\(\S\)19 等価原理
- 式\((19.8)^{\prime}\)の導出
ドラゴン人間 作成
（19.8）より、ｆはuの関数なので
\begin{align*}
\frac{dX^{\mu}}{d\tau}&=\frac{df^{\mu}}{d\tau}=\frac{df^{\mu}}{du^{\rho}}\frac{du^{\rho}}{d\tau}\\\\
\frac{d^2X^{\mu}}{d\tau^2}&=\frac{d}{d\tau}(\frac{df^{\mu}}{du^{\rho}}\frac{du^{\rho}}{d\tau})\\
&=\frac{du^{\rho}}{d\tau}\frac{d}{d\tau}(\frac{df^{\mu}}{du^{\rho}})+\frac{df^{\mu}}{du^{\rho}}\frac{d}{d\tau}(\frac{du^{\rho}}{d\tau})\\
&=\frac{du^{\rho}}{d\tau}\frac{d^2f^{\mu}}{du^{\rho}du^{\sigma}}\frac{du^{\sigma}}{d\tau}+\frac{df^{\mu}}{du^{\rho}}\frac{d^2u^{\rho}}{d\tau^2}\\
&=0
\end{align*}
最後の等式に両辺に\(\frac{du^{\rho}}{df^{\mu}}\)をかけて、
\begin{align*}
\frac{d^2u^{\rho}}{d\tau^2}+\frac{du^{\rho}}{df^{\mu}}\frac{d^2f^{\mu}}{du^{\rho}d^{\sigma}}\frac{du^{\rho}}{d\tau}\frac{du^{\sigma}}{d\tau}=0
\end{align*}
ここで\(X^{\mu}=f^{\mu}\)より、（19.8）となる。