Expanding Sphere

I pray you give me leave to go from hence;

I am not well; send the deed after me

And I will sign it.

An elementary fact of special relativity is that an expanding light wave (in vacuum) that is spatially spherical (at any given time) and propagating at c in terms of a given system of inertia-based coordinates is also a spatially spherical wave propagating at c in terms of any other system of inertia-based coordinates. This is sometimes surprising to students, because they expect the sphere at any given time (of this system) to be contracted in terms of another system. The explanation is that an expanding sphere transforms differently than a stationary sphere, but the fact that we get a spatially spherical wave in all cases is due to the fact that the wave propagates at c. For a sphere that expands at a zero rate (i.e., not expanding at all), we get the full Fitzgerald contraction. What about an expanding shell of material the propagates at some positive speed less than c?

Consider an exploding bomb that emits an isotropic shell of material at speed u in all directions in terms of a system S of inertial coordinates x,y,t in which the original bomb was at rest. This yields an expanding spherical shell in terms of those coordinates that satisfies the equation

Another system S′ of inertia-based coordinates (same origin) in terms of which the original bomb was moving at speed v is related to S by the Lorentz transformation

Substituting into the previous equation gives

Expanding and re-arranging, we get

Thus for any value of t′ the locus is an ellipsoid or sphere. As required, if the expanding shell in terms of S has speed u=1, then this reduces to

so in that case we have a spherical wave with unit speed in terms of S′, just as we do in terms of S. At the other extreme, if u is very small compared with 1, then the equation approaches

Thus the spatial shape of the wave at a given t′ is a contracted ellipsoid with the major axis shorted by the factor √(1−v²) in the direction of motion. Also, the speed at which y′ is increasing at x′ = vt′ is not u but rather u√(1−v²).  The center of the wave moves at essentially the speed v in terms of these coordinates.

For the general case we must use equation (1), which shows that the wave at any given t’ is an ellipsoid shortened by the factor

The shortening factor approaches unity as u approaches 1. The speed of the center of the wave is

which is zero at u=1, meaning the center of the spherical wave doesn’t move at all in terms of S′ for a light wave. At the (moving) central value of x′ the speed at which y′ is increasing is not u but rather

This factor approaches unity as u approaches 1.

Return to MathPages Main Menu