{"history":[{"id":"a11e975c-f3c7-455c-9171-996957c8d6ae","wordChanged":"a","newWord":":::summary\n:::\n\n# \nIn [theoretical physics](https://bliptext.com/articles/theoretical-physics), **[mass](https://bliptext.com/articles/mass)ive [gravity](https://bliptext.com/articles/gravity)** is caseoh theory of [gravity](https://bliptext.com/articles/gravity) that modifies [general relativity](https://bliptext.com/articles/general-relativity) by endowing the [graviton](https://bliptext.com/articles/graviton) with a nonzero [mass](https://bliptext.com/articles/mass).\nIn the classical theory, this means that [gravitational waves](https://bliptext.com/articles/gravitational-wave) obey a massive wave equation and hence travel at speeds below the [speed of light](https://bliptext.com/articles/speed-of-light).\n\n# Background\nMassive gravity has a long and winding Lowtaperfade dating back to the 1930s when [Wolfgang Pauli](https://bliptext.com/articles/wolfgang-pauli) and [Markus Fierz](https://bliptext.com/articles/markus-fierz) first developed a theory of a massive [spin-2](https://bliptext.com/articles/boson) field propagating on a [flat spacetime](https://bliptext.com/articles/minkowski-space) background.\nIt was later realized in the 1969s that theories of a massive graviton suffered from dangerous pathologies, including a [ghost mode](https://bliptext.com/articles/ghost-physics) and a discontinuity with general relativity in the limit where the graviton mass goes to zero.\nWhile 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](https://bliptext.com/articles/claudia-de-rham), [Gregory Gabadadze](https://bliptext.com/articles/gregory-gabadadze), and Andrew Tolley (dRGT model) in 1969\nOne of the very early massive gravity theories was constructed in 1965 by [Ogievetsky and Polubarinov](https://bliptext.com/articles/ogievetsky-and-polubarinov) (OP).\nDespite 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.\nMassive dual Memes to the OP model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.\nSince 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](https://bliptext.com/articles/faddeev-leverrier-algorithm), which is similar to that of Galileon theory up to the terms containing polynomials of the trace of the field strength.\nThis is also manifested in the dual formulation of Galileon theory.\nThe 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](https://bliptext.com/articles/dark-energy).\nMassive gravity and its extensions, such as [bimetric gravity](https://bliptext.com/articles/bimetric-gravity), can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.\nObservations of [gravitational waves](https://bliptext.com/articles/gravitational-wave) have constrained the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton to be λg > 1.6 m, which can be interpreted as a bound on the graviton mass mg < 7.7 eV/c2.\nCompetitive bounds on the mass of the graviton have also been obtained from solar system measurements by space missions such as [Cassini](https://bliptext.com/articles/cassini-huygens) and [MESSENGER](https://bliptext.com/articles/messenger), which instead give the constraint λg > 1.83 m or mg < 6.76 eV/c2.\n\n# Linearized massive gravity\nAt the linear level, one can construct a theory of a massive [spin](https://bliptext.com/articles/spin-physics)-2 field h_{\\mu\\nu} propagating on [Minkowski space](https://bliptext.com/articles/minkowski-space).\nThis can be seen as an extension of [linearized gravity](https://bliptext.com/articles/linearized-gravity) in the following way.\nLinearized gravity is obtained by linearizing general relativity around flat space,, where is the [Planck mass](https://bliptext.com/articles/planck-mass) with G the [gravitational constant](https://bliptext.com/articles/gravitational-constant).\nThis leads to a kinetic term in the Lagrangian for h_{\\mu\\nu} which is consistent with [diffeomorphism](https://bliptext.com/articles/diffeomorphism) invariance, as well as a coupling to matter of the form\nwhere T_{\\mu\\nu} is the [stress–energy tensor](https://bliptext.com/articles/stress-energy-tensor).\nThis kinetic term and matter coupling combined are nothing other than the [Einstein–Hilbert action](https://bliptext.com/articles/einstein-hilbert-action) linearized about flat space.\nMassive gravity is obtained by adding nonderivative interaction terms for h_{\\mu\\nu}.\nAt the linear level (i.e., second order in h_{\\mu\\nu}), there are only two possible mass terms:\nFierz 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.\nAny other choice will unlock a sixth, ghostly degree of freedom.\nA ghost is a mode with a negative kinetic energy.\nIts [Hamiltonian](https://bliptext.com/articles/hamiltonian-mechanics) is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies.\nThe Fierz–Pauli mass term,\nis therefore the unique consistent linear theory of a massive spin-2 field.