Conventions and external domain

The physical tensors are TT relative to their declared references. The longitudinal insertion need not be trace free.

For $k\ne 0$, the map $\xi \mapsto G(k,\xi )$ is injective: choose $a$ with $k_a\ne 0$. If $G=0$, its $aa$ component gives ${\xi }_a=0$, and its $ab$ components then give all ${\xi }_b=0$. In particular,

There is no quotient $\xi \sim \xi +\lambda k$ in this domain.