2.10 A spacetime orbifold in two dimensions

A First Course in String Theory

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 $\displaystyle{e^\lambda \equiv \sqrt{\frac{1+\beta}{1-\beta}}}$ .

~~~

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

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

— Me@2021-05-04 10:48:28 PM

Physics Town

continues

2.10 A spacetime orbifold in two dimensions

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