[6][7] 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.
[10][11] 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.
[16] 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 > 1.83×1016 m or mg < 6.76×10−23 eV/c2.
), there are only two possible mass terms: Fierz and Pauli[18] 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
Its Hamiltonian is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies.
In the 1970s Hendrik van Dam and Martinus J. G. Veltman[19] and, independently, Valentin I. Zakharov[20] 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.
It was argued by Vainshtein[21] 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
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.
As a response to Freund–Maheshwari–Schonberg finite-range gravity model,[22] and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and Stanley Deser found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode;[23] the tuning
For a massless graviton, this process converges and the result is well-known: one simply arrives at general relativity.
In fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost[24] despite endeavors to prove that such belief is invalid.
[25] It is worth noting that the dRGT model is the best way to single out and "bust" the BD ghost since both are developed using Hamiltonian treatments and ADM variables.
[26] In 2010 a breakthrough was achieved when de Rham, Gabadadze, and Tolley constructed, order by order, a theory of massive gravity with coefficients tuned to avoid the Boulware–Deser ghost by packaging all ghostly (i.e., higher-derivative) operators into total derivatives which do not contribute to the equations of motion.
[27][28] The complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by Fawad Hassan and Rachel Rosen.
[29][30] The action for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by[31] or, equivalently, The ingredients require some explanation.
The principle of gauge-invariance renders redundant expressions in any field theory provided with its corresponding gauge(s).
[32] The absence of vDVZ discontinuity in this approach motivated the development of dRGT resummation of massive gravity theory as follows.
This means that, for instance, nonlinearly completing the Fierz–Pauli theory around Minkowski space given above will lead to dRGT massive gravity with
[39] Interestingly, the covariantization tensor was originally introduced by Maheshwari in a solo authored paper sequel to helicity-
The dRGT was developed inspired by applying the previous technique to the 5D DGP model after considering the deconstruction of higher dimensional Kaluza-Klein gravity theories,[43] in which the extra-dimension(s) is/are replaced by series of N lattice sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.
Note that massive gravity in the metric and vierbein formulations are only equivalent if the symmetry condition is satisfied.
While this is true for most physical situations, there may be cases, such as when matter couples to both metrics or in multimetric theories with interaction cycles, in which it is not.
In these cases the metric and vierbein formulations are distinct physical theories, although each propagates a healthy massive graviton.
The novelty in dRGT massive gravity is that it is a theory of gauge invariance under both local Lorentz transformations, from assuming the reference metric
More importantly, the diffeomorphism transformations help manifesting the dynamics of the helicity-0 and helicity-1 modes, hence the easiness of gauging them away when the theory is compared with its version with the only
In fact this is allowed because the variation of the vierbein action with respect to the locally Lorentz transformed Stueckelberg fields yields this nice result.
[45] Moreover, we can solve explicitly for the Lorentz invariant Stueckelberg fields, and on substituting back into the vierbein action we can show full equivalence with the tensorial form of dRGT massive gravity.
, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration.
[47] 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?