- #1
unscientific
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I was reading through hobson and my notes where the covariant acts on contravariant and covariant tensors as
[tex]\nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma[/tex]
[tex]\nabla_\alpha V_\mu = \partial_\alpha V_\mu - \Gamma^\gamma_{\alpha \mu} V_\gamma [/tex]
Why is there a minus sign in the second equation?
I mean, if you simply multiply ##g_{\mu \gamma}## on both side of the equation:
[tex]\nabla_\alpha V^\mu g_{\mu \gamma} = \partial_\alpha V^\mu g_{\mu \gamma} + \Gamma^\mu_{\alpha \gamma} V^\gamma g_{\mu \gamma}[/tex]
[tex]\nabla_\alpha V_{\gamma} = \partial_\alpha V_\gamma + \Gamma^\mu_{\alpha \gamma} V_\mu [/tex]
Swapping ##\mu## and ##\gamma## we have
[tex]\nabla_\alpha V_{\mu} = \partial_\alpha V_\mu + \Gamma^\gamma_{\alpha \mu} V_\gamma [/tex]
[tex]\nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma[/tex]
[tex]\nabla_\alpha V_\mu = \partial_\alpha V_\mu - \Gamma^\gamma_{\alpha \mu} V_\gamma [/tex]
Why is there a minus sign in the second equation?
I mean, if you simply multiply ##g_{\mu \gamma}## on both side of the equation:
[tex]\nabla_\alpha V^\mu g_{\mu \gamma} = \partial_\alpha V^\mu g_{\mu \gamma} + \Gamma^\mu_{\alpha \gamma} V^\gamma g_{\mu \gamma}[/tex]
[tex]\nabla_\alpha V_{\gamma} = \partial_\alpha V_\gamma + \Gamma^\mu_{\alpha \gamma} V_\mu [/tex]
Swapping ##\mu## and ##\gamma## we have
[tex]\nabla_\alpha V_{\mu} = \partial_\alpha V_\mu + \Gamma^\gamma_{\alpha \mu} V_\gamma [/tex]