Superluminal Expanding Gas Shells |

It's well known that in many cases, the shell appears to be expanding

Of course, according to relativity theory, that can't happen. Instead, what we're seeing must be a "trick of the light". Due to the trigonometry of the situation, the shell

The cause of this effect is hard to picture. Furthermore, it doesn't depend on the distance to the star -- "perspective" doesn't enter into it. On this page, I've worked out the details of what's going on.

There is different but related effect, in which a star contained in a spherical nebula explodes, and a "shell of light" expands through the nebula at

A star is located at the origin. We are observing it from a location a large distance away along the 'y' axis (we're "above" it in the diagram). Our exact distance doesn't matter.

The star throws off a spherical shell of glowing gas. The velocity of the shell is '

At time T=0 the shell has already expanded to diameter

As the sight plane moves away from the star, it slices through ever nearer parts of the expanding spherical shell. However, the shell is expanding as this takes place. Since the shell's surface is perpendicular to the plane at the moment when the sight plane contains the star itself, it's clear that initially, the "image" of the shell which is contained in the plane -- and which we will eventually see -- is

In the diagram, I have shown the sight plane at time T=

The "sight plane" consists of the photons which will reach us at one single moment in time, and it travels at

At time T=

Substituting

Multiplying out and combining terms,

We want to know the

Setting the derivative to zero, we get

or finally,

Now, we need to be cautious, because (v

The first term in eq (9) contains v

Note that this is necessarily

Since the shell is moving at v, and the "sight plane" is moving at 1, and the sight plane needs to "catch up" a distance of r

Continuing on, let's plug t

or, expanding t

This looks messy, and it's still not quite what we want. We want the ratio of x

Note that this depends

Now, at this point let's revert to "real world" units in which

We have our answer, and now we can ask a couple of interesting questions about it.

At what velocity will the shell

Solving for the actual velocity when the apparent velocity is 1, we find

or, putting the 'c' back in, we have

We can also ask what happens if v=c. In that case, we have

But actually, this makes sense! Consider that if v=c, then we see the star burst at the same moment the expanding shell reaches us. Therefore, the apparent diameter of the shell will be infinite as soon as we see the explosion: we can look 90 degrees off axis in any direction and see light from the shell shining on us, since it's passing us at that moment.