Difference between revisions of "Forest Relativity Notes"

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;Proper Time : The time measured in the rest frame of the clock.  The time interval is measured at the same x,y,z coordinates because the clock chose is in a frame which is not moving (rest frame).
 
;Proper Time : The time measured in the rest frame of the clock.  The time interval is measured at the same x,y,z coordinates because the clock chose is in a frame which is not moving (rest frame).
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The time given in any frame (t) = \gamma \Tau
  
 
== Proper Length==
 
== Proper Length==

Revision as of 16:53, 30 October 2007

Lorentz Transformations

The picture below represents the relative orientation of two different coordinate systems [math](S, S^{\prime})[/math] . [math]S[/math] is at rest (Lab Frame) and [math]S^{\prime}[/math] is moving at a velocity v to the right with respect to frame [math]S[/math].

ForestRelativityLorentzFrame.jpg

The relationship between the coordinate[math] (x,y,z,ct)[/math] of an object in frame [math]S[/math] to the same object described using the coordinates [math](x^{\prime},y^{\prime},z^{\prime},ct^{\prime})[/math] in frame [math]S^{\prime}[/math] is geven by the Lorentz transformation:

[math]x^{\mu^{\prime}} = \sum_{\nu=0}^3 \Lambda_{\nu}^{\mu} x^{\nu}[/math]

where

[math] x^0 \equiv ct[/math]
[math]x^1 \equiv x[/math]
[math]x^2\equiv y[/math]
[math]x^3\equiv z[/math]
[math]\Lambda = \left [ \begin{matrix} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma &0 &0 \\ 0 &0 &1 &0 \\ 0 &0 &0 &1\end{matrix} \right ][/math]
[math]\beta = \frac{v}{c}[/math]
[math]\gamma = \frac{1}{\sqrt{1 -\beta^2}}[/math]
example
[math]x^{0^{\prime}} = \sum_{\nu=0}^2 \Lambda_{\nu}^0 x^{\nu} = \Lambda_0^0 x^0 + \Lambda_1^0 x^1 \Lambda_2^0 x^2 + \Lambda_3^0 x^2[/math]
[math]ct^{\prime}= \gamma x^0 - \gamma \beta x^1 + 0 x^2 + 0 x^3 = \gamma ct - \gamma \beta x = \gamma(ct -\beta x)[/math]
Or in matrix form the tranformation looks like
[math]\left ( \begin{matrix} ct^{\prime} \\ x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{matrix} \right )= \left [ \begin{matrix} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma &0 &0 \\ 0 &0 &1 &0 \\ 0 &0 &0 &1\end{matrix} \right ] \left ( \begin{matrix} ct \\ x \\ y \\ z \end{matrix} \right )[/math]
Note
Einstein's summation convention drops the [math]\sum[/math] symbols and assumes it to exist whenever there is a repeated subscript and uperscript
ie; [math]x^{\mu^{\prime}} = \Lambda_{\nu}^{\mu} x^{\nu}[/math]
in the example above the[math] \nu[/math] symbol is repeated thereby indicating a summation over [math]\nu[/math].

Proper Time and Length

Proper Time

Proper Time
The time measured in the rest frame of the clock. The time interval is measured at the same x,y,z coordinates because the clock chose is in a frame which is not moving (rest frame).

The time given in any frame (t) = \gamma \Tau

Proper Length

Proer Length
An object length in the object's rest frame.

Invariant Length