# If the light velocity is a vector quantity, why vector addition cannot be applied to it?

If the light velocity is a vector quantity, why vector addition cannot be applied to it?

Or the light velocity is not a vector quantity?

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Update 2: It might also be useful add the relation to usual Galilean physics. This will probably only make sense after reading Update 1.

Recall that one can parametrize boosts in (1+1) by $[\cosh(\eta), \sinh(\eta)]$. As it so happens this equals to $[\gamma, \gamma {v \over c}]$. The classical physics correspond to the asymptotic case $c \to \infty$ (i.e. no limit on the speed of light). So this reduces to $\gamma \to 1$, ${v \over c} \to 0$ and that means $\eta \to 0$. So in Galilean case all boosts degenerate to trivial transformation and this is why time and space separate and addition of velocities starts to work.

First, there exists a concept of rapidity. This is a natural variable for parametrization of boosts. First consider a circle. Why circle? Because it has to do with rotations and rotations are very similar to boosts.

Circle is an object given by an equation $x^2 + y^2 = 1$. You can decide that you'll parametrize (part of) it by coordinates $[x, \sqrt{1 - x^2}]$. Now what if you want to rotate the circle by some angle? Can you figure out how will the parametrization change? Maybe you can but I assure you this is not pretty. But there is a nicer way. Let's try parametrizing the circle by an angle $\phi$ so that it would become set of points $[\cos(\phi), \sin(\phi)]$ and now the rotation by angle $\psi$ corresponds to parametrization $[\cos(\phi + \psi), \sin(\phi + \psi)]$. So our parameter is additive! You won't find any better parametrization of the circle than this.

Okay, so we understand circles a little better now. But as already said, rotation in two space dimensions is almost the same thing as boost in (1+1) space-time dimensions. In the same way that rotations (around origin) preserve circles, boosts preserve hyperbolas. So instead of working with $\sin$ and $\cos$ you'll work with hyperbolic functions $\sinh$ and $\cosh$ and instead of $\phi$ you'll obtain rapidity $\eta$.

Now, this only works this nicely in (1+1)-dimensions. In (3+1) you'll have many more interesting effects (similarly to like rotations are strange beasts in 3 dimensions as opposed in 2). But it's still true that like the general 3-dimensional rotations are nicely parametrized by an axis and rotation angle, the boosts are nicely parametrized by the direction of boost and rapidity. So if you perform two rotations about the same axis it's the same as rotation by a sum of the angles and if you perform two boosts in one direction it's the same as doing a boost with added rapidities.

I'll assume you're talking about Galilean principle of relativity whereby the velocities transform by pure addition. This concept breaks down when the speeds one is dealing with are too large. Speed of light is an extreme case of such speed. Then one has to use Special Relativity and instead consider four-vectors transforming by Lorentz transformation. Now, this transformations preserve the Minkowski length of four-vectors (in the same way that rotations preserve length of usual vectors).

The point is that velocity of light corresponds to zero Minkowski length and so light moves at the speed of light in every inertial frame. This is the famous Einstein's postulate.

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Also check out this wikipedia article which contains some actual computations. – Marek Nov 27 '10 at 12:58
You can also mention that there is a kind of adapted addition law. – Raskolnikov Nov 27 '10 at 13:52
Yeah, I was about to submit my own comment on that before I saw yours, @Raskolnikov. Vector addition does apply perfectly well to velocities (including that of light) if you modify the definition of "addition" to the law explained in the Wikipedia article. – David Z Nov 27 '10 at 20:59
@Raskolnikov, @David: yeah, so you want to just add up four-vectors and pretend that it is just "modified" addition of usual vectors; do I get that right? I always hated this approach and I never understood SR until Lorentz transformations and four-vectors where presented to me as what they really where (instead of just some "modified" laws). – Marek Nov 28 '10 at 5:40

The speed of light is a vector quantity and vector summation works perfectly well for it (at least in Special Relativity). You just cannot change the frame of reference.

For example if you have one object moving at c in one direction and another object moving at 1/2c in the opposite direction, then the middle between the two will move at c/2 in the same direction as the first. This is from point of view of a stationary observer.

The distance between the two objects grows as 3/2c. This is in the stationary reference frame of course, the objects themselves will see each other moving at speed of light.

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