Numerical Study of Fluid Flow in a Kerr Space

Abstract

The equation of motion of a perfect gas is solved on a computer for several cases of matter falling into a Kerr black hole. In most cases, the gas falls into the hole with no appreciable heating. When the material falling in is corotating with the metric, shock waves are formed if the specific angular momentum of the holej/c/GAf 2 , is close to 1 and if the specific angular momentum of the infalling material, U^c/GM, is greater than 2. I. INTRODUCTION Following the gravitational collapse of a star, presumably matter would be left outside the singular region, and would subsequently fall in. The purpose of the present calculation is to understand how gaseous material falling into an already formed black hole behaves-in particular, whether shock waves could be formed or material ejected. The calculations reported here are probably more appropriate to the accretion by a black hole in a gas cloud. If shock waves are formed at a low radius, then large X-ray emission could be expected. Only situations that have axial symmetry about the metric axis and plane symmetry through the equator are studied. II. EQUATIONS The equation we are solving is T v ß]v = 0, with the energy momentum tensor T ßv = [P + p(l + WU.U, + Pfa , (1) where P is the pressure, p is the proper particle density, p(l + e) is total proper mass density, and U ß is the velocity ((¡"U" +1 = 0). Particle conservation is expressed by (P^);" = 0. (2) To make the equations more suitable for computations, the four-momentum density S ß = [P + p(l + í)]17 4 í7^, velocity V ß = U^/U*, density D = pU*, and energy E-eD are introduced. The equation of state will be a gamma law, P = pe(y-1) with 7 a constant. The equations of motion are 1 V-gdt (S" V~g) + V .) + ^ + i¿IÍÍA_o ^ 2 dx* S 4 (3) 1 a V-g dt (D V~g) + (4) I (E V~g) + (EV< V-g) +P£~ a (U^V* V~g) = 0 , (5) where i goes over the three space indices only. For a metric, we have chosen the form of equation (10) from Carter (1968), which, in * This work was performed under the auspices of the U.S. Atomic Energy Commission. 431