It's easy to see that the apparent effect is (approximately) a rotation of the sphere.

We are discussing "apparent effects" here because we are interested in what an observer at a single point

sees at a given instant - i.e. in the light rays intersecting at the observers particular point in spacetime - rather than in the actual position of objects in the observer's reference frame.

The light rays intersecting a given spacetime point are invariant w.r.t. changes in reference frame, so we may consider first what another observer at the same point sees, in whos reference frame the sphere is stationary: Trivially, they see the sphere as a disk, and its outline as a circle. They see its underside when it is positioned directly above them.

Now consider the observer at the same point in spacetime, but who is moving at high speed relative to the sphere. As already discussed on p429 and in Exercise [18.10], their visual field consists of the

same set of impinging light rays, but conformally distorted. Since circles are preserved by conformal mappings, the outline of the sphere still appears circular!

(In general the center point of this circle will be different bewteen the two observers, however, as the interior of the disk is distorted for the moving observer. However, the smaller the angular diameter of the disk, the less this distortion - which is why the exercise specifies "small angular diameter".)

Furthermore, when the sphere appears centered directly overhead to the first (stationary) observer, it will appear to be centered at some angle to the vertical, for the moving observer (again, because of the conformal transformation of the visual field),

even though its visible face is the same in both cases. Thus, for the moving observer, the angular position of the sphere in their field of vision does not match the angular orientation of the visible face of the sphere in the actual coordinate system - i.e. the sphere

appears rotated.

This approach to analysing the appearance of a moving sphere is discussed in more depth here:

Can You See the Lorentz-Fitzgerald Contraction? Or: Penrose-Terrell Rotationand in Terrell's original 1959 paper:

Invisibility of the Lorentz Contraction (PDF)(There's also an interesting animated presentation of the situation, similar to Figure 18.15 that can aid understanding,

here).

...However, this approach seems to sidestep what the problem is

actually asking us to do, which is to "do the math" to show that the contraction effect and the light path-length effect cancel. This suggests to me that it's intended you do all the calculations

in the observer's reference frame (in which the sphere is moving, and contracted).

Well, I tried (at some length) to do that: I derived the expression for visual angle (an (X,Y,Z) unit vector)

observed at (observer-)time T, for any given (x,y,z) coordinate on the surface of the sphere (or anywhere in its reference frame for that matter).

However, whenever I attempted to solve for the visible "rim" of the sphere, (or even for it's leading and trailing edges in a 2-D crossection - i.e. Figure 18.15), I quickly ran into equations that appeared intractably complex, and that I couldn't seem to solve even with Mathematica's help (I was using version 8.0).

Even attempting to figure out, from simple non-relativistic geometry, the angle around the (horizontally compressed) moving sphere marking the boundary at which light can escape towards an observer yeilded an equation I couldn't solve analytically!

This seems odd to me, because if you tried the first approach I described - i.e. did all the calculations for a

stationary sphere, and then applied a conformal Lorentz transformation to the resulting Riemann sphere of direction vectors - I would

think that that ought to be quite tractable (though I haven't attempted it).

Honestly I've spent too much time on this one problem already, so I'm giving up (at least for now). But, I was wondering, if anyone else has attempted this exercise:

* How? i.e. What approach did you take?

* Did you run into similar problems as I did?

* Did you get any insight as to why?

I'd be interested to hear from anyone else who's tried this!