Born Coordinates - Radar Distance in The Small

Radar Distance in The Small

As was mentioned above, for various reasons the family of Langevin observers admits no family of orthogonal hyperslices. Therefore these observers simply cannot be associated with any slicing of spacetime into a family of successive "constant time slices".

However, because the Langevin congruence is stationary, we can imagine replacing each world line in this congruence by a point. That is, we can consider the quotient space of Minkowski spacetime (or rather, the region 0 < R < 1/ω) by the Langevin congruence, which is a three-dimensional topological manifold. Even better, we can place a Riemannian metric on this quotient manifold, turning it into a three dimensional Riemannian manifold, in such a way that the metric has a simple operational significance.

To see this, consider the Born line element

Setting ds2 = 0 and solving for dt we obtain

The elapsed proper time for a roundtrip radar blip emitted by a Langevin observer is then

Therefore, in our quotient manifold, the Riemannian line element

corresponds to distance between infinitesimally nearby Langevin observers. We will call it the Langevin-Landau-Lifschitz metric, and we can call this notion of distance radar distance "in the small".

This metric was first given by Langevin, but the interpretation in terms of radar distance "in the small" is due to Lev Landau and Evgeny Lifshitz, who generalized the construction to work for the quotient of any Lorentzian manifold by a stationary timelike congruence.

If we adopt the coframe

we can easily compute the Riemannian curvature tensor of our three dimensional quotient manifold. It has only one independent nontrivial component,

Thus, in some sense, the geometry of a rotating disk is curved, as Theodor Kaluza claimed (without proof) as early as 1910. In fact, to fourth order in ω it has the geometry of the hyperbolic plane, just as Kaluza claimed.

Warning: as we have seen, there are many possible notions of distance which can be employed by Langevin observers riding on a rigidly rotating disk, so statements referring to "the geometry of a rotating disk" always require careful qualification.

To drive home this important point, let us use the Landau-Lifschitz metric to compute the distance between a Langevin observer riding a ring with radius R₀ and a central static observer. To do this, we need only integrate our line element over the appropriate null geodesic track. From our earlier work, we see that we must plug

into our line element and integrate. This gives

Because we are now dealing with a Riemannian metric, this notion of distance is of course symmetric under interchanging the two observers, unlike radar distance "in the large". The values given by this notion are intermediate between the radar distances computed in the previous section. For example, for r₀ = 1, ω = 1/2, we find approximately Δ = 1.047, which can be compared with 1.155 for the distance from the ring-riding observer to the central observer, or 1 for the central observer to the ring-riding observer. Also, because up to second order the Landau-Lifschitz metric agrees with radar distance "in the large", we see that the curvature tensor we just computed does have operational significance: while radar distance "in the large" between pairs of Langevin observers is certainly not a Riemannian notion of distance, the distance between pairs of nearby Langevin observers does correspond to a Riemannian distance, given by the Langevin-Landau-Lifschitz metric. (In the felicitous phrase of Howard Percy Robertson, this is kinematics im kleinem.)

One way to see that all reasonable notions of spatial distance for our Langevin observers agree for nearby observers is to show, following Nathan Rosen, that for any one Langevin observer, an instantaneously comoving inertial observer will also obtain the distances given by the Langevin-Landau-Lifschitz metric, for very small distances.