Van Stockum Dust - Properties

Properties

Computing the Einstein tensor with respect to our frame shows that in fact the pressure vanishes, so we have a dust solution. The mass density of the dust turns out to be

Happily, this is finite on the axis of symmetry, but the density increases with radius, a feature which unfortunately severely limits possible astrophysical applications.

Solving the Killing equations shows that this spacetime admits a three dimensional abelian Lie algebra of Killing vector fields, generated by

Here, has nonzero vorticity, so we have a stationary spacetime invariant under translation along the world lines of the dust particles, and also under translation along the axis of cylindrical symmetry and rotation about that axis.

Note that unlike the Gödel dust solution, in the van Stockum dust the dust particles are rotating about a geometrically distinguished axis.

As promised, the expansion and shear of the timelike geodesic congruence vanishes, but the vorticity vector is

This means that even though in our comoving chart the world lines of the dust particles appear as vertical lines, in fact they are twisting about one another as the dust particles swirl about the axis of symmetry. In other words, if we follow the evolution of a small ball of dust, we find that it rotates about its own axis (parallel to ), but does not shear or expand; the latter properties define what we mean by rigid rotation. Notice that on the axis itself, the magnitude of the vorticity vector becomes simply .

The tidal tensor is

which shows that observers riding on the dust particles experience isotropic tidal tension in the plane of rotation. The magnetogravitic tensor is