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Somewhat surprisingly, in many of the widely used monographs and reviewarticles the term Transverse-Traceless modes of linearized gravitational wavesis used to denote two entirely different notions. These treatments generallybegin with a decomposition of the metric perturbation that is local in themomentum space (and hence non-local in physical space), and denote theresulting transverse traceless modes by $h_{ab}^{\TT}$. However, whilediscussing gravitational waves emitted by an isolated system --typically in alater section-- the relevant modes are extracted using a `projection operator'that is local in physical space. These modes are also calledtransverse-traceless and again labeled $h_{ab}^{\TT}$, implying that this isjust a reformulation of the previous notion. But the two notions areconceptually distinct and the difference persists even in the asymptoticregion. We show that this confusion arises already in Maxwell theory that isoften discussed as a prelude to the gravitational case. Finally, we discuss whythe distinction has nonetheless remained largely unnoticed, and also point outthat there are some important physical effects where only one of the notionsgives the correct answer.

On the ambiguity in the notion of transverse traceless modes of gravitational waves