Stationary axisymmetric perfect fluid solutions in general relativity (Q1101715)

A new class of exact solutions of Einstein's field equations with the energy-momentum tensor of a perfect fluid is given. The class of solutions is invariantly characterized by means of the following properties: (i) The energy-momentum tensor describes a perfect fluid. (ii) There are two commuting Killing vectors \(\xi\) and \(\eta\) which form an abelian group \(G_ 2\) of motion. (iii) There is a timelike Killing vector parallel to the four-velocity of the fluid (rigid rotation of the fluid). (iv) The four-vector of the angular velocity of the fluid is a gradient \(\Omega^ i=-(1/4c)\epsilon^{irkl}U_ l(U_{r;k}- U_{k;r})=\chi^{'i}.\) The last assumption is the reason that all solutions of this class can be found by solving an ordinary differential equation of the second order.