Born Coordinates - Null Geodesics

Null Geodesics

We wish to compare the appearance of null geodesics in the cylindrical chart and the Born chart.

In the cylindrical chart, the geodesic equations read

We immediately obtain the first integrals

Plugging these into the expression obtained from the line element by setting, we obtain

from which we see that the minimal radius of a null geodesic is given by

We can now solve to obtain the null geodesics as curves parameterized by an affine parameter, as follows:

More useful for our purposes is the observation that the trajectory of a null geodesic (its projection into any spatial hyperslice ) is of course a straight line, given by

To obtain the minimal radius of the line through two points (on the same side of the point of closest approach to the origin), we solve

which gives

Now consider the simplest case, the radial null geodesics. An outward bound radial null geodesic may be written in the form

Transforming to the Born chart, we find that the trajectory can be written as

Similarly for inward bound radial null geodesics. The tracks turn out to appear slightly bent in the Born chart (see figure at right). (We will see in a later section that in the Born chart, we cannot properly refer to these "tracks" as "projections", however.)

Notice that, just as a duck hunter would expect, to send a laser pulse toward the stationary observer at R = 0, the Langevin obsevers have to aim slightly ahead to correct for their own motion. Turning things around, to send a laser pulse toward a Langevin observer riding a counterclockwise rotating ring, the central observer has to aim, not at this observer's current position, but at the position at which he will arrive just in time to intercept the signal. These families of inward and outward bound radial null geodesics represent very different curves in spacetime, but their projections do agree.

Similarly, null geodesics between ring-riding Langevin observers appear slightly bent inward in the Born chart. To see this, write the equation of a null geodesic in the cylindrical chart in the form

Transforming to Born coordinates, we obtain the equations

Eliminating φ gives

which shows that the geodesic does indeed appear to bend inward. We also find that

This completes the description of the appearance of null geodesics in the Born chart, since every null geodesic is either radial or else has some point of closest approach to the axis of cylindrical symmetry.

Note (see figure) that a ring-riding observer trying to send a laser pulse to another ring-riding observer must aim slightly ahead of his angular coordinate as given in the Born chart, in order to compensate for the rotational motion of the target. Note too that the picture presented here is fully compatible with our expectation (see appearance of the night sky) that a moving observer will see the apparent position of other objects on his celestial sphere to be displaced toward the direction of his motion.