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Okay. So I've been told that the speed of light is constant and cannot violate Galileo's dictum, but even if it weren't constant (in a vacuum), how would it violate it anyway?

Say you are on a train and you are stationary. Another train is traveling towards you with velocity v and shoots a bullet with a velocity j. The apparent speed from your point of view would be v+j.

If you were instead traveling towards the train, at a velocity v, and they shot the bullet at velocity j, the perceived velocity of the bullet would still be v+j.

Thus, it must be impossible to tell if you are moving towards something or if something is moving towards you, even if the bullet's speed from an observer's point of view is different (in one example, it is v+j, in the other it is just j)

So if you replace the bullet with light, it would not have to be measured as being the same speed by an observer for the people on the trains to not be able to tell whether they were moving.

Can someone explain why it has to be constant not to violate Galileo's Dictum?

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I assume that with "Galileo's Dictum" you mean the principle of relativity. In short, this principle states that only relative motion is detectible, and that 'absolute motion' has no place in physics.

Galilean relativity (the summation of velocities you refer to) embodies this principle of relativity for mechanics.

However, electromagnetics is at odds with such Galilean relativity. Einstein demonstrated that Galilean relativity can be extended into a more generic form of relativity (special relativity) that embodies electromagnetics. It is this special relativity that forces us to conclude that te speed of light is invariant (the same for each obeserver, regardless her/his state of motion).

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Could you please elaborate on how you could tell if an object is in absolute motion if the speed of light was NOT constant? (I know it is, but I'm just asking about the consequences if it weren't) – Bguy92 Aug 19 '12 at 18:03
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If light propagated through (relative to) some background substance, like the 'aether' theories of years gone by, we would be able to measure the speed of our frame of reference relative to the aether, simply by measuring the speed of light in our frame. Note that this is subtly different to measuring 'absolute motion', since you can still only measure relative to the aether. In any case, the aether doesn't exist so we needn't worry too much about this sort of thing – poorsod Aug 19 '12 at 21:17
@Bguy92 -- I interpret your question as "could one invent a theory of relativity in which, despite the speed of light NOT being invariant, one could still detect only relative motions?" Answering questions assuming such hypothetical situations is always awkward, as it is not clear what else can be assumed to 'give'. This particular question can be answered with "NO", at least within any theoretical framework in which Maxwell's laws of elctromagnetics are valid. By measuring how the speed of light would deviate from the theoretical (absolute) value, one could infer absolute velocities. – Johannes Aug 20 '12 at 3:56

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