Longitudinal projector of a vector field
= Longitudinal projector of a vector field
{title2=$P_L^{\mu\nu}$}
At a nonzero Euclidean <momentum> $k$, the <longitudinal projector of a vector field> is $P_L^{ab}=k^ak^b/k^2$. Its Lorentzian mixed-index form is $(P_L)^a{}_b=p^ap_b/p^2$ for non-null <four-momentum>. Together with the <transverse projector of a vector field>, it obeys $P_T+P_L=I$, $P_TP_L=0$, $P_T^2=P_T$, and $P_L^2=P_L$. This formula is not a pointwise decomposition at null <four-momentum>.