Derivations of the Lorentz transformations
They all have something in common though, the transformations:
If we can show that these equations are wrong then all the derivations must be wrong as well.
If we solve for t in the first equation we get:
t = x/v - x'/vγ
And substitute this into the second equation:
t' = γ(x/v - x'/vγ - vx/c2)
t' = γx(1/v - v/c2) - x'/v
If we now set x' = -vt' (I'm not sure what his means, but I'm sure we can do it,) then:
1/v - v/c2 = 0
v2 = c2
So v = c, I think this is because all the different derivations make the mistake of saying that x = ct at the start and then substituting in x= vt later on. This means that it must be true that v = c.
If you believe that the speed of light is the same regardless of the velocity of the observer (personnally I think it is, I also think the speed of light depends on certain properties of the ether, but I'm open minded,) such that:
dx/dt = dx'/dt' = c
dt'/dt = dx'/dx
And light is a wave so:
c = ω/k
Where ω is the angular frequency and k is the wave number. The elecromagnetic wave equation for the electric field (the magnetic field is the same) is now:
k2(d2E/dx2) = ω2(d2E/dt2)
We can now transform this to the O' refernce frame using the chain rule:
k2(dx'/dx)2(d2E/dx'2) = ω2(dt'/dt)2(d2E/dt'2)
So,
ω'2 = ω2(dt'/dt)2
k'2 = k2(dx'/dx)2
Which just gives:
dt'/dt = ω'/ω
dx'/dx = k'/k
What I think this means is that if an observer is using light to observe something, the changes in times and distances they observe will correlate with the Doppler effect. I think this makes sense because if the peaks of a light wave are in sync with the ticks of a clock in one inertial reference frame, and an observer is observing the clock, the the ticks of the clock should be in sync in every inertial reference frame.
