# Massive gravity

In theoretical physics, massive gravity is a particular generalization of general relativity studied by Hendrik van Dam, Martinus J. G. Veltman (1) , and Vladimir E. Zakharov.(2)

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

$h^{\mu\nu}T_{\mu\nu}$

where T is the stress-energy tensor.

The adjective "massive" means that there are also mass terms in the Lagrangian proportional to

$h^{\mu\nu}h_{\mu\nu}$ .

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 Albert Einstein obtained in general relativity. This is known as the vDVZ discontinuity.

More technically stated by E. Babichev, et al. (3), "The (quadratic) Pauli-Fierz theory is known to suffer from the van Dam-Veltman-Zakharov (vDVZ) discontinuity,i.e. the fact that when one lets the mass m of the graviton vanish, one does not recover predictions of general relativity. E.g., if one adjusts the parameters (namely the Planck scale) such that the Newton constant agrees with the one measured by some type of Cavendish experiment, then the light bending as predicted by Pauli-Fierz theory (and for a vanishingly small graviton mass) will be 3/4 of the one obtained by linearizing GR."

The fact that it is smaller is easy to understand: the essential difference between Pauli-Fierz theory and linearized GR comes from an extra propagating scalar mode present in the massive theory. This mode exerts an extra attraction in the massive case compared to the massless case. Hence, if one wants measurements of the force exerted between non relativistic masses to agree, the coupling constant of the massive theory should be smaller than that of the massless theory. But light bending is blind to the scalar sector - because the light energy momentum tensor is traceless. Hence, provided the two theories agree on the force between non relativistic probes, the massive theory would predict a smaller light bending than the massless one.

It has been recently argued[by whom?] 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.

## References

1. H. van Dam and M. G. Veltman, “Massive and massless Yang-Mills and gravitational field,” Nucl. Phys. B 22, 397 (1970)

2. V. I. Zakharov, “Linearized gravitation theory and the graviton mass,” JETP Lett. 12, 312 (1970) [Pisma Zh. Eksp. Teor. Fiz. 12, 447 (1970)]

3. E. Babichev, C. Deffayet, and R. Ziour, “ The recovery of general relativity in massive gravity via the Vainshtein mechanism,” arXiv:1007.4506 [gr-qc] (2010)