Massive gravity

Theory of gravity in which the graviton has nonzero mass
In theoretical physics, massive gravity is a theory of gravity that modifies general relativity by endowing the graviton with a nonzero mass. In the classical theory, this means that gravitational waves obey a massive wave equation and hence travel at speeds below the speed of light.

Massive gravity has a long and winding history, dating back to the 1930s when Wolfgang Pauli and Markus Fierz first developed a theory of a massive spin 2 field propagating on a flat spacetime background. It was later realized in the 1970s that theories of a massive graviton suffered from dangerous pathologies, including a ghost mode and a discontinuity with general relativity in the limit where the graviton mass goes to zero. While solutions to these problems had existed for some time in three spacetime dimensions, they were not solved in four dimensions and higher until the work of Claudia de Rham, Gregory Gabadadze, and Andrew Tolley (dRGT model) in 2010. One of the very early massive gravity theories was constructed in 1965 by Ogievetsky and Polubarinov (OP). Despite the fact that the OP model coincides with the ghost free massive gravity models rediscovered in dRGT, the OP model has been almost unknown among contemporary physicists who work on massive gravity, perhaps because the strategy followed in that model was quite different from what is generally adopted at present. Massive dual gravity to the OP model can be obtained by coupling the dual graviton field to the curl of its own energy momentum tensor. Since the mixed symmetric field strength of dual gravity is comparable to the totally symmetric extrinsic curvature tensor of the Galileons theory, the effective Lagrangian of the dual model in 4 D can be obtained from the Faddeev–LeVerrier recursion, which is similar to that of Galileon theory up to the terms containing polynomials of the trace of the field strength. This is also manifested in the dual formulation of Galileon theory. 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, can yield cosmological solutions which do in fact display late time acceleration in agreement with observations. Observations of gravitational waves have constrained the Compton wavelength of the graviton to be λg > 1.6, which can be interpreted as a bound on the graviton mass mg < 7.7. Competitive bounds on the mass of the graviton have also been obtained from solar system measurements by space missions such as Cassini and MESSENGER, which instead give the constraint λg > 1.83 or mg < 6.76.

At the linear level, one can construct a theory of a massive spin 2 field propagating on Minkowski space. This can be seen as an extension of linearized gravity in the following way. Linearized gravity is obtained by linearizing general relativity around flat space, , where is the Planck mass with the gravitational constant. This leads to a kinetic term in the Lagrangian for which is consistent with diffeomorphism invariance, as well as a coupling to matter of the form

where is the stress–energy tensor. This kinetic term and matter coupling combined are nothing other than the Einstein–Hilbert action linearized about flat space. Massive gravity is obtained by adding nonderivative interaction terms for At the linear level (i. e., second order in ), there are only two possible mass terms:

Fierz and Pauli showed in 1939 that this only propagates the expected five polarizations of a massive graviton (as compared to two for the massless case) if the coefficients are chosen so that Any other choice will unlock a sixth, ghostly degree of freedom. A ghost is a mode with a negative kinetic energy. Its Hamiltonian is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies. The Fierz–Pauli mass term,

is therefore the unique consistent linear theory of a massive spin 2 field.

In the 1970s Hendrik van Dam and Martinus J. G. Veltman and, independently, Valentin I. Zakharov discovered a peculiar property of Fierz–Pauli massive gravity: its predictions do not uniformly reduce to those of general relativity in the limit In particular, while at small scales (shorter than the Compton wavelength of the graviton mass), Newton's gravitational law is recovered, the bending of light is only three quarters of the result Albert Einstein obtained in general relativity. This is known as the vDVZ discontinuity. We may understand the smaller light bending as follows. The Fierz–Pauli massive graviton, due to the broken diffeomorphism invariance, propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity. These three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field. This scalar mode exerts an extra attraction in the massive case compared to the massless case. Hence, if one wants measurements of the force exerted between nonrelativistic 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 stress energy tensor of light is traceless. Hence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.

It was argued by Vainshtein two years later that the vDVZ discontinuity is an artifact of the linear theory, and that the predictions of general relativity are in fact recovered at small scales when one takes into account nonlinear effects, i. e., higher than quadratic terms in Heuristically speaking, within a region known as the Vainshtein radius, fluctuations of the scalar mode become nonlinear, and its higher order derivative terms become larger than the canonical kinetic term. Canonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius. Because the extra force mediated by the scalar is proportional to (minus) its gradient, this leads to a much smaller extra force than we would have calculated just using the linear Fierz–Pauli theory. This phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as DGP and certain scalar tensor theories, where it is crucial for hiding the effects of modified gravity in the solar system. This allows these theories to match terrestrial and solar system tests of gravity as well as general relativity does, while maintaining large deviations at larger distances. In this way these theories can lead to cosmic acceleration and have observable imprints on the large scale structure of the Universe without running afoul of other, much more stringent constraints from observations closer to home.

If the graviton mass is comparable to the Hubble rate , then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration. Because, roughly speaking, the enhanced diffeomorphism symmetry in the limit protects a small graviton mass from large quantum corrections, the choice is in fact technically natural. Massive gravity thus may provide a solution to the cosmological constant problem: why do quantum corrections not cause the Universe to accelerate at extremely early times? However, it turns out that flat and closed Friedmann–Lemaître–Robertson–Walker cosmological solutions do not exist in dRGT massive gravity with a flat reference metric. Therefore, viable cosmologies can only be found in massive gravity if one abandons the cosmological principle that the Universe is uniform on large scales, or otherwise generalizes dRGT. For instance, cosmological solutions are better behaved in bigravity, those instabilities might find a resolution in the nonlinear dynamics (through a Vainshtein like mechanism) or by pushing the era of instability to the very early Universe. and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all. Following the GW170104 event, the graviton's Compton wavelength was found to be at least 1.6, or about 1.6 light years, corresponding to a graviton mass of no more than 7.7. This relation between wavelength and energy is calculated with the same formula (the Planck–Einstein relation) that relates electromagnetic wavelength to photon energy. However, photons, which have only energy and no mass, are fundamentally different from massive gravitons in this respect, since the Compton wavelength of the graviton is not equal to the gravitational wavelength. Instead, the lower bound graviton Compton wavelength is about 9 times greater than the gravitational wavelength for the GW170104 event, which was ~ 1,700 km. This is because the Compton wavelength is defined by the rest mass of the graviton and is an invariant scalar quantity.