We investigated black hole thermodynamics in flat space in the context of holography. It is very tempting to consider the "holographic principle" as a simple organizing principle for quantum gravity, analogous, for example, to the invariance of the speed of light in Einstein’s theory of special relativity.

String theory provides a concrete realization of the holographic principle for spacetimes with negative cosmological constant, namely the AdS/CFT duality. However, it is of great interest to understand holography in flat spacetimes and our paper is a step further toward developing these ideas. We also use our method to investigate black ring thermodynamics, which continues to be a "hot topic" among the high energy/string theory research community.

Inspired by the AdS/CFT duality and the counterterm method of Balasubramanian and Kraus, we proposed a divergence-free boundary stress-tensor for stationary, locally asymptotically adS or locally asymptotically flat spacetimes.

The thermodynamical properties of a dipole black ring are derived using our method. We provide a physical interpretation for the complex Euclidean geometry of the dipole ring. We also find that the dipole charge appears in the first law in the same manner as a global charge.

Subsequently, Mann and Marolf generalized this method to arbitrary flat spacetimes and showed the equivalence of the conserved charges and the older definitions known in the literature. Also, Robb Mann, Cristi Stelea, and I presented explicit expressions for the action and the associated conserved quantities in the so called "cylindrical cut-off"—which are very useful for practical computations.

The goal of our paper was two-fold:

1. to develop an efficient procedure for studying thermodynamical properties of black objects—black holes and black rings—in flat space.
   
2. to apply this method to non-trivial examples with toroidal horizon topology (black rings) and to understand why a non-conserved charge—the dipole—appears in the first law.

Let us discuss these points in some detail.

1. The problem of defining conserved quantities in gravity, which is a generally covariant theory, is notoriously subtle. The general idea is to study the asymptotic values of the gravitational field far away from an isolated object and compare them with those in its absence. However, most of these proposals will provide results that are relative to the choice of a reference background.
   
   The choice of the background is not unequivocally fixed, and there also might be cases for which the topological properties of the solution rule out any natural choice of the background. We have proposed a novel method to calculate the stress-energy of gravity for the asymptotically flat spacetimes which does not require a reference background.
   
2. Gravity in higher dimensions—an important active research area in both string theory and particle physics—has a much richer spectrum of black objects than in four dimensions. For example, in five dimensions there exist solutions with a toroidal horizon topology (black rings). The ring creates a field analogous to a dipole, with no net charge measured at infinity.
   
   Black rings provide a novel theoretical laboratory for studying physics associated with event horizons. We have explicitly shown that the first law of black dipole ring mechanics expresses the conservation energy by relating the change in the ring mass to the change in its area, angular momentum, and the dipole charge.

At the Perimeter Institute for Theoretical Physics, I've discussed the role of dipoles in the first law of thermodynamics with Roberto Emparan. On the other hand, I was also familiar with the quasilocal formalism and counterterms in anti-de Sitter space and the idea was to apply a similar method to black rings. This was the starting point of the project.

Conceptual questions and the ways to approach them were quite clear from the very beginning. However, there were several technical obstacles and, as expected, there was also pressure from other groups at work on related projects.

This research is pure fundamental science.