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Wald, Problem 5 Chapter 4

  1. Dec 12, 2013 #1
    This question has been asked two years ago, but it wasn't resolved (I think). Here goes

    This problem is Problem 5 in Chapter 4. It is that [itex]T_{ab}[/itex] is a symmetric, conserved field ([itex]T_{ab}=T_{ba}, \partial ^aT_{ab}=0[/itex]) in Minkowski spacetime. Show that there is a tensor field [itex]U_{acbd}[/itex] with the symmetries [itex]U_{acbd}=U_{[ac]bd}=U_{ac[bd]}=U_{bdac}[/itex] such that [itex]T_{ab}=\partial^c\partial^dT_{acbd}[/itex].

    Wald gave a hint: For any vector field [itex]v^a[/itex] in Minkowski spacetime satisfying [itex]\partial_av^a=0[/itex] there is a tensor field [itex]s^{ab}=-s^{ba}[/itex] such that [itex]v^a=\partial_bs^{ab}[/itex]. Use this fact to show that [itex]T_{ab}=\partial^cW_{cab}[/itex] with [itex]W_{cab}=W_{[ca]b}[/itex]. The use the fact that [itex]\partial^cW_{c[ab]}=0[/itex] to derive the desired result.

    I have an idea what to do but I've been thinking I'm not sure where to place the indices, so if someone could help me with that, it would be great.

    We start with the vector field [itex]T^{a\mu}[/itex] then [itex]\partial aT^{a\mu}=0[/itex] so by the hint we have [itex]T^{a\mu}=\partial cW^{ac\mu}[/itex] with [itex]W^{ac\mu}=-W^{ca\mu}[/itex]. Is this correct? if so, why is it like this and not [itex]W^{a\mu c}[/itex] with [itex]W^{a\mu c}=-W^{c\mu a}[/itex] in the above?

    any help much appreciated!
     
  2. jcsd
  3. Dec 12, 2013 #2

    Bill_K

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    Science Advisor

    The two answers are equivalent. Let the first answer given be W1acμ. The second answer is W2aμc ≡ W1acμ. There's no law that tells you where to put the indices, you just need a rank 3 tensor that's antisymmetric on a and c.
     
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