Masses at null infinity for Einstein's equations in harmonic coordinates

In this work, we give a complete picture of how to, in a direct simple way, define the mass at null infinity in harmonic coordinates in three different ways that we show satisfy the Bondi mass loss law. The first and second way involve only the limit of metric (Trautman mass) respectively the null second fundamental forms along asymptotically characteristic surfaces (asymptotic Hawking mass) that only depend on the ADM mass. The last involves construction of special characteristic coordinates at null infinity (Bondi mass). The results here rely on asymptotics of the metric derived in $\href{https://doi.org/10.1007/s00220-017-2876-z}{[27]}$.