# Derivation of Lorentz transformations

JT7
It seems that the common approach to obtain the equations for the Lorentz transformations is to guess at its form and then, by considering four seperate situations, determining the values for the constants. From these equations, things like time dilation and length contraction can be worked out. Now, my goal was to go the other way around: starting from time dilation and length contraction, arrive at the Lorentz equations.

Suppose that in the reference frame O, the reference frame O' is moving at a speed v in the x-direction, with their origins coinciding at t = 0. An event E occurs at (x, y, z, t) in the O frame. It's straightforward to show that y' = y and z' = z. Next I considered x'. In the O frame, the distance between O' and E is x - vt. Because the ruler O' uses is shortened by a factor of γ, she will then measure the distance x - vt as being greater and O measures it, by a factor of γ. Thus, x' = γ(x - vt) (if this approach is incorrect, let me know!).

However, I'm having trouble with t'. I know that t' = γ(t - vx/c2). I assume that the -vx/c2 term comes from that, because O' believes that she's at rest, when the light emitted from the event reaches her, she doesn't treat herself as moving into the light, and thus there's a discrepancy as to how long before the light reaches O' did the event actually occur. Unfortunately, I can't arrive algebraically at this term. Finally, the gamma factor. I assume this comes from time dilation, but the wouldn't O' 's clock be running slower? So wouldn't the term have to be 1/γ (because less time transpires on her clock)? Can someone please tell me how to get the final Lorentz term this way? I know there are probably other, easier routes, but for personal reasons I would like to know how to do it this way. Thanks a lot!

Remember that you have to include the possibility that clocks which are synchronized in one frame will be out-of-sync in another (relativity of simultaneity). In post #14 of this thread I derived the Lorentz transformation using an approach like the one you're suggesting, by putting in variables for length contraction, time dilation and the amount by which two clocks would be out-of-sync, and then solving for these variables:

JT7
Hey thanks a lot for the reply. I read through it, and I seem to understand it (later tonight I'll go through it more thoroughly). Some questions: first, if instead of considering there to be out-of-sync clocks in the O' frame, would taking into account the fact that, in the O frame, light hitting O' will take less time than O' thinks it should (because she's moving into the beam of light in the O' frame) be the same thing (as that, I believe, is the cause of loss of simultaneity)? Because if that's true, then (for personal tastes; I believe this to be more pleasing a route) how would you be able to account for this difference? It should turn out to be vx/c^2, but I can't seem to get the algebra to work. I know that the many clocks method is similar, but I would like to be able to do it this way too. Thanks!

JM
JT7, Look at Einsteins original derivation in his 1905 paper 'On the electrodynamics of moving bodies' ( in the Dover book The Principle of Relativity). There is no guesswork and the use of the Postulate of Constant Light Speed is shown.
JM

Hey thanks a lot for the reply. I read through it, and I seem to understand it (later tonight I'll go through it more thoroughly). Some questions: first, if instead of considering there to be out-of-sync clocks in the O' frame, would taking into account the fact that, in the O frame, light hitting O' will take less time than O' thinks it should (because she's moving into the beam of light in the O' frame) be the same thing (as that, I believe, is the cause of loss of simultaneity)?
When you say "take less time", time between what two events? Is it between the event of the light being emitted from one end of the ruler and the event of the light hitting the observer O' at the other end? If so how does the observer O' measure this time? Are you saying that instead of O' having synchronized clocks at either end of the ruler and measuring the time of each event locally, O' could instead just note the length L of the ruler in her frame, and then based on the assumption that she is at rest and that the light moves at c, conclude that the time between the events must have been L/c? That would be fine.

By the way, I realized in retrospect that what I proved there was not quite the same as what you were asking--I showed that starting from the two basic postulates of SR, one could derive the familiar formulas for length contraction and time dilation. But you're asking how, given length contraction and time dilation, we can the derive the Lorentz transformation equations. OK, suppose an event E happens at (x1,t1) in the O frame. Since the ruler used to mark position in the O' frame is moving at speed v, any point on the O' ruler will move a distance of vt1 between t=0 and t=t1. So, the marking M on the O' ruler that's at position x=x1 at time t=t1 in the O frame must have been at position x1 - vt1 at time t=0 in the O frame. And we know that at time t=0 the marking x'=0 on the O' ruler coincided with position x=0 in the O frame, so in the O frame the distance between M and the x'=0 mark on the O' ruler must be x1 - vt1. But since we know that in the O frame the O' ruler is shrunk by a factor of sqrt(1 - v2/c2), that means that in the O' frame the distance between M and x'=0 is larger than this by a factor of 1/sqrt(1 - v2/c2) = gamma, so the mark M must have a reading of gamma*(x1 - vt1) on the O' ruler.

