Explicit form of the Mann-Marolf surface term in ($3+1$) dimensions

The Mann-Marolf surface term is a specific candidate for the “reference background term” that is to be subtracted from the Gibbons-Hawking surface term in order make the total gravitational action of asymptotically flat spacetimes finite. That is, the total gravitational action is taken to be: $(\mathrm{\text{Einstein}}\mathrm{\text{−}}\mathrm{\text{Hilbert}}\mathrm{\text{bulk}}\mathrm{\text{term}})+(\mathrm{\text{Gibbons}}\mathrm{\text{−}}\mathrm{\text{Hawking}}\mathrm{\text{surface}}\mathrm{\text{term}})-(\mathrm{\text{Mann}}\mathrm{\text{−}}\mathrm{\text{Marolf}}\mathrm{\text{surface}}\mathrm{\text{term}})$. As presented by Mann and Marolf, their surface term is specified implicitly in terms of the Ricci tensor of the boundary. Herein, I demonstrate that, for the physically interesting case of a ($3+1$)-dimensional bulk spacetime, the Mann-Marolf surface term can be specified explicitly in terms of the Einstein tensor of the ($2+1$)-dimensional boundary.