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## Homework Statement

(a) Show acceleration is perpendicular to velocity

(b)Show the following relations

(c) Show the continuity equation

(d) Show if P = 0 geodesics obey:

## Homework Equations

## The Attempt at a Solution

__Part (a)__

[tex]U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} + \Gamma_{v\epsilon}^{\mu} U^{\epsilon} \right] [/tex]

[tex] = \delta_\mu^{v} \left[ \partial_v U^{\mu} + \Gamma_{v\epsilon}^{\mu} U^{\epsilon} \right] [/tex]

[tex] = \partial_{\mu} U^{\mu} + \Gamma_{\mu \epsilon}^{\mu} U^{\epsilon} [/tex]

[tex] = \partial_{\mu} U^{\mu} + \Gamma_{\mu mu}^{\mu} U^{\mu} [/tex]

[tex] = \partial_{\mu} U^{\mu} + \left[ \frac{1}{2} g^{\mu \mu} (\partial_{\mu} g_{\mu \mu} + \partial_{\mu} g_{\mu \mu} - \partial_\mu g_{\mu\mu} ) \right] U^{\mu}[/tex]

[tex] = -\partial_{\mu} U^{\mu} - g^{\mu\mu} \partial_{\mu} g_{\mu\mu} V^{\mu}[/tex]

How do I show it equals 0?