# 快子電話

## 用快子傳訊號到過去

${\displaystyle (t,x)=\left(t,\alpha t\right)}$

${\displaystyle (t^{\prime },x^{\prime })=\left(\gamma (1-\alpha \beta )t,\gamma (\alpha -\beta )t\right)}$

${\displaystyle t={\frac {L^{\prime }}{\gamma (\alpha -\beta )}}}$

${\displaystyle t^{\prime }={\frac {1-\alpha \beta }{\alpha -\beta }}L^{\prime }}$

${\displaystyle T^{\prime }={\frac {L^{\prime }}{\alpha }}+t^{\prime }=\left({\frac {1}{\alpha }}+{\frac {1-\alpha \beta }{\alpha -\beta }}\right)L^{\prime }}$

${\displaystyle \beta >{\frac {2\alpha }{1+\alpha ^{2}}}}$${\displaystyle T^{\prime }<0}$。也就是甲在發出訊號之前就收到乙回信。 整個過程的世界線在右圖。

${\displaystyle (t_{0}^{\prime },x_{0}^{\prime })=\left(-{\frac {L^{\prime }}{\alpha }},L^{\prime }\right)}$

${\displaystyle (t_{0},x_{0})=\left(\gamma (\beta -{\frac {1}{\alpha }})L^{\prime },\gamma (1-{\frac {\beta }{\alpha }})L^{\prime }\right)}$

${\displaystyle \beta >{\frac {1}{\alpha }}}$${\displaystyle t_{0}>0}$。也就是乙收到的訊號來自他的未來。

## 參考文献

1. ^ R. C. Tolman. Velocities greater than that of light. The theory of the Relativity of Motion. University of California Press. 1917: 54. OCLC 13129939.
Reprint: R. C. Tolman. Velocities greater than that of light. The Theory of the Relativity of Motion. BiblioLife. 2009: 54. ISBN 978-1-103-17233-7.