Now let's say the observer O' sits at position x'=0 on the O' ruler, and at time t=0 in the O frame this was at x=0, after which she moved in the positive x direction with speed v. Suppose that when the event E happens at x=x1 and t=t1 in the O frame, it sends a beam of light towards O'. In the O frame, O' will be at position x=vt1 at time t1, so the initial distance between O' and the event E is x1 - vt1 if E happened further in the +x direction than O' was at that point, or vt1 - x1 if O' was further in the +x direction at that time. If E happened further in the +x direction, then O' is moving at v in the +x direction while the light is moving at c in the -x direction, so the distance between them is shrinking at a speed of (c + v), meaning the light will take a time of (x1 - vt1)/(c + v) to reach O' after being emitted at E. And since it was emitted at t1, the time t that it reaches O' as seen in the O frame will be:

[t1] + [(x1 - vt1)/(c + v)]
= [(ct1 + vt1)/(c + v)] + [(x1 - vt1)/(c + v)]
= (x1 + ct1)/(c + v)
= (c - v)*(x1 + ct1)/(c2 - v2)
= (c - v)*(x1 + ct1)/((c2)*(1 - v2/c2))

On the other hand, if O' was further in the +x direction, then O' is moving at v in the +x direction while the light is moving at c in the +x direction too trying to catch up with O' from behind, so in this case the distance between them is only shrinking at a speed of (c - v), so the light will take a time of (vt1 - x1)/(c - v) to reach O' after being emitted at E. And again, it was emitted at t1, so the time t it reaches O' as seen in the O frame will be:

[t1] + [(vt1 - x1)/(c - v)]
= [(ct1 - vt1)/(c - v)] + [(vt1 - x1)/(c - v)]
= (ct1 - x1)/(c - v)
= (c + v)*(ct1 - x1)/(c2 - v2)
= (c + v)*(ct1 - x1)/((c2)*(1 - v2/c2))

Now the clock of O' is slowed down by a factor of sqrt(1 - v2/c2) in the O frame, and her clock read t'=0 at time t=0 in the O frame, so at any later time t in the O frame her clock reads t*sqrt(c2 - v2)/c. So in the first scenario where E was further in the +x direction, the light reached O' when her own clock read:

sqrt(1 - v2/c2)*(c - v)*(x1 + ct1)/((c2)*(1 - v2/c2))
= (c - v)*(x1 + ct1)/((c2)*sqrt(1 - v2/c2))

In the second scenario where O' was further in the +x direction, the light reached O' when her own clock read:

sqrt(1 - v2/c2)*(c + v)*(ct1 - x1)/((c2)*(1 - v2/c2))
= (c + v)*(ct1 - x1)/((c2)*sqrt(1 - v2/c2))

In both cases, if the event E happened at the x' = gamma*(x1 - vt1) mark on her ruler, she must subtract a time of gamma*(x1 - vt1)/c from the time she observed light from E to get the actual time of E in her frame (that's if the mark is at a positive value of her x' ruler, if it's at a negative value then she would subtract a time of -gamma*(x1 - vt1)/c. It will be at a positive value in the first scenario where the event E happened further in the +x direction, and at a negative value in the second scenario where O' was further in the +x direction when E occurred). And gamma*(x1 - vt1)/c = c*(x1 - vt1)/(c2*sqrt(1 - v2/c2)). So if we subtract this from the observed time in the first scenario to get the actual time of E in the O' frame, we get:

[(c - v)*(x1 + ct1)/((c2)*sqrt(1 - v2/c2))] - [c*(x1 - vt1)/(c2*sqrt(1 - v2/c2))] =
(cx1 + c2*t1 - vx1 - cvt1 - cx1 + cvt1)/(c2*sqrt(1 - v2/c2)) =
(c2*t1 - vx1)/(c2*sqrt(1 - v2/c2)) =
(t1 - vx1/c2)/sqrt(1 - v2/c2)

And if we subtract -c*(x1 - vt1)/(c2*sqrt(1 - v2/c2)) from the observed time in the second scenario to get the actual time of E in the O' frame, we get:

[(c + v)*(ct1 - x1)/((c2)*sqrt(1 - v2/c2))] - [-c*(x1 - vt1)/(c2*sqrt(1 - v2/c2))] =
(c2*t1 - cx1 + cvt1 - vx1 + cx1 - cvt1)/(c2*sqrt(1 - v2/c2)) =
(c2*t1 - vx1)/(c2*sqrt(1 - v2/c2)) =
(t1 - vx1/c2)/sqrt(1 - v2/c2)

So, in both scenarios we find that the time of E in the O' frame is gamma*(t1 - vx1/c2).

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