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qinglong.1397

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## Homework Statement

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.

## The Attempt at a Solution

Based on his hint, I got a solution [itex]T_{ab}=\partial^c\partial^dU_{acbd}[/itex]. Like [itex]s^{ab}=-s^{ba}[/itex], I required that [itex]U_{acbd}=-U_{adbc}[/itex], but this condition would lead to the result [itex]T_{ab}=0[/itex]!

So what is wrong with my solution? I need your help, Thank you!