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Lorentz Transformation

  • #1

Homework Statement



Let [tex] \Lambda^{\bar{\alpha}}_{\beta} [/tex] be the matrix of the Lorentz transformation from O to [tex] \bar{O} [/tex], given as: [tex] \bar{t} = \frac{t-vx}{\sqrt{1-v^2}}, \bar{x} = \frac{-vt+x}{\sqrt{1-v^2}}, \bar{z} = z, \bar {y} = y [/tex]. Let [tex] \vec{A} [/tex] be an arbitrary vector with components [tex] (A_0, A_1, A_2, A_3) [/tex] in frame O

A) Write down the matrix [tex] \Lambda^{\alpha}_{\bar{\beta}}(-v) [/tex]

B) Write down the Lorentz transformation matrix from [tex] \bar{O} [/tex] to O, justifying each entry.


Homework Equations



None other than that given:

The Attempt at a Solution



For A) would it just be:

[tex] \[ \left( \begin{array}{cccc}
- \gamma & v \gamma & 0 & 0 \\
v \gamma & -\gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array} \right)\] [/tex]

Because we're just switching signs?

And for B) would it just be the same as the transformation from O to [tex] \bar{O} [/tex] as your just switching "names"?
 

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