- #1
lichen1983312
- 85
- 2
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?
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]?
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]?