Van Stockum Dust - Derivation

Derivation

One way of obtaining this solution is to look for a cylindrically symmetric perfect fluid solution in which the fluid exhibits rigid rotation. That is, we demand that the world lines of the fluid particles form a timelike congruence having nonzero vorticity but vanishing expansion and shear. (In fact, since dust particles feel no forces, this will turn out to be a timelike geodesic congruence, but we won't need to assume this in advance.)

A simple Ansatz corresponding to this demand is expressed by the following frame field, which contains two undetermined functions of r:

To prevent misunderstanding, we should emphasize that taking the dual coframe

gives the metric tensor in terms of the same two undetermined functions:

Multiplying out gives

We compute the Einstein tensor with respect to this frame, in terms of the two undetermined functions, and demand that the result have the form appropriate for a perfect fluid solution with the timelike unit vector everywhere tangent to the world line of a fluid particle. That is, we demand that

This gives the conditions

Solving for f and then for h gives the desired frame defining the van Stockum solution:

Note that this frame is only defined on .