected by these contour lines.