About Massive Gravity

There is a lot of ongoing publication of academic papers on massive gravity theories. There have been sixty-four pre-prints on the topic at arXiv in the last 12 months alone. This research is driven by the desire to develop a theory of quantum gravity, the absence of which is one of the most obvious defects of "core theory" (i.e. the Standard Model plus General Relativity) that remains one of the most important unsolved problems in physics.

It is one part of a larger project in quantum gravity and general relativity research to better understand gravity by exploring modifications of canonical classical general relativity and its most naive quantum generalizations, to see what they imply and to better understand the mathematics of gravity and general relativity more generally,

By crude analogy, abstract algebra, parts of which are vital tools in physics, is basically a working out of what happens if you remove or add rules to rules like the commutative and associative properties of ordinary algebra and seeing what it looks like and whether you can gain insights or tools from that exercise.

We, of course, don't have the instrumental capacity to detect individual gravitons the way that we can detect individual photons, because the coupling constant of gravity at the scale of individual fundamental particles is so much weaker than the three Standard Model forces. So, we can't simply directly measure the mass of a graviton, and even if we could, we couldn't do so with perfect precision (this isn't just a practical difficulty, it is theoretically impossible to do so). Therefore, we can't rule out the possibility of a massive graviton with a mass of less than the uncertainty of our most precise measurement by direct measurement alone.

So, the only way to distinguish between the two possibilities is to work out theoretically the observable implications of each possibility, to learn how they differ, and with this knowledge to try to see if we can use indirect evidence to distinguish between the possibilities.

Indeed, one reason to explore it is that it provides an alternative to the massless spin-2 graviton approach to quantum gravity (the overwhelmingly conventional wisdom) which allows you to quantify how far experimental data requires the conventional wisdom to be true, rather than the alternatives.

It is one thing to say that experimental data is not inconsistent with the hypothesis of a massive graviton. It is another to say that experimental evidence constrains any massive graviton theory to have a graviton with a mass of not more than 6*10^-32 eV at the two sigma confidence level, pursuant to a study of weak gravitational lensing data done in 2004, and that thirteen other papers analyzing different astronomy observations in an effort to constrain this parameter experimentally have imposed no other boundaries that are more strict.

Quantifying the allowed parameter space of massive gravity theories allows us to quickly rule out BSM theories which aren't within the limits of those parameters. So, this is an active area of experimental as well as theoretical investigation, although there haven't been a lot of really big breakthroughs on the experimental side recently.

More generally, comparing the math involved in quantum gravity with a massive graviton to the math involved in massive gravity with a massless graviton, helps you understand what is going on in each case better.

As a practical matter, one of the main barriers to a theory of quantum gravity is that the naive mathematical description of a massless spin-2 graviton that couples in proportion to the mass-energy of what it interacts with is non-renormalizable (and is also a non-Abelian gauge theory, i.e. its math doesn't obey the commutative law mathematically, and is highly non-linear) and no one has figured out how to do the non-perturbative math (or use not yet discovered #mathtricks) that are necessary to get meaningful answers out of this formulation in a more general case as opposed to some very specific, highly idealized and symmetric cases. So, looking at the closely related cases of massive graviton theories may help you to gain insight into why you can't solve the massless graviton case.

To take one concrete example of that, the graviton matter coupling in massive gravity is arguably easier to formulate than in the massless graviton case. Similarly, it is arguably to describe the interaction of quantum electrodynamics with gravity in a massive graviton formulation, and some of the insights that result from that analysis may generalize to both massive and massless graviton cases.

Also, it would be hubris to claim to know for sure that the massless graviton case is really true, when we can't even realize it mathematically in any way that we can use practically. This isn't Platonic knowledge that we are born with. While the massless graviton case is more attractive for many reasons, we can't rule out the possibility that the massless graviton mass is not just hard, but impossible to solve and non-physical. If so, perhaps the seemingly unlikely massive graviton case is actually correct, so we may as well pursue both possibilities theoretically.

On the other hand, if we pursue massive graviton theory to the point where we can definitively rule out it as a possibility due to some theoretical inconsistency that exists for all of the available parameter space, then we could indirectly establish that quantum gravity must arise from a massless graviton, even though we can't directly measure that fact.

Another reason to explore it is that even in the conventional massless spin-2 graviton approach to quantum gravity, gravitons still emit and absorb gravitons, because gravitons couple to mass-energy rather than mass alone, and gravitons have energy even though they lack rest mass in the conventional quantum gravity analysis. Gravitational fields have self-interactions in General Relativity, but the way that this occurs in GR is not very transparent or illuminating when GR is formulated in terms of Einstein's equations, but is much more transparent and obvious when developing an understanding of these self-interactions in a massive gravity quantum gravity context. A massive graviton approach can be used as a way to understand these self-interactions by viewing massless gravity as a limiting case of the massive graviton theory that is mathematically less vexed in some respects because any time you try to do math with zeros everywhere, you will usually end up with infinities that are mathematically hard to work with sooner or later (because division by zero is undefined and approaches "infinity" in the limit).

The problem with using a massive gravity theory to explore the limiting case of the massless graviton, however, is that there are qualitative differences between the way that bosons with even tiny masses behave compared to the way that a truly massless boson behaves.

For example, in layman's terms, massless bosons basically don't experience time and always move at exactly the speed of light regardless of how much energy they carry. But a massive boson does experience time, must move at either less than the speed of light (or more than the speed of light if it is a tachyon), and as a consequence of special relativity, it takes increasingly more energy to increase its speed by the same amount, as it approaches the relativistic regime near the speed of light.

Lensing effects are also not continuous between the massive graviton case and the massless graviton case, so the lensing effects of massive gravitons in the limit as the mass of the graviton falls in the direction of zero from above is not equal to the lensing effects created by a massless graviton.

In a massless graviton theory, tachyonic gravitons (i.e. those traveling at more than the speed of light) can be ruled out almost automatically by assumption. In a massive graviton theory, this isn't a foregone conclusion, and if it is true, you have to work a lot harder to reach that conclusion.

In general, it is quite challenging to formula a massive gravity theory that has desirable properties such as being "ghost free", and the discovery in 2010 that it appeared to be possible to devise a ghost free massive gravity theory rebooted interest in the theory that had gone dormant not long after it was determined in 1972 that a large class of massive gravity theories produce mathematical "ghosts" that it is impossible to remove from this class of theories. While this wasn't a true "no go" theorem, and the conclusion had loopholes that were later successfully exploited, interest in massive gravity theories waned to a trickle for a generation as a result of this discovery.

There are even efforts to explore the possibility of dark matter phenomena as a function of quantum gravity with a massive graviton. Likewise:

The fact that general relativity is modified at large distances in massive gravity provides a possible explanation for the accelerated expansion of the Universe that does not require any dark energy. Massive gravity and its extensions, such as bimetric gravity,[12] can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.[13][14][15]

This is attractive, because while it is trivial to insert a cosmological constant into Einstein's equations in classical GR as an integration constant, it is highly non-trivial to produce dark energy in a graviton based quantum gravity theory in which all global phenomena must arise from the local properties of a graviton (it is easier, at least in principle, to reproduce dark energy in quantum gravity theories like loop quantum gravity, that are quantizing space-time rather than merely inserting a graviton into a smooth and continuous space-time).

Theorists are also looking at massive gravity theories to address other problems in cosmology and black hole and neutron star physics that have gone unsolved in the massless graviton/massless gravitational field paradigm (all illustrated by the list of pre-prints linked above).