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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2021.04.07 Wednesday (c) All rights reserved by ACHK