BiGyElLoWhAt said:
Another question that I feel is important to my conceptual understanding of relativity.
Can theses conclusions (relativity) be reached a "Doppler effect" method?
Well, there is an interesting connection between relativity and the nonrelativistic Doppler shift.
Suppose that we have two observers, Alice, who is at rest in some frame, and Bob, who is moving at speed v away from Alice, according to that frame. Let Alice send a light signal to Bob. Let f_A be the frequency of the light as sent by Alice, and let f_B be the frequency of the light as received by Bob. Using the nonrelativistic Doppler shift formula, we find:
f_B/f_A = 1 - v/c
Now, suppose that Bob sends a light signal toward Alice. Let f_B' be the frequency as sent by Bob, and let f_A' be the frequency as received by Alice. Then the Doppler shift formula is:
f_A'/f_B' = 1/(1+v/c)
They aren't the same formulas. So if the nonrelativistic Doppler shift formula were correct, then you could tell whether it was Alice or Bob who was really moving. That would violate the principle of relativity.
But now, let's introduce a time dilation factor F. Assume that, from the point of view of Alice's rest frame, Bob's clock is running slow by a factor of F. Then when Bob receives a light signal from Alice, it will seem to have a higher frequency by a factor of F. Why is that? Because the measured frequency is the number of oscillations in one second. If Bob's clock is running slow by a factor of F, then that means that what he thinks of as one second is actually F seconds. So he'll measure F times as many oscillations. So, instead of
f_B/f_A = 1 - v/c
we'll have
f_B/f_A = F (1-v/c)
Now, if Bob is the sender, then signals from Bob will have a lower frequency because of time dilation, by the same factor F. So instead of
f_A'/f_B' = 1/(1+v/c)
we'll have
f_A'/f_B' = (1/(1+v/c))/F
Now, if the time dilation factor F just happens to be the right amount, it's possible to make those two ratios equal (and so it would be consistent with relativity--it wouldn't matter whether it was Bob who was moving, or Alice).
f_A'/f_B' = f_B/f_A
To get this to work out, it must be that
F (1-v/c) = (1/(1+v/c))/F
or
F^2 = 1/(1-v^2/c^2)
So if the time dilation factor F is \gamma = 1/\sqrt{1-v^2/c^2}, then the principle of relativity is satisfied by Doppler shift, and it's impossible to determine from Doppler shift whether Alice or Bob is at rest.
