2.10 A spacetime orbifold in two dimensions

A First Course in String Theory

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Consider a two-dimensional world with coordinates \displaystyle{x^0} and \displaystyle{x^1}.

A boost with velocity parameter \displaystyle{\beta} along the \displaystyle{x^1} axis is described by the first two equations in (2.36). We want to understand the two-dimensional space that emerges if we identify

\displaystyle{(x^0, x^1) \sim ({x'}^0, {x'}^1)}.

We are identifying spacetime points whose coordinates are related by a boost!

(a) Use the result of Problem 2.2, part (a), to recast (1) as

\displaystyle{(x^+, x^-) \sim \left( e^{-\lambda} x^+, e^{\lambda} x^- \right)}, where \displaystyle{e^\lambda \equiv \sqrt{\frac{1+\beta}{1-\beta}}}.

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\displaystyle{  \begin{aligned}  (x')^0 &= \gamma (x^0 - \beta x^1) \\  (x')^1 &= \gamma (- \beta x^0 + x^1) \\  \end{aligned}}

\displaystyle{  \begin{aligned}  \begin{bmatrix} (x^+)' \\ (x^-)' \end{bmatrix}   &= \begin{bmatrix}         \gamma (1-\beta) & 0 \\         0 & \gamma (1+\beta) \\      \end{bmatrix}   \begin{bmatrix} x^+ \\ x^- \end{bmatrix} \\  &=     \begin{bmatrix}         \frac{1}{\sqrt{1 - \beta^2}} (1-\beta) x^+ \\         \frac{1}{\sqrt{1 - \beta^2}} (1+\beta) x^- \\      \end{bmatrix} \\  &=     \begin{bmatrix}         e^{- \lambda} x^+ \\         e^{\lambda} x^- \\      \end{bmatrix} \\  \end{aligned}}

— Me@2021-05-04 10:48:28 PM

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2021.05.05 Wednesday (c) All rights reserved by ACHK