Potential flows in general relativity: Nonlinear and time-dependent solutions

We present several new calculations of subsonic relativistic fluid flows in fixed background spacetimes. For irrotational, isentropic, perfect-fluid flows, the relativistic Euler and continuity equations can be formulated as a potential problem which is very accessible to numerical and analytic techniques. Schwarzschild and Kerr background spacetimes are considered with the inner boundary condition determining whether the gravitating source is a black hole or a hard sphere. For stationary flows with a polytropic equation of state $P=K{n}^{\gamma }$, the potential equation is nonlinear and elliptic. We compute several examples of stationary, axisymmetric flows about a hard sphere moving through an asymptotically homogeneous medium. For a $P=\rho$ (i.e., $\gamma =2\mathrm{}$) equation of state the potential equation is linear and can be solved analytically for steady-state flows. We solve the hyperbolic system to calculate various time-dependent two-dimensional flows past a hard sphere. For hard spheres larger than a critical size $\frac{R_{\lim }}{M}$, there is an asymptotic velocity (of the sphere through the medium) above which the steady state cannot be achieved. We use the time-dependent code to study the behavior of the stationary and nonstationary cases. The code is also employed to examine the transition to steady-state accretion flow onto a black hole moving through a homogeneous medium. We also present a new, analytic, three-dimensional solution for flow past a rotating hard sphere. These flows represent excellent test problems for numerical relativity. Multidimensional fluid solutions may be used to benchmark codes as well as provide realistic numerical data for developing three-dimensional visualization methods. Because of the numerical tractability of the potential formulation, great precision can be achieved with modest computational resources.