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

Somewhat surprisingly, in many of the widely used monographs and review articles the term Transverse-Traceless modes of linearized gravitational waves is used to denote two entirely different notions. These treatments generally begin with a decomposition of the metric perturbation that is local in the momentum space (and hence non-local in physical space), and denote the resulting transverse traceless modes by $h_{ab}^{\TT}$. However, while discussing gravitational waves emitted by an isolated system --typically in a later section-- the relevant modes are extracted using a `projection operator' that is local in physical space. These modes are also called transverse-traceless and again labeled $h_{ab}^{\TT}$, implying that this is just a reformulation of the previous notion. But the two notions are conceptually distinct and the difference persists even in the asymptotic region. We show that this confusion arises already in Maxwell theory that is often discussed as a prelude to the gravitational case. Finally, 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.