Observers riding on a rigidly rotating disk will conclude from measurements of small distances between themselves that the geometry of the disk is non-Euclidean. Regardless of which method they use, they will conclude that the geometry is well approximated by a certain Riemannian metric, namely the Langevin-Landau-Lifschitz metric. This is in turn very well approximated by the geometry of the hyperbolic plane (with the constant negative curvature -3 ω2). But if these observers measure larger distances, they will obtain different results, depending upon which method of measurement they use! In all such cases, however, they will most likely obtain results which are inconsistent with any Riemannian metric. In particular, if they use the simplest notion of distance, radar distance, owing to various effects such as the asymmetry already noted, they will conclude that the "geometry" of the disk is not only non-Euclidean, it is non-Riemannian.
To motivate the Born chart, we first consider the family of Langevin observers represented in an ordinary cylindrical coordinate chart for Minkowski spacetime. The world lines of these observers form a timelike congruence which is rigid in the sense of having a vanishing expansion tensor. They represent observers who rotate rigidly around an axis of cylindrical symmetry.
From the line element
we can immediately read off a frame field representing the local Lorentz frames of stationary (inertial) observers
Here, is a timelike unit vector field while the others are spacelike unit vector fields; at each event, all four are mutually orthogonal and determine the infinitesimal Lorentz frame of the static observer whose world line passes through that event.
Simultaneously boosting these frame fields in the direction, we obtain the desired frame field describing the physical experience of the non-inertial LNIF Langevin observers, namely
This frame was apparently first introduced (implicitly) by Paul Langevin in 1935; its first explicit use appears to have been by T. A. Weber, as recently as 1997! It is defined on the region 0 < R < 1/ω; this limitation is fundamental, since near the outer boundary, the velocity of the Langevin observers approaches the speed of light.
This points radially inward and it depends only on the (constant) radius of each helical world line. The expansion tensor vanishes identically, which means that nearby Langevin observers maintain constant distance from each other. The vorticity vector is
which is parallel to the axis of symmetry. This means that the world lines of the nearest neighbors of each Langevin observer are twisting about its own world line, as suggested by the figure at right. This is a kind of local notion of "swirling" or vorticity.
In contrast, note that projecting the helices onto any one of the spatial hyperslices orthogonal to the world lines of the static observers gives a circle, which is of course a closed curve. Even better, the coordinate basis vector is a spacelike Killing vector field whose integral curves are closed spacelike curves (circles, in fact), which moreover degenerate to zero length closed curves on the axis R = 0. This expresses the fact that our spacetime exhibits cylindrical symmetry, and also exhibits a kind of global notion of the rotation of our Langevin observers.
In the figure, the magenta curve shows how the spatial vectors are spinning about (which is suppressed in the figure since the Z coordinate is inessential). That is, the vectors are not Fermi-Walker transported along the world line, so the Langevin frame is spinning as well as non-inertial. In other words, in our straightforward derivation of the Langevin frame, we kept the frame aligned with the radial coordinate basis vector . By introducing a constant rate rotation of the frame carried by each Langevin observer about , we could, if we wished "despin" our frame to obtain a gyrostabilized version.
To obtain the Born chart, we straighten out the helical world lines of the Langevin observers using the simple coordinate transformation
The new line element is
Notice the "cross-terms" involving , which show that the Born chart is not an orthogonal coordinate chart. The Born coordinates are also sometimes referred to as rotating cylindrical coordinates.
In the new chart, the world lines of the Langevin observers appear as vertical straight lines. Indeed, we can easily transform the four vector fields making up the Langevin frame into the new chart. We obtain
These are exactly the same vector fields as before--- they are now simply represented in a different coordinate chart!
Needless to say, in the process of "unwinding" the world lines of the Langevin observers, which appear as helices in the cylindrical chart, we "wound up" the world lines of the static observers, which now appear as helices in the Born chart! Note too that, like the Langevin frame, the Born chart is only defined on the region 0 < r < 1/ω.
If we recompute the kinematic decomposition of the Langevin observers, that is of the timelike congruence
we will of course obtain the same answer that we did before, only expressed in terms of the new chart. Specifically, the acceleration vector is
the expansion tensor vanishes, and the vorticity vector is
The dual covector field of the timelike unit vector field in any frame field represents infinitesimal spatial hyperslices. However, the Frobenius integrability theorem gives a strong restriction on whether or not these spatial hyperplane elements can be "knit together" to form a family of spatial hypersurfaces which are everywhere orthogonal to the world lines of the congruence. Indeed, it turns out that this is possible, in which case we say the congruence is hypersurface orthogonal, if and only if the vorticity vector vanishes identically. Thus, while the static observers in the cylindrical chart admits a unique family of orthogonal hyperslices , the Langevin observers admit no such hyperslices. In particular, the spatial surfaces in the Born chart are orthogonal to the static observers, not to the Langevin observers. This is our second (and much more pointed) indication that defining "the spatial geometry of a rotating disk" is not as simple as one might expect.
To better understand this crucial point, consider integral curves of the third Langevin frame vector
which pass through the radius . (For convenience, we will suppress the inessential coordinate z from our discussion.) These curves lie in the surface
shown in the figure. We would like to regard this as a "space at a time" for our Langevin observers. But two things go wrong.
First, the Frobenius theorem tells us that are tangent to no spatial hyperslice whatever. Indeed, except on the initial radius, the vectors do not lie in our slice. Thus, while we found a spatial hypersurface, it is orthogonal to the world lines of only some our Langevin observers. Because the obstruction from the Frobenius theorem can be understood in terms of the failure of the vector fields to form a Lie algebra, this obstruction is differential, in fact Lie theoretic. That is, it is a kind of infinitesimal obstruction to the existence of a satisfactory notion of spatial hyperslices for our rotating observers.
Second, as the figure shows, our attempted hyperslice would lead to a discontinuous notion of "time" due to the "jumps" in the integral curves (shown as a coral colored discontinuity). Alternatively, we could try to use a multivalued time. Neither of these alternatives seems very attractive! This is evidently a global obstruction. It is of course a consequence of our inability to synchronize the clocks of the Langevin observers riding even a single ring---say the rim of a disk--- much less an entire disk.
Imagine that we have fastened a fiber-optic cable around the circumference of a ring which is rotating with steady angular velocity ω. We wish to compute the round trip travel time, as measured by a ring-riding observer, for a laser pulse sent clockwise and counterclockwise around the cable. For simplicity, we will ignore the fact that light travels through a fiber optic cable at somewhat less than the speed of light in a vacuum, and will pretend that the world line of our laser pulse is a null curve (but certainly not a null geodesic!).
In the Born line element, let us put . This gives
We obtain for the round trip travel time
Putting , we find so that the ring-riding observers can determine the angular velocity of the ring (as measured by a static observer) from the difference between clockwise and counterclockwise travel times. This is known as the Sagnac effect. It is evidently a global effect.