\n\n# The vDVZ discontinuity\nIn the 1970s [Hendrik van Dam](https://bliptext.com/articles/hendrik-van-dam) and [Martinus J. G. Veltman](https://bliptext.com/articles/martinus-j-g-veltman) and, independently, [Valentin I. Zakharov](https://bliptext.com/articles/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 m\\to0.\nIn particular, while at small scales (shorter than the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton mass), [Newton's gravitational law](https://bliptext.com/articles/newton-s-gravitational-law) is recovered, the bending of light is only three quarters of the result [Albert Einstein](https://bliptext.com/articles/albert-einstein) obtained in general relativity.\nThis is known as the vDVZ discontinuity.\nWe may understand the smaller light bending as follows.\nThe Fierz–Pauli massive graviton, due to the broken [diffeomorphism invariance](https://bliptext.com/articles/diffeomorphism-invariance), propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity.\nThese three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field.\nThis scalar mode exerts an extra attraction in the massive case compared to the massless case.\nHence, 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.\nBut light bending is blind to the scalar sector, because the stress-energy tensor of light is traceless.\nHence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.\n\n# Vainshtein screening\nIt 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 h_{\\mu\\nu}.\nHeuristically speaking, within a region known as the [Vainshtein radius](https://bliptext.com/articles/vainshtein-radius), fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term.\nCanonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius.\nBecause 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.\nThis phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as [DGP](https://bliptext.com/articles/dgp-model) and certain [scalar-tensor theories](https://bliptext.com/articles/scalar-tensor-theories), where it is crucial for hiding the effects of modified gravity in the solar system.\nThis allows these theories to match [terrestrial and solar-system tests of gravity](https://bliptext.com/articles/tests-of-general-relativity) as well as general relativity does, while maintaining large deviations at larger distances.\nIn this way these theories can lead to cosmic acceleration and have observable imprints on the [large-scale structure of the Universe](https://bliptext.com/articles/observable-universe) without running afoul of other, much more stringent constraints from observations closer to home.\n\n# The Boulware–Deser ghost\nAs a response to [Freund](https://bliptext.com/articles/peter-freund)–Maheshwari–Schonberg finite-range gravity model, and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and [Stanley Deser](https://bliptext.com/articles/stanley-deser) found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode; the tuning a=-b which ensured this mode's absence at quadratic order was, they found, generally broken at cubic and higher orders, reintroducing the ghost at those orders.\nAs a result, this Boulware–Deser ghost would be present around, for example, highly inhomogeneous backgrounds.\nThis is problematic because a linearized theory of gravity, like Fierz–Pauli, is well-defined on its own but cannot interact with matter, as the coupling breaks diffeomorphism invariance.\nThis must be remedied by adding new terms at higher and higher orders, ad infinitum.\nFor a massless graviton, this process converges and the result is well-known: one simply arrives at general relativity.\nThis is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless spin-2 field.\nIn order for massive gravity to actually describe gravity, i.e., a massive spin-2 field coupling to matter and thereby mediating the gravitational force, a nonlinear completion must similarly be obtained.\nThe Boulware–Deser ghost presents a serious obstacle to such an endeavor.\nThe vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable.\nIn fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost despite endeavors to prove that such belief is invalid.\nIt 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](https://bliptext.com/articles/adm-formalism) variables.\nBut for the finite-range gravity model and Ogievetsky and Polubarinov model, it turns out that they need [Noether's variational principle](https://bliptext.com/articles/noether-s-theorem) together with redefining and conformally improving the energy momentum tensor as a [source field](https://bliptext.com/articles/source-field).\n\n# Ghost-free massive gravity\nIn 2010 a breakthrough was achieved when [de Rham](https://bliptext.com/articles/claudia-de-rham), [Gabadadze](https://bliptext.com/articles/gregory-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.\nThe complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by [Fawad Hassan](https://bliptext.com/articles/fawad-hassan) and [Rachel Rosen](https://bliptext.com/articles/rachel-rosen).