# 2.9 Lightlike compactification, c

A First Course in String Theory

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… Find the velocity parameter of $\displaystyle{S'}$ with respect to $\displaystyle{S}$ and the compactification radius in the Lorentz frame $\displaystyle{S'}$.

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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 \sqrt{R^2 + R_S^2} \\ \end{bmatrix} \end{aligned} }

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

\displaystyle{ \begin{aligned} R + \beta \sqrt{R^2 + R_S^2} &= 0 \\ \beta^2 &= \frac{R^2}{R^2 + R_S^2} \\ \gamma &= \sqrt{\frac{R^2 + R_S^2}{R_S^2}} \\ \end{aligned} }

\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \end{bmatrix} &\sim \begin{bmatrix} ct' \\ x' \\ \end{bmatrix} + \begin{bmatrix} 0 \\ R_S \\ \end{bmatrix} 2 \pi \\ \end{aligned} }

— Me@2021-04-07 07:02:14 AM

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