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Solutions of the Einstein Field Equations for a Rotating Perfect Fluid, Part 1 - Presentation of the Flow-Stationary and Vortex-Homogeneous Solutions

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The equations of isentropic rotational motion of a perfect fluid are investigated with use of the Darboux theorem. It is shown that, together with the equation of continuity, they ensure the existence of four scalar functions which constitute a dynamically distinguished set of coordinates. If in this system of coordinates the metric tensor is constant along the lines tangent to velocity and vorticity fields, then the field equations with $T_{ij}=(\in\pm p)\mu_i\mu_j-pg_{ij}$ can be completely integrated. The resulting metrics divide into 3 families, first of which contains 6 types of new solutions with non-zero pressure. All of them are given explicitly in terms of hypergeometric or confluent hypergeometric functions, type IV being the only one containing entirely elementary functions. The second family contains only the solution of G\" odel, and the third one --- only the solution of Lanczos.

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