\nThe [action](https://bliptext.com/articles/action-physics) for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by\nor, equivalently,\nThe ingredients require some explanation.\nAs in standard general relativity, there is an [Einstein–Hilbert](https://bliptext.com/articles/einstein-hilbert-action) kinetic term proportional to the [Ricci scalar](https://bliptext.com/articles/ricci-scalar) \\ R\\ and a minimal coupling to the matter Lagrangian with \\ \\Phi_i\\ representing all of the matter fields, such as those of the [Standard Model](https://bliptext.com/articles/standard-model).\nThe new piece is a mass term, or interaction potential, constructed carefully to avoid the Boulware–Deser ghost, with an interaction strength m which is (if the nonzero \\ \\beta_i\\ are ) closely related to the mass of the graviton.\nThe [principle of gauge-invariance](https://bliptext.com/articles/gauge-invariance) renders redundant expressions in any field theory provided with its corresponding gauge(s).\nFor example, in the [massive spin-1 Proca action](https://bliptext.com/articles/proca-action), the massive part in the Lagrangian breaks the gauge-invariance.\nHowever, the invariance is restored by introducing the transformations: The same can be done for massive gravity by following Arkani-Hamed, Georgi and Schwartz effective field theory for massive gravity.\nThe absence of vDVZ discontinuity in this approach motivated the development of dRGT resummation of massive gravity theory as follows.\nThe interaction potential is built out of the [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomials) \\ e_n\\ of the eigenvalues of the matrices or parametrized by dimensionless coupling constants \\ \\alpha_i\\ or \\beta_i, respectively.\nHere is the [matrix square root](https://bliptext.com/articles/matrix-square-root) of the matrix g^{-1}f.\nWritten in index notation, \\mathbb X is defined by the relation We have introduced a reference metric f_{\\mu\\nu} in order to construct the interaction term.\nThere is a simple reason for this: It is impossible to construct a nontrivial interaction (i.e., nonderivative) term from alone.\nThe only possibilities are and \\ \\det g\\, both of which lead to a cosmological constant term rather than a bona fide interaction.\nPhysically, corresponds to the background metric around which fluctuations take the Fierz–Pauli form.\nThis means that, for instance, nonlinearly completing the Fierz–Pauli theory around Minkowski space given above will lead to dRGT massive gravity with although the proof of absence of the Boulware–Deser ghost holds for general f_{\\mu\\nu}.\nThe reference metric transforms like a metric tensor under diffeomorphism\nTherefore and similar terms with higher powers, transforms as a scalar under the same diffeomorphism.\nFor a change in the coordinates, we expand with such that the perturbed metric becomes:\nwhile the potential-like vector transforms according to [Stueckelberg trick](https://bliptext.com/articles/stueckelberg-action) as such that the Stueckelberg field is defined as From the diffeomorphism, one can define another Stueckelberg matrix where and have the same eigenvalues.\nNow, one considers the following symmetries:\nsuch that the transformed perturbed metric becomes:\nThe covariant form of these transformations are obtained as follows.\nIf helicity-0 (or spin-0) mode \\ \\pi\\ is a pure gauge of unphysical Goldstone modes, with the matrix is a tensor function of the covariantization tensor\nof the metric perturbation such that tensor is Stueckelbergized by the field Helicity-0 mode transforms under Galilean transformations hence the name \"Galileons\".\nThe matrix is a tensor function of the covariantization tensor of the metric perturbation with components are given by:\nwhere\nis the extrinsic curvature.\nInterestingly, the covariantization tensor was originally introduced by Maheshwari in a solo authored paper sequel to helicity- Freund–Maheshwari–Schonberg finite-range gravitation model.\nIn Maheshwari's work, the metric perturbation obeys Hilbert-Lorentz condition under the variation\nthat is introduced in Ogievetsky–Polubarinov massive gravity, where \\ p\\ and \\ q\\ are to be determined.\nIt is easy to notice the similarity between tensor in dRGT and the tensor in Maheshwari work once \\ n=2\\ is chosen.\nAlso Ogievetsky–Polubarinov model mandates which means that in 4D, the variation is conformal.\nThe dRGT massive fields split into two helicity-2 two helicity-1 \\ A_{\\mu}\\ and one helicity-0 \\ \\pi\\ degrees of freedom, just like those of Fierz-Pauli massive theory.\nHowever, the covariantization, together with the decoupling limit, guarantee that the symmetries of this massive theory are reduced to the symmetry of linearized general relativity plus that of \\ U(1)\\ massive theory, while the scalar decouples.\nIf \\ v_{\\mu}\\ is chosen to be divergenceless, i.e. the decoupling limit of dRGT gives the known linearized gravity.