Parameter Space of Shock Formation in Adiabatic Flows

We study shock formation in a stationary, axisymmetric, adiabatic flow of a perfect fluid in the equatorial plane of a Kerr geometry. For such a flow, there exist two intrinsic constants of motion along a fluid world line, namely the specific total energy, E = −hut, and the specific angular momentum, l = −uφ/ut, where the uμ’s are the four velocity components, h is the specific enthalpy, i.e., h = (P + ε)/ρ, with P, ε, and ρ being the pressure, the mass-energy density, and the rest-mass density, respectively.As shown in Fig. 1 (Fig. la is for a Schwarzschild black hole, i.e. the hole’s specific angular momentum a = 0; Fig. lb is for a rapid Kerr hole, i.e. a = 0.99M, where M is the black-hole mass, and prograde flows: and Fig. 1c is for a = 0.99M and retrograde flows), in the parameter space spanned by E and l there is a strictly defined region bounded by four lines: three characteristic functional curves lk(E), lmax(E), and lmin(E), and the vertical line E = 1. Only such a flow with parameters located within this region can have two physically realizable sonic points, the inner one rin, and the outer one rout. In between there is still one more, but unrealizable, sonic point, rmid. The region is divided by another characteristic functional curve lc(E) into two parts: in region I (= Ia + Ib) only τout is realized in a shock-free global solution (i.e., that joining the black-hole horizon to large distances), while in region II (= IIa + IIb) only rin is realized.