- #1

ergospherical

- 1,020

- 1,300

S[x,e] = \dfrac{1}{2} \int d\lambda e^{-1} \dot{x}^{\mu} \dot{x}^{\nu} g_{\mu \nu}(x)

\end{align*}Let ##\xi## be a conformal Killing vector of ##ds^2##, then under a transformation ##x^{\mu} \rightarrow x^{\mu} + \alpha \xi^{\mu}## and ##e \rightarrow e + \dfrac{1}{4} \alpha e g^{\mu \nu} (L_{\xi} g)_{\mu \nu}## the action changes (to first order in ##\alpha##) as\begin{align*}

S[x,e] &\rightarrow \dfrac{1}{2} \int d\lambda e^{-1} \left(1- \dfrac{1}{4} \alpha g^{\rho \sigma} (L_{\xi} g)_{\rho \sigma} \right)(\dot{x}^{\mu} + \alpha \dot{\xi}^{\mu})(\dot{x}^{\nu} + \alpha \dot{\xi}^{\nu}) g_{\mu \nu}(x) \\ \\

&= S[x,e] + \dfrac{1}{2} \int d\lambda e^{-1} \left( \alpha [\dot{x}^{\mu} \dot{\xi}^{\nu} + \dot{x}^{\nu} \dot{\xi}^{\mu}] - \dfrac{1}{4}\alpha g^{\rho \sigma} (L_{\xi} g)_{\rho \sigma} \dot{x}^{\mu} \dot{x}^{\nu} \right) g_{\mu \nu}(x)

\end{align*}Since ##\xi## is a conformal Killing vector it satisfies ##(L_{\xi} g)_{\mu \nu} = \Omega^2 g_{\mu \nu}## therefore ##\dfrac{1}{4} g^{\rho \sigma} (L_{\xi} g)_{\rho \sigma} = \dfrac{1}{4} \Omega^2 {\delta^{\rho}}_{\rho} = \Omega^2## so\begin{align*}

S[x,e] \rightarrow S[x,e] + \dfrac{1}{2} \int d\lambda e^{-1} \alpha \left( [\dot{x}^{\mu} \dot{\xi}^{\nu} + \dot{x}^{\nu} \dot{\xi}^{\mu}] - \Omega^2 \dot{x}^{\mu} \dot{x}^{\nu} \right)

\end{align*}This transformation should leave ##S## invariant, so how does one show that the remaining integral vanishes? Is it a case of integrating by parts and neglecting a boundary term, in which case, which one?