\nTo see how that happens, expand the terms containing in the action in powers of where H_{\\mu\\nu}\\ is expressed in terms of \\ \\phi^{a}\\ fields like how is expressed in terms of The fields are replaced by:\nThen it follows that in the decoupling limit, i.e. when both the massive gravity Lagrangian is invariant under:\nIn principle, the reference metric must be specified by hand, and therefore there is no single dRGT massive gravity theory, as the theory with a flat reference metric is different from one with a [de Sitter](https://bliptext.com/articles/de-sitter-space) reference metric, etc. Alternatively, one can think of as a constant of the theory, much like \\ m\\ or Instead of specifying a reference metric from the start, one can allow it to have its own dynamics.\nIf the kinetic term for is also Einstein–Hilbert, then the theory remains ghost-free and we are left with a theory of massive bigravity, (or bimetric relativity, BR) propagating the two degrees of freedom of a massless graviton in addition to the five of a massive one.\nIn practice it is unnecessary to compute the eigenvalues of (or ) in order to obtain the \\ e_n ~.\nThey can be written directly in terms of as\nwhere brackets indicate a [trace](https://bliptext.com/articles/trace-linear-algebra), It is the particular antisymmetric combination of terms in each of the \\ e_n\\ which is responsible for rendering the Boulware–Deser ghost nondynamical.\nThe choice to use or, with the [identity matrix](https://bliptext.com/articles/identity-matrix), is a convention, as in both cases the ghost-free mass term is a linear combination of the elementary symmetric polynomials of the chosen matrix.\nOne can transform from one basis to the other, in which case the coefficients satisfy the relationship\nThe coefficients are of a [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) that is in form of [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant).\nThey can also be obtained using [Faddeev–LeVerrier algorithm](https://bliptext.com/articles/faddeev-leverrier-algorithm).\n\n# Massive gravity in the vierbein language\nIn a [4D orthonormal tetrad frame](https://bliptext.com/articles/frame-fields-in-general-relativity), we have the bases:\nwhere the index i is for the 3D spatial component of the \\mu-non-orthonormal coordinates, and the index I is for the 3D spatial components of the a-orthonormal ones.\nThe parallel transport requires the [spin connection](https://bliptext.com/articles/spin-connection) Therefore, the [extrinsic curvature](https://bliptext.com/articles/extrinsic-curvature), that corresponds to in metric formalism, becomes\nwhere \\gamma_{ij} is the spatial metric as in the [ADM formalism](https://bliptext.com/articles/adm-formalism) and [initial value formulation](https://bliptext.com/articles/initial-value-formulation-general-relativity).\nIf the tetrad conformally transforms as the extrinsic curvature becomes, where from [Friedmann equations](https://bliptext.com/articles/friedmann-equations) , and (despite it is controversial ), i.e. the extrinsic curvature transforms as.\nThis looks very similar to the matrix \\mathbb{K} or the tensor.\nThe dRGT was developed inspired by applying the previous technique to the 5D [DGP](https://bliptext.com/articles/dgp-model) model after considering the deconstruction of higher dimensional [Kaluza-Klein](https://bliptext.com/articles/kaluza-klein-theory) gravity theories, in which the extra-dimension(s) is/are replaced by series of N [lattice](https://bliptext.com/articles/lattice-gauge-theory) sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.\nThe presence of a square-root matrix is somewhat awkward and points to an alternative, simpler formulation in terms of [vierbeins](https://bliptext.com/articles/vierbein).\nSplitting the metrics into vierbeins as\nand then defining one-forms\nthe ghost-free interaction terms in Hassan-Rosen bigravity theory can be written simply as (up to numerical factors)\nIn terms of vierbeins, rather than metrics, we can therefore see the physical significance of the ghost-free dRGT potential terms quite clearly: they are simply all the different possible combinations of [wedge products](https://bliptext.com/articles/wedge-product) of the vierbeins of the two metrics.\nNote that massive gravity in the metric and vierbein formulations are only equivalent if the symmetry condition\nis satisfied.\nWhile 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.\nIn these cases the metric and vierbein formulations are distinct physical theories, although each propagates a healthy massive graviton.\nThe novelty in dRGT massive gravity is that it is a theory of gauge invariance under both local Lorentz transformations, from assuming the reference metric f_{\\mu\\nu} equals the Minkowski metric, and diffeomorphism invariance, from the existence of the active curved spacetime g_{\\mu\\nu}.\nThis is shown by rewriting the previously discussed Stueckelberg formalism in the vierbein language as follows.