Van Stockum Dust - A Genuine Paradox

A Genuine Paradox

Let us draw the light cones for some typical events in the van Stockum dust, to see how their appearance (in our comoving cylindrical chart) depends on the radial coordinate:

As the figure shows, as, the cones become tangent to the coordinate plane, and we obtain a closed null curve (the red circle). Note that this is not a null geodesic.

As we move further outward, we can see that horizontal circles with larger radii are closed timelike curves. The paradoxical nature of these CTCs was apparently first pointed out by van Stockum: observers whose world lines form a closed timelike curve can apparently revisit or affect their own past. Even worse, there is apparently nothing to prevent such an observer from deciding, on his third lifetime, say, to stop accelerating, which would give him multiple biographies.

These closed timelike curves are not timelike geodesics, so these paradoxical observers must accelerate to experience these effects. Indeed, as we would expect, the required acceleration diverges as these timelike circles approach the null circles lying in the critical cylinder .

Closed timelike curves turn out to exist in many other exact solutions in general relativity, and their common appearance is one of the most troubling theoretical objections to this theory. However, very few physicists refuse to use general relativity at all on the basis of such objections; rather most take the pragmatic attitude that using general relativity makes sense whenever one can get away with it, because of the relative simplicity and well established reliability of this theory in many astrophysical situations. This is not unlike the fact that many physicists use Newtonian mechanics every day, even though they are well aware that Galilean kinematics has been "overthrown" by relativistic kinematics.