For a stationary-ish object like a ruler, this effect is pretty irrelevant. That’s because the same bit of ruler was in the same place whether or not you saw it a nanosecond before or after or whatever. But when an object stops being stationary, you can hope to see some pretty weird things once you start accounting for the finite speed of light.

I’ve been wondering what an observer would see if a sphere expanded faster than the speed of light. I’m not talking about the effect of relativistic effects like length contraction, only the effect of a finite speed of light, i.e. the fact that the light from the nearer side of the sphere would reach the observer before simultaneously-emitted light from the far side – the fact that when you see something, you see it where it was when the light left it, not where it is at the time the light gets to your eyes.

Although this doesn’t use special relativity, the sphere expanding faster than light doesn’t violate it necessarily. It doesn’t need to be a sphere of physical “stuff”, for example we could build a huge grid of beacons pre-programmed to flash in such a way that the flashes form a sphere expanding at any speed we wish – no Lorentz violation needed.

So let’s say a sphere expanded about a point a with velocity *v*. After time *t*, we would have a sphere of radius *vt*, so the set of points would be on the “locus” defined by

|**x**–**b**|=*vt*.

This is what *actually* happens (in a particular inertial frame), but what would an observer at another point (say **b**) see? If the speed of light is *c*, then light would take time

Δ*t* = *d* / *c*

to travel a distance *d*. This means that at time *t*, the observer at **b** will see something at point **x** if there was /actually/ something there at the earlier time

*t*‘ = *t* – |**x**–**b**| / *c*

(where |**x**–**b**| is the distance between the observer and the point they are looking at).

For our expanding sphere, we can rearrange the first equation and see that there would have been something at point **x** at time *t*‘ if

*t*‘ = |**x**–**b**| / *v*.

Substituting this expression for *t*‘, we see that the set of points that *look* occupied according to the observer after a time *t* are given by the locus

|**x**–**b**| / *v* + |**x**–**a**| / *c* = *t*.

Now, if *v* = *c* then this is a prolate spheroid, i.e. a surface of revolution about the line through **a** and **b** of an ellipse with foci at **a** and **b** and with major diameter *ct*, exactly half the diameter of the ‘real’ sphere (2*ct*).

In the more general case with *v* not equal to *c*, this equation describes a Cartesian ovoid.

I made some animations for various values of *v* and *c* using Wolfram Mathematica to visualise these shapes. The setup is rotationally symmetric about the axis through **a** and **b** so we can plot only a 2*d* cross-section without losing anything. I have also plotted circles showing the “real” position of the sphere for comparison.

In this animation, the pink circle is expanding from the point (0,0) (the lower dot) at one-tenth of the speed of light. The blue locus is what an observer at the point (0,1) (the upper dot) *thinks* they can see. See how it looks almost like a circle (it isn’t, though) that emanates from the same point slightly after the pink circle. It catches up with the real circle exactly when it crosses over the observer at (0,1), then begins to drop behind again.

In this animation, the pink circle is still expanding slower than light-speed, but it’s nearly there – 0.9 times as fast. The ‘real’ circle has to get 9/10 of the way to the observer at (0,1) before the ‘apparent’ locus appears to pop out of (0,0), a pointy Cartesian ovoid quickly rushing up to catch up with the real pink circle exactly as it crosses over the observer.

Now the circle is expanding at *exactly* the speed of light. The observer at (0,1) sees nothing whatsoever until it passes over them, and then suddenly all the light coming from the nearest point of the circle hits them at exactly the same time – they see nothing but a straight line connecting them and the origin of the circle. This expands into an ellipse whose foci are the observer at (0,1) and the origin of the ‘real’ circle at (0,0). Its major axis – here the distance between the top-most point and the bottom-most point, is equal to *ct*, which means is half the diameter of the real circle.

This is what I really wanted to visualize. Now the real (pink) circle is expanding faster than the speed of light. The observer at (0,1) sees nothing at all until it hits them. Because it’s travelling faster than light, though, they still don’t see anything from the circle’s journey thus far. When the real circle has passed the observer a little, the observer sees the same “bit” of circle *twice* – once from light emitted before the circle crossed them; once from light emitted after. Altogether, the observer sees a Cartesian ovoid, this time apparently emanating from *themselves*, its point heading rapidly towards the origin of the ‘real’ sphere.

There is a nice symmetry there: a sphere expanding slower than light looks like a Cartesian ovoid emanating from the origin of the sphere and pointing towards the observer; a sphere expanding faster than light looks like a Cartesian ovoid emanating from the *observer* and pointing towards the origin of the sphere.

There is a significant and nifty difference though: in the case where the sphere is expanding slower than light, once the observer can see the sphere, they can see all of it. It looks a little distorted due to the light taking different amounts of time to reach the observer from different points, but they can still see the whole thing exactly once.

But in the case of the sphere expanding faster than light, that’s not what the observer sees: between the time the sphere passes them, and the time they see the sphere leaving the origin, they will see some parts of it *twice* and some parts of it *not at all*. The sphere is hacked in half, photocopied and glued to itself. But this is done in an elegant and smooth way such that the *two* copies of one *half* of the sphere appear to be the same ovoid shape as the whole sphere does when travelling less fast than light.

But I think the neatest thing is imagining how this mashed-together double-half-sphere smoothly transitions into a slightly distorted copy of the actual sphere once the light from the origin reaches the observer. By the symmetry of the problem, the two copies of the visible chunk of sphere will appear to be divided by a circle, from which the as-yet invisible chunk of sphere will seem to be emerging into view, dividing itself into two copies, one copy sliding onto the “inner” side of the circle and one on the “outer” side. The circle will begin as a point located at the observer as the super-luminal sphere crosses over them, it will grow and slide along the ovoid until it ends back up at the origin, once more as a point, when finally all the ‘bits’ of the whole sphere will come into view.

At the top of this post I’ve put a 3*d* render comparing the Cartesian ovoid with the expanding sphere of which it’s an apparition, as seen by the little chappy.