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Tangent87
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Hi, I'm stuck on the last bit the attached question where we're given the metric [tex]ds^2=-du^2+u^2dv^2[/tex] and have to use equation (*) to find the geodesic equations.
They tell us to use [tex]V^a=\dot{x}^a[/tex] the tangent vector to the geodesic and presumably we use the three killing vectors they gave us, so then from (*) we have:
[tex]\left(V^ak_a\right)_{,b}V^b=\left(\dot{x}^ak^cg_{ac}\right)_{,b}\dot{x}^b=0[/tex]
But then using the killing vector (0,1) and the metric I get the equation [tex]2u\dot{v}\dot{u}=0[/tex] which doesn't seem right to me. Am I correct in thinking that when we partial differentiate w.r.t to u say we leave the [tex]\dot{u}[/tex] term alone right?
Thanks.
They tell us to use [tex]V^a=\dot{x}^a[/tex] the tangent vector to the geodesic and presumably we use the three killing vectors they gave us, so then from (*) we have:
[tex]\left(V^ak_a\right)_{,b}V^b=\left(\dot{x}^ak^cg_{ac}\right)_{,b}\dot{x}^b=0[/tex]
But then using the killing vector (0,1) and the metric I get the equation [tex]2u\dot{v}\dot{u}=0[/tex] which doesn't seem right to me. Am I correct in thinking that when we partial differentiate w.r.t to u say we leave the [tex]\dot{u}[/tex] term alone right?
Thanks.
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