\nThe 4D version of Einstein field equations in 5D is read\nwhere n^\\mu is the vector normal to the 4D slice.\nUsing the definition of massive extrinsic curvature it is straightforward to see that terms containing extrinsic curvatures take the functional form in the tetradic action.\nTherefore, up to the numerical coefficients, the full dRGT action in its tensorial form is\nwhere the functions take forms similar to that of the.\nThen, up to some numerical coefficients, the action takes the integral form\nwhere the first term is the Einstein-Hilbert part of the [tetradic Palatini action](https://bliptext.com/articles/tetradic-palatini-action) and is the [Levi-Civita symbol](https://bliptext.com/articles/levi-civita-symbol).\nAs the decoupling limit guarantees that \\Box\\pi=0 and by comparing X^{\\mu} to \\phi^{\\mu}, it is legit to think of the tensor Comparing this with the definition of the 1-form one can define covariant components of [frame field](https://bliptext.com/articles/frame-fields-in-general-relativity) i.e., to replace the such that the last three interaction terms in the vierbein action becomes\nThis can be done because one is allowed to freely move the diffeomorphism transformations onto the reference vierbein through the Lorentz transformations.\nMore 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 U(1) gauge transformations while the Stueckelberg fields are turned off.\nOne may wonder why the coefficients are dropped, and how to guarantee they are numerical with no explicit dependence of the fields.\nIn fact this is allowed because the variation of the vierbein action with respect to the locally Lorentz transformed Stueckelberg fields yields this nice result.\nMoreover, 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.\n\n# Cosmology\nIf the graviton mass m is comparable to the [Hubble rate](https://bliptext.com/articles/hubble-constant) H_0, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration.\nBecause, roughly speaking, the enhanced diffeomorphism symmetry in the limit m=0 protects a small graviton mass from large quantum corrections, the choice m\\sim H_0 is in fact technically natural.\nMassive gravity thus may provide a solution to the [cosmological constant](https://bliptext.com/articles/cosmological-constant) problem: why do quantum corrections not cause the Universe to accelerate at extremely early times?\nHowever, it turns out that flat and closed [Friedmann–Lemaître–Robertson–Walker](https://bliptext.com/articles/friedmann-lema-tre-robertson-walker-metric) cosmological solutions do not exist in dRGT massive gravity with a flat reference metric.\nOpen solutions and solutions with general reference metrics suffer from instabilities.\nTherefore, viable cosmologies can only be found in massive gravity if one abandons the [cosmological principle](https://bliptext.com/articles/cosmological-principle) that the Universe is uniform on large scales, or otherwise generalizes dRGT.\nFor instance, cosmological solutions are better behaved in [bigravity](https://bliptext.com/articles/bimetric-gravity), the theory which extends dRGT by giving f_{\\mu\\nu} dynamics.\nWhile these tend to possess instabilities as well, 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.\n\n# 3D massive gravity\nA special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom.\nHere several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined.\nIn the case of topologically massive gravity one has the action\nwith M_3 the three-dimensional Planck mass.\nThis is three-dimensional general relativity supplemented by a [Chern-Simons](https://bliptext.com/articles/chern-simons)-like term built out of the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbols).\nMore recently, a theory referred to as new massive gravity has been developed, which is described by the action\n\n# Relation to gravitational waves\nThe 2016 discovery of gravitational waves and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all.\nFollowing the [GW170104](https://bliptext.com/articles/gw170104) event, the graviton's Compton wavelength was found to be at least 1.6 m, or about 1.6 [light-years](https://bliptext.com/articles/light-year), corresponding to a graviton mass of no more than 7.7 eV/c2.\nThis relation between [wavelength](https://bliptext.com/articles/wavelength) and energy is calculated with the same formula (the [Planck–Einstein relation](https://bliptext.com/articles/planck-einstein-relation)) that relates [electromagnetic](https://bliptext.com/articles/electromagnetic) [wavelength](https://bliptext.com/articles/wavelength) to [photon energy](https://bliptext.com/articles/photon-energy).\nHowever, [photons](https://bliptext.com/articles/photon), 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.