I am trying to follow Nakahara's book. From the context, it seems that the author is trying to say if moving a point along a flow always give a isometry, the corresponding vector field [tex]X[/tex] is a Killing vector field. am I right?(adsbygoogle = window.adsbygoogle || []).push({});

then the book gives a proof. It only considers a linear approximation

[tex]f:{x^\mu } \mapsto {x^\mu } + \varepsilon {X^\mu }[/tex]

in each step ignoring terms containing higher orders of [tex]\varepsilon [/tex]

[tex]\begin{array}{l}

{g_{\mu \nu }}(x) = \frac{{\partial ({x^\kappa } + \varepsilon {X^\kappa })}}{{\partial {x^\mu }}}\frac{{\partial ({x^\lambda } + \varepsilon {X^\lambda })}}{{\partial {x^\nu }}}{g_{\kappa \lambda }}(x + \varepsilon X)\\

= (\delta _\mu ^\kappa + \varepsilon {\partial _\mu }{X^\kappa })(\delta _\nu ^\lambda + \varepsilon {\partial _\nu }{X^\lambda })[{g_{\kappa \lambda }}(x) + \varepsilon {X^\xi }{\partial _\xi }{g_{\kappa \lambda }}(x)]\\

\approx (\delta _\mu ^\kappa \delta _\nu ^\lambda + \delta _\mu ^\kappa \varepsilon {\partial _\nu }{X^\lambda } + \varepsilon {\partial _\mu }{X^\kappa }\delta _\nu ^\lambda )[{g_{\kappa \lambda }}(x) + \varepsilon {X^\xi }{\partial _\xi }{g_{\kappa \lambda }}(x)]\\

\approx \delta _\mu ^\kappa \delta _\nu ^\lambda {g_{\kappa \lambda }}(x) + \delta _\mu ^\kappa \delta _\nu ^\lambda \varepsilon {X^\xi }{\partial _\xi }{g_{\kappa \lambda }}(x) + \delta _\mu ^\kappa \varepsilon {\partial _\nu }{X^\lambda }{g_{\kappa \lambda }}(x) + \varepsilon {\partial _\mu }{X^\kappa }\delta _\nu ^\lambda {g_{\kappa \lambda }}(x)\\

\approx {g_{\mu \nu }}(x) + \varepsilon {X^\xi }{\partial _\xi }{g_{\mu \nu }}(x) + \varepsilon {\partial _\nu }{X^\lambda }{g_{\mu \lambda }}(x) + \varepsilon {\partial _\mu }{X^\kappa }{g_{\kappa \nu }}(x)

\end{array}[/tex]

then we obtain the Killing equation

[tex]{X^\xi }{\partial _\xi }{g_{\mu \nu }}(x) + {\partial _\nu }{X^\lambda }{g_{\mu \lambda }}(x) + {\partial _\mu }{X^\kappa }{g_{\kappa \nu }}(x) = 0[/tex]

I feel uncomfortable here because the Killing equation only looks a necessary condition for the equation

[tex]{g_{\mu \nu }}(x) = \frac{{\partial ({x^\kappa } + \varepsilon {X^\kappa })}}{{\partial {x^\mu }}}\frac{{\partial ({x^\lambda } + \varepsilon {X^\lambda })}}{{\partial {x^\nu }}}{g_{\kappa \lambda }}(x + \varepsilon X)[/tex]

to be true, how about the terms contianing higher order of [tex]\varepsilon [/tex]?

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# A Question about Killing vector fields

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