Problem 2.2c

A First Course in String Theory

.

2.2 Lorentz transformations for light-cone coordinates.

Consider coordinates \displaystyle{x^\mu = ( x^0, x^1, x^2, x^3 )} and the associated light-cone coordinates \displaystyle{x^\mu = ( x^+, x^-, x^2, x^3 )}. Write the following Lorentz transformations in terms of the light-cone coordinates.

(c) A boost with velocity parameter \displaystyle{\beta} in the \displaystyle{x^3} direction.

\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ z' \\ x' \\ y' \end{bmatrix} &= \begin{bmatrix}         \gamma & -\beta \gamma & 0 & 0 \\         -\beta \gamma & \gamma & 0 & 0 \\              0 & 0 & 1 & 0 \\              0 & 0 & 0 & 1 \\     \end{bmatrix} \begin{bmatrix} c\,t \\ z \\ x \\ y \end{bmatrix}  \\ \end{aligned} }

~~~

\displaystyle{ \begin{aligned}  \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix}  &= \begin{bmatrix}         \gamma & 0 & 0 &  -\beta \gamma \\         0 & 1 & 0 & 0 \\              0 & 0 & 1 & 0 \\          -\beta \gamma & 0 & 0 & \gamma \\     \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}     \\ \end{aligned} }

\displaystyle{ \begin{aligned}  \begin{bmatrix}  1  & 1 & 0 & 0 \\  1  & -1 & 0 & 0 \\  0  & 0 & \sqrt{2} & 0 \\  0  & 0 & 0 & \sqrt{2} \\     \end{bmatrix}  \begin{bmatrix} (x^+)' \\ (x^-)' \\ y' \\ z' \end{bmatrix}  &= \begin{bmatrix}         \gamma & 0 & 0 &  -\beta \gamma \\         0 & 1 & 0 & 0 \\              0 & 0 & 1 & 0 \\          -\beta \gamma & 0 & 0 & \gamma \\    \end{bmatrix} \begin{bmatrix}  1  & 1 & 0 & 0 \\  1  & -1 & 0 & 0 \\  0  & 0 & \sqrt{2} & 0 \\  0  & 0 & 0 & \sqrt{2} \\     \end{bmatrix} \begin{bmatrix} x^+ \\ x^- \\ y \\ z \end{bmatrix} \\   \end{aligned} }

\displaystyle{ \begin{aligned}  \begin{bmatrix} (x^+)' \\ (x^-)' \\ y' \\ z' \end{bmatrix}  &= \frac{1}{2}  \begin{bmatrix} \gamma + 1 & \gamma - 1 & 0 & -\sqrt{2}\,\beta\,\gamma \\ \gamma - 1 & \gamma + 1 & 0 & -\sqrt{2}\,\beta\,\gamma \\  0 & 0 & 2 & 0 \\  - \sqrt{2}\,\beta\,\gamma & - \sqrt{2}\,\beta\,\gamma & 0 & 2 \gamma \end{bmatrix}  \begin{bmatrix} x^+ \\ x^- \\ y \\ z \end{bmatrix}  \\   \end{aligned} }

— Me@2020-04-19 11:52:09 PM

.

.

2020.04.21 Tuesday (c) All rights reserved by ACHK