193. On massive gravity in three dimensions

This is a guest blog post by Olaf Hohm from the University of Groningen. Dmitry.

Dmitry kindly asked me to write about my recent paper with Eric Bergshoeff and Paul Townsend on massive gravity in three dimensions (0901.1766). In the last two years there has been quite some interest in three-dimensional gravity, most notably due to the paper by Witten in 2007 (0706.3359) and by Li, Song and Strominger last year (0801.4566). Both deal with the AdS/CFT correspondence in D=3 and the problem of counting the microscopic degrees of freedom of black holes (the so-called BTZ black holes) via some holographic dual in D=2. More precisely, Witten considers pure AdS gravity in D=3, while Strominger and collaborators investigate a “topologically massive” version. In our paper we present a new variant of D=3 gravity, which I will introduce in the following. For those who are not so familiar with three-dimensional gravity, let me first briefly review some basic facts.

In three dimensions pure gravity is topological, meaning that it does not carry propagating degrees of freedom. There are many ways to see this, but the easiest is perhaps to note that the Riemann tensor has the same number of components as the Ricci tensor. In other words, given a Ricci tensor the Riemann tensor is uniquely determined. Since the vacuum equations of pure gravity set the Ricci tensor equal to zero, it follows that also the Riemann tensor vanishes and so space-time is locally flat. Up to topological effects there is accordingly only one solution of the Einstein equations and that’s what we mean when we say that a theory doesn’t have local degrees of freedom. To put the same thing differently: there are no propagating massless spin-2 fields in D=3.

However, the situation changes once we consider massive spin-2 fields. In any dimension there is a unique mass term that can be added to the linearized Einstein-Hilbert term, giving rise to the so-called Pauli-Fierz action. This action yields propagating massive degrees of freedom, even in D=3. More precisely, it describes two states of helicities . In fact, the representation theory for the three-dimensional Poincare group in the massive case is quite similar to the representation theory in D=4 for the massless case, since both refer to the same little group SO(2). Thus, one may compare with the massless graviton in D=4, which also carries the two helicities . However, a massive spin-2 theory defined via the Pauli-Fierz action a priori makes only sense as a free, linearized theory since the mass term cannot be promoted to a non-linear theory, and therefore in general dimensions it is very hard to get a sensible interacting theory for massive spin-2.

In three dimensions, however, there are more possibilities. One of them has been introduced in a remarkable paper by Deser, Jackiw and Templeton in 1982. They show that extending the Einstein-Hilbert action by a gravitational Chern-Simons term, that is, a Chern-Simons action based on either the Christoffel symbols or the spin connection, yields a theory that describes a single, parity-violating massive degree of freedom of helicity +2 or -2, where the sign depends on the sign of the Chern-Simons term. Remarkably, even though this Chern-Simons theory is of third-derivative order, it defines a unitary theory that makes perfect sense at the non-linear level.

Now, the question we addressed in our recent paper is whether one can define a non-linear theory that describes both the two helicities , i.e., which is equivalent to the parity-invariant Pauli-Fierz theory at the linearized level, but which can be promoted to a non-linear theory. It turns out that this can be achieved by a particular fourth-order extension in derivatives. Specifically, we extend the Einstein-Hilbert action by the following terms quadratic in the curvature,

where is a mass parameter. Here I should note that the Einstein-Hilbert term has the “wrong” sign compared to what one would naively expect from a ghost-free theory in D=4, a feature that it shares with the topologically massive theory above. To see that this action indeed describes the two spin-2 helicities, let us consider the field equations

where is the Einstein tensor and the higher-derivative contribution. Then we linearize around flat Minkowski space, which yields

Here we have used that the tensor satisfies the “miraculous” property that its trace is proportional to the original higher-derivative Lagrangian. (That’s in fact what fixes the coefficients in the higher-derivative term.) As a consequence, taking the trace of the field equations yields an expression for the Ricci scalar in terms of curvature-square terms. Most importantly, it does not contain derivatives of curvatures, which implies that the Ricci scalar is set to zero by the linearized field equations.

Inspecting these equations more closely one infers that they look exactly the same as the massive Pauli-Fierz equations, except that the spin-2 field is replaced by the Einstein tensor (being a second derivative of the fundamental field). However, in three dimensions this doesn’t make a big difference, for the Einstein tensor is invertible, up to irrelevant gauge transformations. (That’s just another way of saying that pure gravity doesn’t have degrees of freedom.) Thus, the action describes pure spin-2. I should stress that adding generic higher-derivative terms yields an additional ghost-like spin-0 excitation. In fact, it is the constraint setting the Ricci scalar to zero which removes precisely this ghost, thus leaving a unitary theory. A different and perhaps more direct way to prove this result is to show the equivalence of Pauli-Fierz and the above higher-derivative theory via a master action, as we do in the paper.

In three dimensions we have therefore the situation that the free massive Pauli-Fierz theory can be equivalently formulated as a (linearized) curvature correction to the Einstein-Hilbert term. However, in contrast to Pauli-Fierz, the latter has an obvious extension to a non-linear theory, which was in fact our starting point above.

Now, what could this non-linear theory be good for? First of all, there is also a cosmological extension, which admits AdS vacua and BTZ black hole solutions, and so one may readdress the microstate counting in AdS/CFT. Another interesting point I would like to mention is related to the issue of renormalizability. In four dimensions it is a classic result due to Kelly Stelle that gravity with curvature-square terms is renormalizable. However, in D=4 those theories contain ghosts (as do the general theories in D=3, except the special combination we consider). Therefore, this is not a satisfactory theory of quantum gravity as it violates unitarity. However, the three-dimensional theory does maintain unitarity and so possibly makes sense as a quantum theory as well (which has also been conjectured by Deser and Yang for topologically massive gravity). I hope we will be able to come back to this theory in the future.