2.9 Lightlike compactification, b

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A First Course in String Theory

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Consider coordinates \displaystyle{(ct', x')} related to \displaystyle{(ct, x)} by a boost with velocity parameter \displaystyle{\beta}. Express the identifications in terms of the primed coordinates.

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

\displaystyle{ \begin{aligned} \begin{bmatrix} ct \\ x \\ \end{bmatrix} &\sim \begin{bmatrix} ct - 2 \pi R \\ x + 2 \pi R \\ \end{bmatrix} \end{aligned}}

\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \end{bmatrix} &\sim \begin{bmatrix} c t' \\ x' \end{bmatrix} + \begin{bmatrix} - \gamma - \beta \gamma \\ \gamma + \beta \gamma \\ \end{bmatrix} 2 \pi R \\ \end{aligned}}

— Me@2021-03-30 08:38:16 PM

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