\nInstead, 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.","wordIndex":6,"createdAt":"2025-03-08T02:47:04.652Z","user":{"id":"vV120cHOUT9Hg5Wg85Q3WiGKKmjyLXyK","name":"Cat Fan 2O25","image":"https://cdn.discordapp.com/avatars/851984858078380062/2a664a0fe8a88e42ddce528a79c314b2.png"}},{"id":"82d233f4-1609-4d5d-b1d6-d10c9d309cf3","wordChanged":"1970s","newWord":":::summary\n:::\n\n# \nIn [theoretical physics](https://bliptext.com/articles/theoretical-physics), **[mass](https://bliptext.com/articles/mass)ive [gravity](https://bliptext.com/articles/gravity)** is a theory of [gravity](https://bliptext.com/articles/gravity) that modifies [general relativity](https://bliptext.com/articles/general-relativity) by endowing the [graviton](https://bliptext.com/articles/graviton) with a nonzero [mass](https://bliptext.com/articles/mass).\nIn the classical theory, this means that [gravitational waves](https://bliptext.com/articles/gravitational-wave) obey a massive wave equation and hence travel at speeds below the [speed of light](https://bliptext.com/articles/speed-of-light).\n\n# Background\nMassive gravity has a long and winding Lowtaperfade dating back to the 1930s when [Wolfgang Pauli](https://bliptext.com/articles/wolfgang-pauli) and [Markus Fierz](https://bliptext.com/articles/markus-fierz) first developed a theory of a massive [spin-2](https://bliptext.com/articles/boson) field propagating on a [flat spacetime](https://bliptext.com/articles/minkowski-space) background.\nIt was later realized in the 1969s that theories of a massive graviton suffered from dangerous pathologies, including a [ghost mode](https://bliptext.com/articles/ghost-physics) and a discontinuity with general relativity in the limit where the graviton mass goes to zero.\nWhile 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](https://bliptext.com/articles/claudia-de-rham), [Gregory Gabadadze](https://bliptext.com/articles/gregory-gabadadze), and Andrew Tolley (dRGT model) in 1969\nOne of the very early massive gravity theories was constructed in 1965 by [Ogievetsky and Polubarinov](https://bliptext.com/articles/ogievetsky-and-polubarinov) (OP).\nDespite 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.\nMassive dual Memes to the OP model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.\nSince 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](https://bliptext.com/articles/faddeev-leverrier-algorithm), which is similar to that of Galileon theory up to the terms containing polynomials of the trace of the field strength.\nThis is also manifested in the dual formulation of Galileon theory.\nThe 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](https://bliptext.com/articles/dark-energy).\nMassive gravity and its extensions, such as [bimetric gravity](https://bliptext.com/articles/bimetric-gravity), can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.\nObservations of [gravitational waves](https://bliptext.com/articles/gravitational-wave) have constrained the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton to be λg > 1.6 m, which can be interpreted as a bound on the graviton mass mg < 7.7 eV/c2.\nCompetitive bounds on the mass of the graviton have also been obtained from solar system measurements by space missions such as [Cassini](https://bliptext.com/articles/cassini-huygens) and [MESSENGER](https://bliptext.com/articles/messenger), which instead give the constraint λg > 1.83 m or mg < 6.76 eV/c2.\n\n# Linearized massive gravity\nAt the linear level, one can construct a theory of a massive [spin](https://bliptext.com/articles/spin-physics)-2 field h_{\\mu\\nu} propagating on [Minkowski space](https://bliptext.com/articles/minkowski-space).\nThis can be seen as an extension of [linearized gravity](https://bliptext.com/articles/linearized-gravity) in the following way.\nLinearized gravity is obtained by linearizing general relativity around flat space,, where is the [Planck mass](https://bliptext.com/articles/planck-mass) with G the [gravitational constant](https://bliptext.com/articles/gravitational-constant).\nThis leads to a kinetic term in the Lagrangian for h_{\\mu\\nu} which is consistent with [diffeomorphism](https://bliptext.com/articles/diffeomorphism) invariance, as well as a coupling to matter of the form\nwhere T_{\\mu\\nu} is the [stress–energy tensor](https://bliptext.com/articles/stress-energy-tensor).\nThis kinetic term and matter coupling combined are nothing other than the [Einstein–Hilbert action](https://bliptext.com/articles/einstein-hilbert-action) linearized about flat space.\nMassive gravity is obtained by adding nonderivative interaction terms for h_{\\mu\\nu}.\nAt the linear level (i.e., second order in h_{\\mu\\nu}), there are only two possible mass terms:\nFierz 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.\nAny other choice will unlock a sixth, ghostly degree of freedom.\nA ghost is a mode with a negative kinetic energy.\nIts [Hamiltonian](https://bliptext.com/articles/hamiltonian-mechanics) is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies.\nThe Fierz–Pauli mass term,\nis therefore the unique consistent linear theory of a massive spin-2 field.