On a basic conceptual confusion in gravitational radiation theory

In much of the literature on linearized gravitational waves two completely different notions are called transverse traceless modes and labelled $h_{ab}^{TT}$, often in different sections of the same reference, without realizing the underlying inconsistency. We compare and contrast the two notions and find that the difference persists even at leading asymptotic order near future null infinity $\mathit{I}^+$. We discuss why the distinction has nonetheless remained largely unnoticed, and also point out that there are some important physical effects where only one of the notions gives the correct answer.