Nonzero energy of $2+1$ Minkowski space

We compute the energy of $2+1$ Minkowski space from a covariant action principle. Using Ashtekar and Varadarajan’s characterization of $2+1$ asymptotic flatness, we first show that the $2+1$ Einstein-Hilbert action with Gibbons-Hawking boundary term is both finite on shell (apart from past and future boundary terms) and stationary about solutions under arbitrary smooth asymptotically flat variations of the metric. Thus, this action provides a valid variational principle and no further boundary terms are required. We then obtain the gravitational Hamiltonian by direct computation from this action. The result agrees with the Hamiltonian of Ashtekar and Varadarajan up to an overall additive constant. This constant is such that $2+1$ Minkowski space is assigned the energy $E_{{\mathrm{Mink}}_{2+1}}=-\frac{1}{4G}$, while the upper bound on the energy becomes $E\le 0$. Any variational principle with a boundary term built only from the extrinsic and intrinsic curvatures of the boundary is shown to lead to the same result. Interestingly, our result is not the $\Lambda \to 0$ limit of the corresponding energy $E_{{\mathrm{AdS}}_{2+1}}=-\frac{1}{8G}$ of $2+1$ anti-de Sitter (AdS) space.