How Do You Calculate the Inverse Lorentz Transformation Matrix?

[schwarzschild]

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"?

 

[anathema]

Yes, your answer for A) is correct. For B), the transformation matrix would be the inverse of the matrix in A), so it would be:

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

Justification: The Lorentz transformation matrix from \bar{O} to O is the inverse of the matrix from O to \bar{O}, as you mentioned. This is because if we apply the transformation from \bar{O} to O to a vector \vec{A} in \bar{O}, we should get back the original vector \vec{A} in O.

Additionally, the Lorentz transformation preserves the length of a vector, so the entries in the first two rows and columns should have the same magnitude but opposite signs as in the matrix from A). The last two rows and columns remain unchanged because the transformation does not affect the y and z components of the vector.