How to Derive the Tensor Field U_{acbd} from T_{ab} in Wald's Problem?

[tommyj]

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

Wald gave a hint: For any vector field $v^a$ in Minkowski spacetime satisfying $\partial_av^a=0$ there is a tensor field $s^{ab}=-s^{ba}$ such that $v^a=\partial_bs^{ab}$. Use this fact to show that $T_{ab}=\partial^cW_{cab}$ with $W_{cab}=W_{[ca]b}$. The use the fact that $\partial^cW_{c[ab]}=0$ 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 $T^{a\mu}$ then $\partial aT^{a\mu}=0$ so by the hint we have $T^{a\mu}=\partial cW^{ac\mu}$ with $W^{ac\mu}=-W^{ca\mu}$. Is this correct? if so, why is it like this and not $W^{a\mu c}$ with $W^{a\mu c}=-W^{c\mu a}$ in the above?

any help much appreciated!

 

[Bill_K]

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

The two answers are equivalent. Let the first answer given be W₁^acμ. The second answer is W₂^aμc ≡ W₁^acμ. 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.