stephenmitten
Jul8-08, 12:42 PM
In Hartle's GR book (p. 177), there is a derivation of \xi \cdot u = constant, where \xi is a Killing vector, u is four-velocity along a geodesic in an arbitrary metric, and
L = (-g_{\alpha\beta}\frac{dx^\alpha}{d\sigma}\frac{dx^\ beta}{d\sigma})^\frac{1}{2}
The derivation goes:
\frac{\partial}{\partial \sigma}\frac{\partial L}{\partial \frac{dx^1}{d\sigma}}} = 0 \\ \Rightarrow \frac{\partial L}{\partial \frac{dx^1}{d\sigma}} = -g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma} = ... = -\xi \cdot u
is conserved along the geodesic. (Here the symmetry associated with \xi is in x^1.) It seems to be saying that
\frac{\partial L}{\partial \frac{dx^1}{d\sigma}} = \frac{1}{2L}({-g_{\alpha 1}\frac{1}{L}\frac{dx^\alpha}{d\sigma}-g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma}) = {-g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma}
but it appears to me that \frac{\partial L}{\partial \frac{dx^1}{d\sigma}} has only seven terms, not eight, since -g_{11}\frac{dx^1}{d\sigma}}\frac{dx^1}{d\sigma}} appears only once. I'd appreciate it if someone could point out where I went wrong.
Thanks.
L = (-g_{\alpha\beta}\frac{dx^\alpha}{d\sigma}\frac{dx^\ beta}{d\sigma})^\frac{1}{2}
The derivation goes:
\frac{\partial}{\partial \sigma}\frac{\partial L}{\partial \frac{dx^1}{d\sigma}}} = 0 \\ \Rightarrow \frac{\partial L}{\partial \frac{dx^1}{d\sigma}} = -g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma} = ... = -\xi \cdot u
is conserved along the geodesic. (Here the symmetry associated with \xi is in x^1.) It seems to be saying that
\frac{\partial L}{\partial \frac{dx^1}{d\sigma}} = \frac{1}{2L}({-g_{\alpha 1}\frac{1}{L}\frac{dx^\alpha}{d\sigma}-g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma}) = {-g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma}
but it appears to me that \frac{\partial L}{\partial \frac{dx^1}{d\sigma}} has only seven terms, not eight, since -g_{11}\frac{dx^1}{d\sigma}}\frac{dx^1}{d\sigma}} appears only once. I'd appreciate it if someone could point out where I went wrong.
Thanks.