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Hi all. I'm working on a project that requires me to perform calculations in Fermi normal coordinates to certain orders, mostly 2nd order in the distance along the central worldline orthogonal space-like geodesics. In particular I need a rotating tetrad propagated along the central worldline obeying an arbitrary transport law which is parallel transported along the space-like geodesics so as to generate a tetrad field at each point in the coordinates to 2nd order. I found a very handy paper http://arxiv.org/pdf/gr-qc/0010096.pdf which computes the series expansion I need to arbitrary order but I'm not able to reproduce the calculation fully.
Appendix B of the paper is the relevant section for my question. In (66) the tetrad is expanded in a power series about the central worldline ##\gamma## in terms of the distance ##u## along the space-like geodesics orthogonal to and emanating from the worldline. In (67), ##\xi^{\mu}## is the unit tangent field to each space-like geodesic. The first thing to note is that ##\xi^{\mu}(e_0)_{\mu}|_{u = 0} = 0## by construction hence ##\xi^{\mu}(e_0)_{\mu} = 0## all along the space-like geodesic since both ##\xi^{\mu}## and ##e^{\mu}_0## are parallel transported along the geodesic. Now ## e^{\mu}_0|_{u = 0} = \delta^{\mu}_{0}## so ##\xi^{\mu}(e_0)_{\mu}|_{u = 0} = 0 \Rightarrow \xi^{0}|_{u = 0} = 0## but this does not mean that ##\xi^0 = 0## for all ##u## because ##\xi^{\mu}(e_0)_{\mu} = 0## does not imply ##\xi^{\mu}\delta^0_{\mu} = 0## as is clear both intuitively and from (65). So I will write ##\xi^{i}## in place of ##\xi^{\mu}## below because the final expression involves only ##\xi^{\mu}## and in the end we evaluate at ##u = 0## so we only need to consider the ##\xi^i## terms but you should keep the above in mind.
My first problem then is with (68). We differentiate (67) once with respect to ##\frac{d}{du}## to get [tex]\frac{d^2}{du^2}e^{(\nu)}_{\mu} - \Gamma^{\sigma}_{\mu i, l} e^{(\nu)}_{\sigma} \xi^l \xi^i - \Gamma^{\sigma}_{\mu i}\xi^i \frac{d}{du}e^{(\nu)}_{\sigma} - \Gamma^{\sigma}_{\mu i}e^{(\nu)}_{\sigma} \frac{d \xi^i}{du} \\= \frac{d^2}{du^2}e^{(\nu)}_{\mu} - \Gamma^{\sigma}_{\mu i, l} e^{(\nu)}_{\sigma} \xi^l \xi^i - \Gamma^{\sigma}_{\mu i}\Gamma^{\lambda}_{\sigma l}\xi^i \xi^l e^{(\nu)}_{\lambda} - \Gamma^{\sigma}_{\mu i}e^{(\nu)}_{\sigma} \frac{d \xi^i}{du}\\= \frac{d^2}{du^2}e^{(\nu)}_{\mu} - \Gamma^{\sigma}_{\mu i, l} e^{(\nu)}_{\sigma} \xi^l \xi^i - \Gamma^{\sigma}_{\mu i}\Gamma^{\lambda}_{\sigma l}\xi^i \xi^l e^{(\nu)}_{\lambda} + \Gamma^{\sigma}_{\mu i}e^{(\nu)}_{\sigma} \Gamma^{i}_{l m}\xi^l \xi^m \\ = \frac{d^2}{du^2}e^{(\nu)}_{\mu} - \Gamma^{\nu}_{\mu i, l} \xi^l \xi^i - \Gamma^{\sigma}_{\mu i}\Gamma^{\nu}_{\sigma l}\xi^i \xi^l + \Gamma^{\nu}_{\mu i}\Gamma^{i}_{l m}\xi^l \xi^m = 0[/tex]
where I've used ##\frac{d \xi^i}{du} = -\Gamma^{i}_{lm}\xi^l \xi^m## from the geodesic equation. We thus have [tex]\frac{1}{2!}\frac{d^2 e^{(\nu)}_{\mu}}{du^2}|_{u = 0}u^2 = \frac{1}{2!}(\overset{0}{\Gamma^{\nu}_{\mu i, l}} + \overset{0}{\Gamma^{\sigma}_{\mu i} }\overset{0}{\Gamma^{\nu}_{\sigma l}} - \overset{0}{\Gamma^{\nu}_{\mu m}}\overset{0}{\Gamma^{m}_{i l}})X^i X^l[/tex]
since ##X^i = u \xi^i|_{u = 0}##. As you can see this is clearly not what the paper has in (68). It doesn't have the extra ##- \overset{0}{\Gamma^{\nu}_{\mu m}}\overset{0}{\Gamma^{m}_{i l}}## term. I however do not see how this term necessarily vanishes. Could anyone help me out with this? Why does the aforementioned term vanish in (68)? Thanks in advance.
