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Just a basic question:

I know that if you are traveling at $x$ speed the time will pass for you slower than to an observer that is relatively stopped. That's all just because a photon released at the $x$ speed can't travel faster than the $c$ limit.

I want to know what happens if you have two bodies, $A$ and $B$ moving towards each other. If $A$ releases a light beam, and $B$ measures it (the speed of the photons), the speed measured is still the same? The only difference will be the wave length? And if we have the opposite case, $A$ and $B$ are moving away from each other, we get the red shift, but the speed measured will be still the same?

I just want to know if I got it right...

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The speed of light will be the same, yes. This is the fundamental tenet of special relativity - that all inertial observers see the same laws of physics, including universal constants like the speed of light.

And yes, the wavelength $\lambda$ will change. The frequency $\nu$ will also change, since after all we still must have $$ \lambda \nu = c. $$

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Chris, isn't it perhaps a bit misleading to liken laws of physics to universal constants. Your second sentence almost makes it seem as though the principle that inertial observers see the same physical laws implies that the value of the speed of light is frame-invariant. I suppose that if you define the term "physical law" to encompass values of physical constants, then this is true, but that seems somehow misleading to me. – joshphysics Apr 4 '13 at 19:26
I'm construing "physical law" a bit broadly, I agree, but I felt symmetry groups and Lorentz transformations would make things too complicated, at least for me ;) – Chris White Apr 4 '13 at 20:20

Yes, every observer who makes a local measurement of the speed of light will get the same result, i.e. $c$, regardless of where the light came from.

I've used the word local because in general relativity the speed of light can differ from $c$ if it's not measured at your location. The most famous example of this is probably the fact that light slows down to a halt as it approaches the event horizon of a black hole. However even in general relativity a local measurement always returns the value $c$.

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Hey... I didn't quite expect a GR's intro for this question from you ;-) – Waffle's Crazy Peanut Apr 4 '13 at 17:22

Yes. There are two things you gotta take care of. In order for the speed of light to be constant (SR's postulate) relative to the oppositely rushing observers $A$ and $B$, we can use the velocity addition formula. Another Yes. The wavelength will be contracted or elongated and the light seen by $B$ or $A$ would be blue-shifted or red-shifted according to the Relativistic Doppler effect, by how they're moving relatively (towards or away).

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