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Physics
Special and General Relativity
Understanding killing vectors and transformations of metric
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[QUOTE="PeterDonis, post: 6859146, member: 197831"] Ok, so seeing equation 2.4 it looks like you could invert the transformation ##x^\mu \to y^\mu = x^\mu + \xi^\mu## to obtain ##y^\mu \to x^\mu = y^\mu - \xi^\mu##, and then the inverted version of equation 2.4 should give you (to linear order in ##\xi^\mu##, the same way equation 2.25 was obtained for the forward transformation) the relation ##\xi^\lambda \partial_\lambda \tilde{g}_{\mu \nu} = - \xi^\lambda \partial_\lambda g_{\mu \nu}## (because of the sign change in the ##\xi## term in the inverse transformation). At least, that seems to me to be what the notes are doing. As for how ##\tilde{g}_{\mu \nu}## differs from ##g_{\mu \nu}##, that's defined by the coordinate transformation; the general form of the relationship is equation 2.4, and equation 2.25 is just equation 2.4 to linear order in ##\xi^\mu## for the particular transformation ##y^\mu = x^\mu + \xi^\mu##. [/QUOTE]
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Special and General Relativity
Understanding killing vectors and transformations of metric
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