Derivation of Time Dilation and Length Contraction from Lorentz Transformation

1. Nov 19, 2008

mahapan

Hello there, a simple special relativity question, how can the Time Dilation and Length Contraction Formulas be derived from the Lorentz Transformation Equations?

2. Nov 19, 2008

Naty1

3. Nov 19, 2008

Staff: Mentor

Derive them for yourself. Here are a few hints:
(1) For time dilation, consider a moving clock. Pick two events recorded by that clock: Δx' = 0.
(2) For length contraction, consider a moving stick. Measure the distance between the ends of that stick at the same time: Δt = 0.

4. Nov 19, 2008

Naty1

I'm not sure doc al's suggestion is realistic for most. yet not impossible either...

The people who derive such original insights have usually been immersed in the work full time and spent a lot of time thinking about the issues...and made many,many errors before successfully creating new insights.

As an example, I have read a little about Einstein's initial work on relativity...studying frames of reference and Newton's views as he decided on a framework to develop his own work...He also used Lorentz and Fitzgerald and Riemann's curved geometric insights....In developing relativity, today's physicsts have been able to review Einsteins original notes and in light of what we know today, his many errors are apparently very obvious...but were not to Einstein!!! He just refused to give up, retraced his steps repeatedly until he discovered where he had gone off course, corrected himself and proceeded again to make more erros...eventually he got there!!!!

5. Nov 19, 2008

Fredrik

Staff Emeritus
The Lorentz transformation in units such that c=1 is

$$\gamma\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix}$$

where $\gamma=1/\sqrt{1-v^2}$. Try applying this to the points with coordinates

$$\begin{pmatrix}T\\0\end{pmatrix}$$

and

$$\begin{pmatrix}0\\L\end{pmatrix}$$

6. Nov 19, 2008

Staff: Mentor

Actually, given the hints and the fact that we know the answer that we're looking for, it should be relatively easy. After all, you're starting with the Lorentz transformations already known. (And in case you can't find them, here they are: Basic Equations of Special Relativity)

7. Nov 19, 2008

bernhard.rothenstein

I think that it is worth to know that the Lorentz transformations could be derived from length contraction and time dilation.

8. Nov 20, 2008

Staff: Mentor

I would certainly agree, and that's exactly what's done in many elementary treatments of SR. (After first deriving length contraction, time dilation, and clock desynchronization from the more basic assumptions of SR via simple thought experiments.)

But that's not what was asked for here.

9. Nov 22, 2008

bernhard.rothenstein

Make shure that you have a good understanding of the concepts of PROPER TIME INTERVAL
and PROPER LENGTH.
Present the Lorentz transformations as
dx=g[dx'+Vdt'] (1)
dt=g[dt'+Vdx'/cc] (2)

If dx'=0, the wrist watch of the observer at rest in I' measures a proper time interval dt(0), observers from I measure a coordinate time interval dt related by the time dilation formula
dt=gdt(0) as results from (1). Knowing the conditions under which proper length is measured you obtain from (1) the formula that acconts for length contraction.

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