Question

What is the speed of a photon with respect to another photon if (a) the two photons are going in the same direction and (b) they are going in opposite directions?

Answer 1

(a) In relativity, the relative speed of two objects$\left(v_{rel}\right)$ moving in the same direction with speeds u and v is given by
$v_{\mathrm{rel}}=\frac{u-v}{1-{\displaystyle \frac{uv}{c^2}}}...\left(1\right)$
As the photons are moving with the speed of light, u = c and v = c.
On substituting the values of u and v in equation (1), we get:
$v_{rel}$ = 0
Thus, relative velocity of a photon with respect to another photon will be 0, when they are going in the same direction.
(b) In relativity, relative speed of two objects moving in opposite directions with speeds u and v is given by
$v_{\mathrm{rel}}=\frac{u+v}{1+{\displaystyle \frac{uv}{c^2}}}...\left(2\right)$
We know that a photon travels with the speed of light. Therefore, u = c and v = c
On substituting the values of u and v in equation (2), we get:
$v_{rel}=c$
Thus, the relative velocity of a photon with respect to another photon will be equal to the speed of light when they are going in opposite directions.