Langevin Observers in The Cylindrical Chart
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 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.
Each integral curve of the timelike unit vector field appears in the cylindrical chart as a helix with constant radius (such as the red curve in the figure at right). Suppose we choose one Langevin observer and consider the other observers who ride on a ring of radius R which is rigidly rotating with angular velocity ω. Then if we take an integral curve (blue helical curve in the figure at right) of the spacelike basis vector, we obtain a curve which we might hope can be interpreted as a "line of simultaneity" for the ring-riding observers. But as we see from the figure, ideal clocks carried by these ring-riding observers cannot be synchronized. This is our first hint that it is not as easy as one might expect to define a satisfactory notion of spatial geometry even for a rotating ring, much less a rotating disk!
Computing the kinematic decomposition of the Langevin congruence, we find that the acceleration vector is
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.
Read more about this topic: Born Coordinates
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