Massive gravity

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. 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 in 2010. 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 > 1.6×1016 m, which can be interpreted as a bound on the graviton mass mg < 7.7×10−23 eV/c2. At the linear level, one can construct a theory of a massive spin-2 field h μ ν {displaystyle h_{mu u }} 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, g μ ν = η μ ν + M P l − 1 h μ ν {displaystyle g_{mu u }=eta _{mu u }+M_{mathrm {Pl} }^{-1}h_{mu u }} , where M P l = ( 8 π G ) − 1 / 2 {displaystyle M_{mathrm {Pl} }=(8pi G)^{-1/2}} is the Planck mass with G {displaystyle G} the gravitational constant. This leads to a kinetic term in the Lagrangian for h μ ν {displaystyle h_{mu u }} which is consistent with diffeomorphism invariance, as well as a coupling to matter of the form where T μ ν {displaystyle T_{mu u }} 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 h μ ν {displaystyle h_{mu u }} . At the linear level (i.e., second order in h μ ν {displaystyle h_{mu u }} ), 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 a = − b {displaystyle a=-b} . 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,