A relativistic spherical vortex

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Abstract
This investigation is concerned with stationary relativistic flows of an inviscid and incompressible fluid. In choosing a density-pressure relation to represent relativistic “incompressibility,” it is found that a fluid in which the velocity of sound equals the velocity of light is to be preferred for reasons of mathematical simplicity. In the case of axially symmetric flows, the velocity field can be derived from a stream function obeying a partial differential equation which is nonlinear. A transformation of variables is found which makes the relativistic differential equation linear. An exact solution is obtained for the case of a vortex confined to a stationary sphere. One can make all three of the components of velocity vanish on the surface of the sphere, as in the nonrelativistic Hicks spherical vortex. In the case of an isolated vortex on whose surface the pressure is made to vanish, it is found that the pressure at the center of the sphere becomes negative, as in the nonrelativistic case. A solution is also obtained for a relativistic vortex advancing in a fluid. The sphere is distorted into an oblate spheroid. The maximum possible velocity of advance of the vortex is (2/3) c.

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