Proposition: Conserved improvement ambiguity

On the flat comparison carrier, for every smooth $f$,

is symmetric and satisfies

There are nonzero such tensors. Therefore symmetry and conservation alone do not uniquely identify a stress-energy candidate.

Proof. Symmetry follows from commutation of derivatives. Moreover,

The example $f=(x^0{)}^2$ gives $I_{ii}=-2$. Adding such a tensor to a conserved symmetric candidate preserves those two properties while changing the candidate.