Difference between two massive particles approaching $c$ moving away from us and two photons doing the same

Question

When two massive bodies move oppositely away from us, they can only approach the speed of light. Two photons, on the other hand, have the speed of light wrt to us. Although the situation of two massive particles approaching the speed of light (oppositely away from us) looks similar to the situation where two photons move away from us (apart from the energy contained in both relative to each other), one can always find a restframe for one of the massive particles, from which the other seems to move away approaching the speed of light.

For a photon though, there doesn't exist a restframe. Can we say that the two photons move wrt each other with the speed of light exactly? How can that be if they don't have a restframe?

Can we compare it with a situation where the speed of light is infinite, like in Newtonian mechanics, and masses will never be able to reach infinite speed, and a restframe for light can never be found?

Answer 1

Don't worry about relative speeds. Worry about events. Draw your events on a spacetime diagram and compute the intervals between them.

In your case the interesting events are

1. particles $A$ and $B$ are emitted back-to-back
2. [time passes]
3. $A$ interacts with some detector
4. $B$ interacts with some detector

The intervals $\Delta s_{13}$ and $\Delta s_{14}$ are timelike if the particles are massive, and lightlike (or "null") if the particles are massless. In the symmetric case where the detection events are simultaneous in the experiment's rest frame, the interval $\Delta s_{34}$ is spacelike, and observers moving in different directions would disagree about which detection happened "first."

You would like to add an additional event,

2. $B$ receives information about $A$'s trajectory and computes a relative velocity.

In order for this computation to take place, there are two requirements. First, there has to be information about $A$'s trajectory. That information has to come from some interaction between $A$ and its environment ... which we might as well go ahead and call event #3, the detection of particle $A$. Second, at the time the computation occurs, this detection of $A$ must lie in $B$'s past light cone.

For massive but ultra-relativistic particles, a tiny fraction of $A$'s trajectory lies in $B$'s past, and so it makes sense for $B$ to ask questions like "what is $A$'s velocity relative to me?" But for massless particles with back-to-back velocities, $A$'s entire trajectory (apart from their co-creation) is spacelike-separated from $B$. You cannot assign a speed to two spacelike-separated events, because you cannot establish which of them happens first.

Answer 2

I'd say the velocity of a photon relative to another photon is not well-defined. Making a measurement of velocity requires a frame of reference, but such a frame of reference doesn't exist for a photon.

If we consider your situation such that

$$A\longleftarrow B\longrightarrow C$$

A is a photon moving relative to us (A) with $v=c$. The velocity of C relative to us is $u<c$. Then according to special relativity the velocity of A measured by C is

$$v'=\frac{v+u}{1+\frac{vu}{c^2}}=\frac{c+u}{1+\frac uc}$$

since $v=c$. You see that if $u$ approaches $c$, $v'$ approaches $c$. In fact if we just set $u=c$, we don't get a mathematically indeterminate result (unlike the case for motion in the same direction which gives $v'=0/0$ for $u=c$) but simply $v'=c$.

However this doesn't mean that the relative velocity of two photons is actually $c$. It is, as already explained above, not defined since making a measurement of velocity from a (non-existent!) frame in which light is at rest doesn't have any physical meaning.