Theoretical physicists love to ask "what if" questions, to explore the limits of mathematical self-consistency, even--perhaps especially--when the questions they are asking fly in the face of conventional wisdom. A classic story of theoretical tinkering concerns the graviton, the hypothetical fundamental particle that carries the gravitational force. In general relativity, the graviton--like the photon--has no mass. But as far back as 1939, Markus Fiertz and Wolfgang Pauli became the first to ask a heretical question: could the graviton have a mass?
What they found was a surprisingly subtle answer. They were able to formulate a theory of massive gravity; the surprise was that the theory was mathematically unique. Usually, theoretical physicists can find many mathematically consistent ways to state related theories and have to rely on experiments or even aesthetic criteria to narrow down the scope of the possible. Hence the uniqueness of the Fierz-Pauli answer has been, for decades, a theoretical curiosity that tugs at the theorist's intuition, tempting him to think that this rare mathematical uniqueness is a hint that this is a path worth following.
Following the Fierz-Pauli result, nothing much happened until the early 1970s. Then, there was a spate of study of massive gravity, with a roller coaster of results. First, van Dam, Veltman, and Zakharov looked at the theory and declared that, though self-consistent, the Fierz-Pauli theory could never describe our world. What they had discovered is a funny phenomenon now called the vDVZ discontinuity. This is the fact that the linear theory of Pauli and Fierz is completely disconnected from the physics of general relativity, even when the mass of the graviton is set to zero. The physical fact that underlies this disconnect is simple and surprising: adding a mass to the graviton gives it new degrees of freedom, going from the 2 "tensor" waves that characterize general relativistic waves to 5 modes--2 tensor, 2 vector, and 1 scalar--in the case of massive gravity. Adding the mass, like opening Pandora's box, releases these extra modes in a way that can't be undone: a theory with more degrees of freedom is simply different from the one that lacks them. These extra modes make gravity stronger, so we can tell for sure that gravity near us does not have these extra modes. Hence, if no way around this discontinuity could be found, the theory would be dead for good.
A couple of years later, though, a Russian physicist called Arkady Vainshtein found another unexpected surprise in massive gravity. The linear theory of Pauli and Fiertz was, he reasoned, not enough: linear perturbations are the start of a complete theory, but any real theory will also have non-linearities that can play an important role. So Vainshtein started checking what non-linearities would do to the apparently fatal vDVZ discontinuity. What he found was remarkable: the nonlinearities of the theory become stronger and stronger as the mass of the graviton shrinks. What this means is that when the graviton mass vanishes, the extra modes are completely frozen by their non-linearities, and the theory becomes compatible with general relativity after all! The idea is that if the graviton mass is small, this would force the extra modes to become non-linear in places, like our solar system, where there was a lot of energy density around; and since the non-linear theory looks just like general relativity, gravity could be massive without us having noticed it yet.
But the roller coaster ride wasn't yet over. Later that same year, Boulware and Deser uncovered what everyone thought was a fatal flaw in Vainshtein's solution to the Fiertz-Pauli problem: they proved that any non-linear theory of massive gravity must somewhere possess a ghost, which is theorists' lingo for an intrinsic and uncontrollable instability that means a theory is, if not mathematically inconsistent, then nonetheless irredeemably unphysical. Bouware and Deser's proof was so expansive and thorough that they managed to kill the theory of massive gravity for a few more decades.
In the meantime, two apparently unrelated developments have occurred. The first is a deepening of our understanding of perturbative theories, like Fiertz-Pauli gravity. These were once thought to be little parts of a fundamental, ultimate theory; in this mindset, the discovery of any problem or unavoidable anomaly in a theory was enough to kill that theory dead. However, the intervening years have taught us that theories defined perturbatively are actually much more flexible than that. We now call them effective field theories, and understand that they intrinsically possess regimes of validity and regimes in which they are invalid. That is, all our theories--even the celebrated Standard Model--are theories that describe what's going on for certain energy and length scales, and only those length scales. If there seems to be a problem on length or energy scales where we know them to be invalid, then we don't need to worry: those are probably totally phony problems, ones that will resolve themselves when the full theory is known.
This kind of evolution in thinking has been a big part of the story of modern physics. For instance, one of the theoretical conundrums that led to quantum mechanics was the question of why orbital electrons didn't crash into the nuclei of atoms. After all, when we propel an electron in a circle in an experiment, it emits radiation and loses speed and angular momentum. It turns out, of course, that the regime of validity for the classical physics that describes electron radiation doesn't include the length scales inside of an atom, where we have to use quantum mechanics to understand what's going on.
The other development came with theoretical speculations about infinite extra dimensions, begun in earnest when Dvali, Gabadadze, and Porrati (DGP) proposed a model of the Universe with our familiar 4 dimensions plus one infinite dimension we couldn't see. Theorists soon realized that, for 4D observers like us, a theory with an infinite extra dimension could be turned into an effective field theory for gravity, but one in which the graviton had a kind of "soft", or energy-dependent, mass. This effective theory is related to but slightly different from the Fiertz-Pauli form; however, this kind of "soft" mass doesn't really constitute "massive gravity", which is why it doesn't violate the proofs that the Fiertz-Pauli form was unique.
All of these developments have led to the latest breakthrough, announced quietly in a recent, and very technical, paper by by Claudia de Rham and Gregory Gabadadze (the same Gabadadze as in the infinite extra dimension model), http://arxiv.org/abs/1007.0443v1. Using insights gained from the study of the DGP model and its successors, de Rham and Gabadadze have again returned to the study of massive gravity. The mathematics are complex but the conclusion is simple: the famous Boulware-Deser ghost does exist, but it lives outside of the regime of validity of the massive gravity theory! A ghost outside of the regime of validity is nothing to worry about, just as the electron in the atom doesn't spin out of control. So, if de Rham and Gabadadze are right, the study of massive gravity is yet again being turned on its head. Given the decades of cat and mouse hunting for a final theory of massive gravity, we can't be sure that this newest breakthrough will be followed by the discovery of new problems or will prove to be enduring. In the meantime, we theorists can again permit ourselves to think about what this strangely unkillable theory of massive graviton might mean for physics.
(Graviton plushie image above from www.particlezoo.net.)