# Problem 2.2b

A First Course in String Theory

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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.

(b) A rotation with angle $\displaystyle{\theta}$ in the $\displaystyle{x^1, x^2}$ plane. \displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix} &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta & 0 \\ 0 & \sin \theta & \cos \theta & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix} \\ \end{aligned} }

~~~ \displaystyle{ \begin{aligned} \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} (x^+)' \\ (x^-)' \\ y' \\ z' \end{bmatrix} &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta & 0 \\ 0 & \sin \theta & \cos \theta & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x^+ \\ x^- \\ y \\ z \end{bmatrix} \\ \begin{bmatrix} (x^+)' \\ (x^-)' \\ y' \\ z' \end{bmatrix} &= \frac{1}{2} \begin{bmatrix} \cos\theta + 1 & 1 - \cos\theta & -\sqrt{2} \sin\theta & 0 \\ 1 - \cos\theta & \cos\theta + 1 & \sqrt{2} \sin\theta & 0 \\ \sqrt{2} \sin{\theta} & -\sqrt{2} \sin{\theta} & 2 \cos{\theta} & 0 \\ 0 & 0 & 0 & 2 \\ \end{bmatrix} \begin{bmatrix} x^+ \\ x^- \\ y \\ z \end{bmatrix} \end{aligned} }

— Me@2020-03-22 10:16:09 PM

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