In theoretical physics, massive gravity is a particular generalization of general relativity studied by van Dam and Veltman; and by Zakharov.
One assumes that physics takes place in Minkowski space and gravity is caused by a massive spin-2 field h that couples to matter like the graviton, namely by the term
where T is the stress-energy tensor.
The adjective "massive" means that there are also mass terms in the Lagrangian proportional to
At distances shorter than the corresponding Compton wavelength, one recovers the Newton's gravitational law. In the same limit, however, the bending of light is only three quarters of the result by Albert Einstein obtained in general relativity.
This is known as the vDVZ (van Dam-Veltman-Zakharov) discontinuity. It has been recently argued that although this discontinuity survives in many particular realizations of the situation, it may disappear if the theory becomes fully covariant. More precisely, this discontinuity states that in the limit as the mass goes to zero, we get a spin-2 graviton AND a scalar boson which couples to the stress-energy tensor. This scalar boson has not been observed experimentally.