# Lorentz Transformation

## Homework Statement

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

A) Write down the matrix $$\Lambda^{\alpha}_{\bar{\beta}}(-v)$$

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

## Homework Equations

None other than that given:

## The Attempt at a Solution

For A) would it just be:

$$$\left( \begin{array}{cccc} - \gamma & v \gamma & 0 & 0 \\ v \gamma & -\gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$$

Because we're just switching signs?

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