Longitudinal projector of a vector field ($P_L^{\mu \nu }$)

The longitudinal projector is $P_L^{\mu \nu }=k^{\mu }{k}^{\nu }/k^2$. It obeys $P_T+P_L=I$, $P_T{P}_L=0$, and $P_T^2=P_T$, $P_L^2=P_L$.