The Comeback of Massive Gravity

Massive gravity, a modification of general relativity in which gravitons have mass, has an interesting history. Massive gravity was long believed to be internally inconsistent, but physicists at Stockholm University now claim to have constructed a consistent theory for massive gravity. This theory is a viable alternative to general relativity and can address some of its problems.

In Einstein’s theory of general relativity gravitational waves spread with the speed of light, and the quanta of the gravitational field, the gravitons, are expected to do the same*. To be more precise, gravitons move with the speed of massless particles because they are assumed to be massless. But whether or not a particle is indeed massless is in the end a question of experiment.

Neutrinos were long believed to be massless, but we know today that at least two of them have tiny non-zero masses (whose absolute value has not yet been determined). The mass of the photon is known to be zero to extremely high precision on experimental grounds. But what about gravity? This is a timely question because a small mass would lead to a long-distance modification of general relativity, and present observational evidence left physicists with some puzzles at these long distances, notably dark energy and dark matter.

However, to be able to even properly ask whether gravitons have masses, we need a consistent theory for massive gravity. But making gravitons massive is a challenge for the theoretical physicist. In fact, it was long believed to be impossible.

The problems start when you want to introduce a mass-term into general relativity. For vector fields, you can take a contraction of fields of the form A^νA_ν to stand in front of the mass term. In general relativity the field is the metric tensor, and the only full contractions that you can create without using derivatives are constant: they create a cosmological constant, not a graviton mass. If you want a mass-term in general relativity you need a second two-tensor, that is a field which looks like a metric but isn’t the metric. Theories of this type are also known as ‘bi-metric’. Massive gravity is thus intimately related to bi-metric gravity.

But that’s only the beginning of the problems, a beginning that dates back more than 70 years.

In 1939, Fierz and Pauli wrote down a theory of massive gravity in the perturbative limit. They found that for the theory to be consistent – meaning free of ‘ghosts’ that lead to unphysical instabilities – the parameters in the mass-terms must have specific values. With these values, the theory is viable.

In 1970 however, van Dam and Veltman and, independently, Zukharov, showed that in the Fierz-Pauli approach, the limit in which the mass of the graviton is taken to zero is not continuous and does, contrary to naïve expectations, not reproduce general relativity. Any graviton mass, regardless how small, leads to deviations that can contribute factors of order one to observables, which is in conflict with observation. The Fierz-Pauli theory now seemed theoretically fine, but experimentally ruled out.

Two years later, in 1972, Vainshtein argued that this discontinuity is due to the treatment of the gravitational degrees of freedom in the linearization procedure and can be cured in a full, non-linear, version of massive gravity. Unfortunately, in the same year, Deser and Boulware claimed that any non-linear completion of the Fierz-Pauli approach reintroduces the ghost. So now massive gravity was experimentally fine but theoretically sick.

Nothing much happened in this area for more than 30 years. Then, in the early 2000s, the wormy can was opened again by Arkani-Hamed et al and Creminelli et al, but they essentially confirmed the Deser-Boulware problem.

The situation began to look brighter in 2010, when de Rahm, Gabadadze and Tolley proposed a theory of massive gravity that did not suffer from the ghost-problem in a certain limit. Needless to say, after massive gravity had been thought dead and buried for 40 years, nobody really believed this would work. The de Rahm-Gabadadze approach did not make many friends because the second metric was treated as a fixed background field, and the theory was shown to allow for superluminal propagation (and, more recently, acausality).

However, starting in 2011, Fawad Hassan and Rachel Rosen from Stockholm University (ie next door), succeeded in formulating a theory of massive gravity that does not suffer from the ghost instability. The key to success was a generalization of the de Rahm-Gabadadze approach in which the second metric is also fully dynamic, and the interaction terms between the two metrics take on a specific form. The specific form of the interaction terms is chosen such that it generates a constraint which removes the ghost field. The resulting theory is to best present knowledge fully consistent and symmetric between the two metrics.

(Which, incidentally, explains my involvement with the subject, as I published a paper with a fully dynamic, symmetric, bi-metric theory in 2008, though I wasn’t interested in the massive case and don’t have interaction terms. The main result of my paper is that I ended up in defense committees of Fawad’s students.)

In the last years, the Stockholm group has produced a series of very interesting papers that not only formalizes their approach and shows its consistency, but they also derived specific solutions. This is not a small feat as it is already difficult to find solutions in general relativity if you have only one metric and having two doesn’t make the situation easier. Indeed, not many solutions are presently known, and the known ones have quite strong symmetry assumptions. (More students in the pipe...)

Meanwhile, others have studied how well this modification of general relativity fares as an alternative to ΛCDM. It has been found that massive gravity can fit all cosmological data without the need to introduce an additional cosmological constant. But before you get too excited about this, note that massive gravity has more free parameters than ΛCDM, that being the coupling constants in the interaction terms.

What is missing right now though is a smoking-gun signal, some observation that would allow to distinguish massive gravity from standard general relativity and could be used to distinguish between both. This is presently a very active area of research and one that I’m sure we’ll hear more about.

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* To be precise, in the classical theory we should be speaking of gravitational waves instead. The frequent confusion between gravitational waves and gravitons, the latter of which only exist in quantized gravity, is a bad habit but forgivable. Far worse are people who say ‘gravity wave’ when they refer to a gravitational wave. A gravity wave is a type of cloud formation and has nothing to do with linearized gravity.