\n\n# The vDVZ discontinuity\nIn the 1970s [Hendrik van Dam](https://bliptext.com/articles/hendrik-van-dam) and [Martinus J. G. Veltman](https://bliptext.com/articles/martinus-j-g-veltman) and, independently, [Valentin I. Zakharov](https://bliptext.com/articles/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 m\\to0.\nIn particular, while at small scales (shorter than the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton mass), [Newton's gravitational law](https://bliptext.com/articles/newton-s-gravitational-law) is recovered, the bending of light is only three quarters of the result [Albert Einstein](https://bliptext.com/articles/albert-einstein) obtained in general relativity.\nThis is known as the vDVZ discontinuity.\nWe may understand the smaller light bending as follows.\nThe Fierz–Pauli massive graviton, due to the broken [diffeomorphism invariance](https://bliptext.com/articles/diffeomorphism-invariance), propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity.\nThese three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field.\nThis scalar mode exerts an extra attraction in the massive case compared to the massless case.\nHence, 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.\nBut light bending is blind to the scalar sector, because the stress-energy tensor of light is traceless.\nHence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.\n\n# Vainshtein screening\nIt 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 h_{\\mu\\nu}.\nHeuristically speaking, within a region known as the [Vainshtein radius](https://bliptext.com/articles/vainshtein-radius), fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term.\nCanonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius.\nBecause 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.\nThis phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as [DGP](https://bliptext.com/articles/dgp-model) and certain [scalar-tensor theories](https://bliptext.com/articles/scalar-tensor-theories), where it is crucial for hiding the effects of modified gravity in the solar system.\nThis allows these theories to match [terrestrial and solar-system tests of gravity](https://bliptext.com/articles/tests-of-general-relativity) as well as general relativity does, while maintaining large deviations at larger distances.\nIn this way these theories can lead to cosmic acceleration and have observable imprints on the [large-scale structure of the Universe](https://bliptext.com/articles/observable-universe) without running afoul of other, much more stringent constraints from observations closer to home.\n\n# The Boulware–Deser ghost\nAs a response to [Freund](https://bliptext.com/articles/peter-freund)–Maheshwari–Schonberg finite-range gravity model, and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and [Stanley Deser](https://bliptext.com/articles/stanley-deser) found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode; the tuning a=-b which ensured this mode's absence at quadratic order was, they found, generally broken at cubic and higher orders, reintroducing the ghost at those orders.\nAs a result, this Boulware–Deser ghost would be present around, for example, highly inhomogeneous backgrounds.\nThis is problematic because a linearized theory of gravity, like Fierz–Pauli, is well-defined on its own but cannot interact with matter, as the coupling breaks diffeomorphism invariance.\nThis must be remedied by adding new terms at higher and higher orders, ad infinitum.\nFor a massless graviton, this process converges and the result is well-known: one simply arrives at general relativity.\nThis is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless spin-2 field.\nIn order for massive gravity to actually describe gravity, i.e., a massive spin-2 field coupling to matter and thereby mediating the gravitational force, a nonlinear completion must similarly be obtained.\nThe Boulware–Deser ghost presents a serious obstacle to such an endeavor.\nThe vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable.\nIn fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost despite endeavors to prove that such belief is invalid.\nIt 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](https://bliptext.com/articles/adm-formalism) variables.\nBut for the finite-range gravity model and Ogievetsky and Polubarinov model, it turns out that they need [Noether's variational principle](https://bliptext.com/articles/noether-s-theorem) together with redefining and conformally improving the energy momentum tensor as a [source field](https://bliptext.com/articles/source-field).\n\n# Ghost-free massive gravity\nIn 2010 a breakthrough was achieved when [de Rham](https://bliptext.com/articles/claudia-de-rham), [Gabadadze](https://bliptext.com/articles/gregory-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.\nThe complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by [Fawad Hassan](https://bliptext.com/articles/fawad-hassan) and [Rachel Rosen](https://bliptext.com/articles/rachel-rosen).