Appendix B of the paper is the relevant section for my question. In (66) the tetrad is expanded in a power series about the central worldline ##\gamma## in terms of the distance ##u## along the space-like geodesics orthogonal to and emanating from the worldline. In (67), ##\xi^{\mu}## is the unit tangent field to each space-like geodesic. The first thing to note is that ##\xi^{\mu}(e_0)_{\mu}|_{u = 0} = 0## by construction hence ##\xi^{\mu}(e_0)_{\mu} = 0## all along the space-like geodesic since both ##\xi^{\mu}## and ##e^{\mu}_0## are parallel transported along the geodesic. Now ## e^{\mu}_0|_{u = 0} = \delta^{\mu}_{0}## so ##\xi^{\mu}(e_0)_{\mu}|_{u = 0} = 0 \Rightarrow \xi^{0}|_{u = 0} = 0## but this does not mean that ##\xi^0 = 0## for all ##u## because ##\xi^{\mu}(e_0)_{\mu} = 0## does not imply ##\xi^{\mu}\delta^0_{\mu} = 0## as is clear both intuitively and from (65). So I will write ##\xi^{i}## in place of ##\xi^{\mu}## below because the final expression involves only ##\xi^{\mu}## and in the end we evaluate at ##u = 0## so we only need to consider the ##\xi^i## terms but you should keep the above in mind.
My first problem then is with (68). We differentiate (67) once with respect to ##\frac{d}{du}## to get [tex]\frac{d^2}{du^2}e^{(\nu)}_{\mu} - \Gamma^{\sigma}_{\mu i, l} e^{(\nu)}_{\sigma} \xi^l \xi^i - \Gamma^{\sigma}_{\mu i}\xi^i \frac{d}{du}e^{(\nu)}_{\sigma} - \Gamma^{\sigma}_{\mu i}e^{(\nu)}_{\sigma} \frac{d \xi^i}{du} \\= \frac{d^2}{du^2}e^{(\nu)}_{\mu} - \Gamma^{\sigma}_{\mu i, l} e^{(\nu)}_{\sigma} \xi^l \xi^i - \Gamma^{\sigma}_{\mu i}\Gamma^{\lambda}_{\sigma l}\xi^i \xi^l e^{(\nu)}_{\lambda} - \Gamma^{\sigma}_{\mu i}e^{(\nu)}_{\sigma} \frac{d \xi^i}{du}\\= \frac{d^2}{du^2}e^{(\nu)}_{\mu} - \Gamma^{\sigma}_{\mu i, l} e^{(\nu)}_{\sigma} \xi^l \xi^i - \Gamma^{\sigma}_{\mu i}\Gamma^{\lambda}_{\sigma l}\xi^i \xi^l e^{(\nu)}_{\lambda} + \Gamma^{\sigma}_{\mu i}e^{(\nu)}_{\sigma} \Gamma^{i}_{l m}\xi^l \xi^m \\ = \frac{d^2}{du^2}e^{(\nu)}_{\mu} - \Gamma^{\nu}_{\mu i, l} \xi^l \xi^i - \Gamma^{\sigma}_{\mu i}\Gamma^{\nu}_{\sigma l}\xi^i \xi^l + \Gamma^{\nu}_{\mu i}\Gamma^{i}_{l m}\xi^l \xi^m = 0[/tex]
where I've used ##\frac{d \xi^i}{du} = -\Gamma^{i}_{lm}\xi^l \xi^m## from the geodesic equation. We thus have [tex]\frac{1}{2!}\frac{d^2 e^{(\nu)}_{\mu}}{du^2}|_{u = 0}u^2 = \frac{1}{2!}(\overset{0}{\Gamma^{\nu}_{\mu i, l}} + \overset{0}{\Gamma^{\sigma}_{\mu i} }\overset{0}{\Gamma^{\nu}_{\sigma l}} - \overset{0}{\Gamma^{\nu}_{\mu m}}\overset{0}{\Gamma^{m}_{i l}})X^i X^l[/tex]
since ##X^i = u \xi^i|_{u = 0}##. As you can see this is clearly not what the paper has in (68). It doesn't have the extra ##- \overset{0}{\Gamma^{\nu}_{\mu m}}\overset{0}{\Gamma^{m}_{i l}}## term. I however do not see how this term necessarily vanishes. Could anyone help me out with this? Why does the aforementioned term vanish in (68)? Thanks in advance.