Born Coordinates - Transforming To The Born Chart

Transforming To The Born Chart

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.