# Covariant derivative for four velocity

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1. Oct 27, 2014

### Kyrios

1. The problem statement, all variables and given/known data
Show $U^a \nabla_a U^b = 0$

2. Relevant equations
$U^a$ refers to 4-velocity so $U^0 =\gamma$ and $U^{1 - 3} = \gamma v^{1 - 3}$

3. The attempt at a solution

I get as far as this:

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

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

2. Oct 27, 2014

### Orodruin

Staff Emeritus
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.

3. Oct 27, 2014

### Kyrios

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