Covariant derivative for four velocity

[Kyrios]

Homework Statement

Show [itex] U^a \nabla_a U^b = 0 [/itex]

Homework Equations

[itex] U^a [/itex] refers to 4-velocity so [itex] U^0 =\gamma [/itex] and [itex] U^{1 - 3} = \gamma v^{1 - 3} [/itex]

The Attempt at a Solution

I get as far as this:

[itex] U^a \nabla_a U^b = U^a ( \partial_a U^b + \Gamma^b_{c a} U^c) [/itex]
And I think that the Christoffel symbol is 0 in minkowski space so that gives:
[itex] U^a \partial_a U^b [/itex]

And I'm not certain what to do from this point

 

[Orodruin]

Is this the problem statement exactly as given. At face value it just looks like the 4-acceleration which by no means needs to be zero.

 

[Kyrios]

Yes this is all that was given, unfortunately, so I'm having trouble explaining why exactly it is zero