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Covariant derivative for four velocity

  1. Oct 27, 2014 #1
    1. The problem statement, all variables and given/known data
    Show [itex] U^a \nabla_a U^b = 0 [/itex]

    2. Relevant 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]

    3. 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
     
  2. jcsd
  3. Oct 27, 2014 #2

    Orodruin

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    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.
     
  4. Oct 27, 2014 #3
    Yes this is all that was given, unfortunately, so I'm having trouble explaining why exactly it is zero
     
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