Observing light speed

Question

Consider an observer B is moving with speed $0.8c$ relative to another observer A in standard-configuration (I think it is called that in english: that the system B is only moving along the $x$-axis of system A).

• At what speed will A observe the light leave the system B in the direction in which B travels away from A? Is it just $0.2c$ or would you have to use the velocity-addition formula to calculate it?
• And what about the light leaving system B in the opposite direction (towards A) seen from A relative to B? B would observe the speed as being the same in every direction (c) relative to itself, but would A see the light moving with the same speed relative to B in every directions?
• If that is actually true, then the speed of light is not the same observed by A relative to A, so I guess they would observe different happenings? So how would one calculate the speed of light emitted by B relative to B as observed by A in different directions?

Remember I do not ask about the speed relative to any of the systems themselves observed inside the system as that would always be $c$. I hope you understand the question. I gladly accept mathematical explanations.

Answer 1

All inertial observers (including A) will see light move at exactly $c$. So both A and B see the light move at $c$. But from A's frame, it appears that the light moves with a speed relative to B of $(c - 0.8c) = 0.2c$. Likewise the light in the other direction would be observed by A to be moving relative to B at $(c + 0.8c) = 1.8c$

As long as you are asking questions about relative speeds in a single frame (that of A in the example), we can use simple vector addition.

There is no problem with measured relative speeds in excess of $c$, because nothing physical is really moving at that rate relative to an inertial frame. As an example, two photons fired in opposite directions will have a distance between them that is measured to be increasing at the rate of $2c$.