# Speed of light travel

I have just started special theory of relativity. The limiting speed known as speed of light fascinated me most. I asked my teacher:

Consider two massless objects moving in the same direction at the speed of light. What would be their relative velocity w.r.t each other?

My teacher tells me that their relative velocity would be zero. But then the speed of light is universal constant regardless of the motion of its frame of reference, so shouldn't their relative velocity be $c$? What is their relative velocity and how? Do objects moving at the speed of light obey law of addition of velocities?

• Can you draw a diagram showing the objects you're asking about and what their velocities are. Dec 23, 2013 at 14:22
• I am new here,so i don't know enough latex to draw digrams;( Dec 23, 2013 at 14:25
• The relative velocity is zero, but the relative velocity wrt to YOU is $c$. And you can always use MSPaint or semething like that. Dec 23, 2013 at 14:36
• @ jinawee,speed of light is said to be independent of the motion of frame of refrence,its always the same,so actually its never relative,am i right? Dec 23, 2013 at 14:39
• Are you confusing relative speed and speed? Dec 23, 2013 at 14:42

But then the speed of light is universal constant regardless of the motion of its frame of reference, so shouldn't their relative velocity be $$c$$? What is their relative velocity and how?

EDIT Upon further review (special thanks to Alfred's comment), I think my original answer is incorrect. It turns out that the question of relative velocities of photons moving in the same direction is a meaningless question. The reason is as follows.

For two objects A and B moving as such,

                       v

               u      -------> A
-------> B
------------------ (ground)


The velocity of B in A's frame is then $$u'=\frac{u-v}{1-\frac{uv}{c^2}}$$ Notice the denominator? For $$u=v=c$$, this is zero and we get 0/0 which is an undefined operation, hence the meaningless question.

Do objects moving at the speed of light obey law of addition of velocities?

Not exactly. The Galilean velocity addition, $$s=u+v$$ does not hold for large-velocity objects. We use the "composition law", $$s=\frac{u\pm v}{1\pm\frac{uv}{c^2}}$$ where $$\pm$$ depends on directions/frames. If $$uv\ll c^2$$, then this does reduce to the Galilean transformation.

• @ Kyle Kanos So speed of light is relative? Since speed of 1st particle w.r.t other becomes zero? Dec 23, 2013 at 14:48
• "It is just that, in their co-moving reference frame," There are no reference frames with speed c. Dec 23, 2013 at 14:48
• Can someone explain the downvote? Dec 23, 2013 at 18:58

You state "two massless objects moving in the same direction at the speed of light".

This implies the two objects have the same velocity, so the relative velocity is zero.

If two massless objects travel in opposite directions, both at the speed of light relative to a stationary observer, then the relative velocity of the two objects is still the speed of light.

1. You said that they travel in parallel then you know that the relative speed between them is 0.
2. But suppose that one, or both, intends to measure the relative speed. They send a light beam and wait forever for the returning reflected beam. Obviously they must conclude that the other is at an infinite distance apart. This is the way we measure distances: with light.

note: although the speed of light is constant and measured $$c$$ by any observer it does not invalidate (it is easy to explain) that the speed of light is constant irt to the background, the medium.