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What makes the trace-free tensor (or part of it) so important? As in trace-free Ricci tensor or Weyl tensor.

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Symmetry. A general rank-2 tensor $T\in V\otimes V$ can be decomposed as

$$T = \tau I + S + A,$$

where $n\tau = \operatorname{Tr}(T)$ ($n$ being the dimension of $V$), $I$ is the identity, $S$ is trace-less symmetric and $A$ is skew-symmetric. These "components" obey different transformation laws under a given symmetry. For example, the scalar part $\tau I$ is obviously invariant.

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