The non-zero energy of 2+1 Minkowski space

TL;DR

This paper calculates the energy of 2+1 Minkowski space using a covariant action principle, finding it to be -1/4G, which differs from the anti-de Sitter case and clarifies boundary term contributions.

Contribution

It demonstrates that the Einstein-Hilbert action with Gibbons-Hawking boundary term is finite and stationary for 2+1 asymptotically flat spacetimes, and derives the gravitational Hamiltonian explicitly.

Findings

The energy of 2+1 Minkowski space is -1/4G.

The action principle is finite and stationary without additional boundary terms.

The result differs from the anti-de Sitter case and is consistent with boundary curvature terms.

Abstract

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 addititve constant. This constant is such that 2+1 Minkowski space is assigned the energy E = -1/4G, while the upper bound on the energy is set to zero. 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 flat-space limit of the corresponding energy -1/8G of 2+1 anti-de Sitter space.