# Are components of the velocity of light equal to $c$?

The special theory of relativity says that the speed of light in vacuum is a constant $c$ in all inertial frames. Suppose a ray of light is moving in space. Let the source of light be the origin of our coordinate system and let the ray be moving in some random direction making some finite angles ($\alpha$, $\beta$, $\gamma$) with the $x$, $y$ and $z$ axes. Now, what are the $x$, $y$ and $z$ components of velocity of light in this case? Is the postulate of relativity applicable to components also? If that is the case, then all three components will have to be $c$ and vector sum should exceed $c$!

• you are expected to accept answers if they fulfill the question. – Shashaank Mar 1 '17 at 20:29

What I think is that you are getting the 1st postulate of STR wrong. It's the speed of light that remains invariant.

The velocity of light changes all the time in special inertial frames, even simply because of relative motion. Light can go in different directions.

What stays constant is the speed. The famous Michelson Morley experiment had that diagram in which light goes diagonally. The components add up to c. What stays constant is the speed. And STR says only of the magnitude of light being c.

• I'd clarify that when you say "the components add up to $c$" you mean addition in quadrature, not plain old addition. – David Z Jan 24 '17 at 10:55
• @DavidZ Yes off course vectorial. Thanks for the correction ! – Shashaank Jan 24 '17 at 10:57

If I understand your question correctly, the velocity components of the light ray can be found by using the usual trigonometric calculations you'd use on a typical point particle, but with a magnitude of c. The magnitude of the velocity of the photon/wavefront is what moves at the speed of light. It's components are arbitrary. In fact, breaking down a light ray into its (slower*) components is basically how we prove special relativity.

*relatively

• Thanks for the response. Just to be sure.... does that mean the magnitude of components of light velocity 'çan be' less than c? and postulate of Special theory of relativity does not apply to individual components but only to the net total? – Krishna Jan 24 '17 at 7:15
• Well, I need to make it clear that this only applies in your frame of reference, but yes, the components are either equal to or less than its magnitude c. – guy Jan 24 '17 at 7:23
• And by only applies in your frame, I mean that you may measure c for your y-component but another observer may measure less than c for their y-component. (I was trying to alleviate confusion, but I may have done the opposite :s) – guy Jan 24 '17 at 7:36
• I don't follow your last sentence: is there a motivation for STR along these lines (resolving a velocity vector into components)? – WetSavannaAnimal Jan 24 '17 at 11:11
• @WetSavannaAnimalakaRodVance Sorry for the confusion. I was trying to hint at the proof many intro classes use to introduce time dilation where the path of light moves diagonally in one reference frame but only vertically in another. – guy Jan 24 '17 at 15:43