2.9 Lightlike compactification, b

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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