Difference between revisions of "Forest Relativity Notes"

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=Lorentz Transformations=
 
=Lorentz Transformations=
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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>.
  
 
[[Image:ForestRelativityLorentzFrame.jpg]]
 
[[Image:ForestRelativityLorentzFrame.jpg]]

Revision as of 16:42, 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

[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]