It can be fun and interesting to investigate theories of gravity other than General Relativity, and one popular strategy is to introduce extra degrees of freedom. This paper deals with a family of such theories called bimetric gravity (or "bigravity" as in the paper) where the extra degree of freedom is a (classical) second rank-two tensor field.
Typically one runs into a bimetric theory where the second metric is energy-dependent, and is simply not noticeable in weak gravity. Where one finds strong gravity -- notably, in the very early universe -- the second metric becomes important. A popular approach is that the second metric "steers" the first metric. In our classical expansion g = \eta + h, the perturbations that obey the equations for massive wave propagation instead obey the massless wave equations. This drag on the waves means that (classical) electromagnetic waves outrace (classical) gravitational waves when the second metric is operative. This is often used in models to solve the horizon problem: why are readings of the cosmic microwave background at opposite sides of our view of the universe at the same temperature? Because light was fast enough to carry energy between distantly separated matter, thermalizing it before gravitational collapse could begin in earnest. [1][2]
When we quantize a bimetric theory, we get two gravitons: one that is the standard massless spin-2, and the other which is a massive spin-2. The massive graviton only interacts with the massless graviton, and in weak gravity there simply aren't many of either type of graviton, so you don't see the interaction. In strong gravity the densities of both types of graviton inevitably lead to interactions. As in standard perturbatively quantized gravity, the massless graviton mediates the gravitational interaction of everything in the universe (including both types of graviton).
We can now look at this particular bigravity paper's non-novelty in its Abstract: "... a second ... tensor ... [couples with the standard metric tensor] ... such that one massless and one massive linear combination arise. Only one ... tensor ... [couples] to matter" and have an idea what that means. The novelty here is that the second -- massive -- tensor is not energy-dependent, it's always there but gets "washed out" by the behaviour of the tensor fields.
This novel behaviour, "oscillations", make sense in perturbatively quantized gravity: we get two different gravitons, g_standard and g_massive, where g_massive only interacts with g_standard, and g_standard interacts with everything; when g_standard and g_massive are in the same point in the background spacetime, there is a linear effect [3] . g_standard continues to propagate at the maximum speed (c), while wave packets of g_massive have a lower group velocity. This is a tight analogy with neutrino oscillations where heavier neutrinos move more slowly than the lightest neutrino, and where a heavy sterile (no weak charge) neutrino exists.
This is a pretty wild claim. The authors believe their theory is self-consistent, and propose some observables that are in detectable through gravitational wave astronomy that we are close to being able to do. Roughly, a spinning barbell arrangement of matter (two black holes in orbit can be seen as the weights on the end of a vanishingly thin bar) sheds gravitational waves -- and under the Max-Platscher-Smirnov model there is a drag on their propagation compared under standard General Relativity. As a practical example, this leads to a dependence on several parameters, including travel time for the massive component, of gravitational waves detectable by LIGO: the final "bloop" [3] will sound different for similar-mass mergers at different redshifts.
There are a lot of other likely observables including ones that should appear in the cosmic microwave background. Bimetric theories are notorious for generating modes that seem unreasonable (and thus are unlikely to be there when you look for them). However, the idea is a neat and provocative one that is put across well, and so was worth publishing on that basis.
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[1] https://www.wikiwand.com/en/Horizon_problem#/Variable_speed_...
[2] https://www.wikiwand.com/en/Variable_speed_of_light
[3] https://www.youtube.com/watch?v=TWqhUANNFXw
[4] https://www.wikiwand.com/en/Electromagnetic_tensor ("... an antisymmetric rank-2 tensor field—on Minkowski space.")
[5] https://www.wikiwand.com/en/Electrovacuum_solution ("... in which the only nongravitational mass-energy present is the field energy of an electromagnetic field")