2.10 A spacetime orbifold in two dimensions, 5

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A First Course in String Theory

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(c) Use the expression

\displaystyle{-ds^2 = -2dx^+ dx^-}

for the interval to show that any curve in (3) is spacelike.

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\displaystyle{ \begin{aligned} x^+ x^- &= a^2 \\ x^- &= a^2 \frac{1}{x^+} \\ d x^- &= a^2 \frac{-1}{(x^+)^2} d x^+ \\ \end{aligned}}

\displaystyle{ \begin{aligned} -ds^2 &= -2dx^+ dx^- \\ &= \frac{2 a^2}{(x^+)^2} (d x^+)^2 \\ ds^2 &< 0 \\ \end{aligned}}

— Me@2021-12-27 12:24:09 PM

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