Assume two photons are moving in opposite directions from each other from a common light source. How fast would they be traveling relative to each other? Twice the speed of light? If the speed of light is the ultimate speed limit in the universe, how can something travel twice that speed?
A mosaic image of the M31 galaxy in ultraviolet light.
Image Credit: NASA/Swift/Stefan Immler (GSFC) and Erin Grand (UMCP)
Suppose two particles are moving away from each other in opposite directions, with the same speed, along some line. Let one of them have velocity v1=v, and the other velocity v2=-v.
Then the "usual" way to calculate their relative velocity is simply vrelative = v1 - v2 = 2v. This relative velocity will be greater than the speed of light, c, whenever v is greater than c/2.
But the theory of relativity does more than merely assert by fact that nothing goes faster than light. It sets up a framework in which that is a consistent requirement. One ingredient of that framework is a new formula for how to determine relative velocities. The "usual" formula is no longer the right one.
I'm not going to derive the right formula (you can find derivations elsewhere), but I'm going to tell you what it is. Replace the recipe above for v(relative) with:
vrelative = [v1 - v2]/[1 - (v1*v2/c2)] = 2v/[1 + v2/c2].
Here c2 means "c squared" — c multiplied by itself; it's the same thing that appears in Einstein's famous formula: E = mc2.
For everyday velocities (cars, baseballs, airplanes, bullets, etc.) both v1 and v2 are much less than c, so the additional expression in the denominator of this new equation is very close to 1. Consequently, the "usual" formula works as a very accurate approximation. But for velocities close to the speed of light, the formula given by relativity can vary significantly from the usual one.
Let's apply this formula to the case in your question, when v itself is equal to the speed of light c. In that case,
vrelative =2c/[1 + 1] = 2c/2 = c.
So the photons are not moving away from each other at twice the speed of light. Instead, their relative velocity is still just c.
Alan Chodos, PhD
Associate Executive Officer
American Physical Society
Skip from New Mexico