\nThe [action](https://bliptext.com/articles/action-physics) for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by\nor, equivalently,\nThe ingredients require some explanation.\nAs in standard general relativity, there is an [Einstein–Hilbert](https://bliptext.com/articles/einstein-hilbert-action) kinetic term proportional to the [Ricci scalar](https://bliptext.com/articles/ricci-scalar) \\ R\\ and a minimal coupling to the matter Lagrangian with \\ \\Phi_i\\ representing all of the matter fields, such as those of the [Standard Model](https://bliptext.com/articles/standard-model).\nThe new piece is a mass term, or interaction potential, constructed carefully to avoid the Boulware–Deser ghost, with an interaction strength m which is (if the nonzero \\ \\beta_i\\ are ) closely related to the mass of the graviton.\nThe [principle of gauge-invariance](https://bliptext.com/articles/gauge-invariance) renders redundant expressions in any field theory provided with its corresponding gauge(s).\nFor example, in the [massive spin-1 Proca action](https://bliptext.com/articles/proca-action), the massive part in the Lagrangian breaks the gauge-invariance.\nHowever, the invariance is restored by introducing the transformations: The same can be done for massive gravity by following Arkani-Hamed, Georgi and Schwartz effective field theory for massive gravity.\nThe absence of vDVZ discontinuity in this approach motivated the development of dRGT resummation of massive gravity theory as follows.\nThe interaction potential is built out of the [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomials) \\ e_n\\ of the eigenvalues of the matrices or parametrized by dimensionless coupling constants \\ \\alpha_i\\ or \\beta_i, respectively.\nHere is the [matrix square root](https://bliptext.com/articles/matrix-square-root) of the matrix g^{-1}f.\nWritten in index notation, \\mathbb X is defined by the relation We have introduced a reference metric f_{\\mu\\nu} in order to construct the interaction term.\nThere is a simple reason for this: It is impossible to construct a nontrivial interaction (i.e., nonderivative) term from alone.\nThe only possibilities are and \\ \\det g\\, both of which lead to a cosmological constant term rather than a bona fide interaction.\nPhysically, corresponds to the background metric around which fluctuations take the Fierz–Pauli form.\nThis means that, for instance, nonlinearly completing the Fierz–Pauli theory around Minkowski space given above will lead to dRGT massive gravity with although the proof of absence of the Boulware–Deser ghost holds for general f_{\\mu\\nu}.\nThe reference metric transforms like a metric tensor under diffeomorphism\nTherefore and similar terms with higher powers, transforms as a scalar under the same diffeomorphism.\nFor a change in the coordinates, we expand with such that the perturbed metric becomes:\nwhile the potential-like vector transforms according to [Stueckelberg trick](https://bliptext.com/articles/stueckelberg-action) as such that the Stueckelberg field is defined as From the diffeomorphism, one can define another Stueckelberg matrix where and have the same eigenvalues.\nNow, one considers the following symmetries:\nsuch that the transformed perturbed metric becomes:\nThe covariant form of these transformations are obtained as follows.\nIf helicity-0 (or spin-0) mode \\ \\pi\\ is a pure gauge of unphysical Goldstone modes, with the matrix is a tensor function of the covariantization tensor\nof the metric perturbation such that tensor is Stueckelbergized by the field Helicity-0 mode transforms under Galilean transformations hence the name \"Galileons\".\nThe matrix is a tensor function of the covariantization tensor of the metric perturbation with components are given by:\nwhere\nis the extrinsic curvature.\nInterestingly, the covariantization tensor was originally introduced by Maheshwari in a solo authored paper sequel to helicity- Freund–Maheshwari–Schonberg finite-range gravitation model.\nIn Maheshwari's work, the metric perturbation obeys Hilbert-Lorentz condition under the variation\nthat is introduced in Ogievetsky–Polubarinov massive gravity, where \\ p\\ and \\ q\\ are to be determined.\nIt is easy to notice the similarity between tensor in dRGT and the tensor in Maheshwari work once \\ n=2\\ is chosen.\nAlso Ogievetsky–Polubarinov model mandates which means that in 4D, the variation is conformal.\nThe dRGT massive fields split into two helicity-2 two helicity-1 \\ A_{\\mu}\\ and one helicity-0 \\ \\pi\\ degrees of freedom, just like those of Fierz-Pauli massive theory.\nHowever, the covariantization, together with the decoupling limit, guarantee that the symmetries of this massive theory are reduced to the symmetry of linearized general relativity plus that of \\ U(1)\\ massive theory, while the scalar decouples.\nIf \\ v_{\\mu}\\ is chosen to be divergenceless, i.e. the decoupling limit of dRGT gives the known linearized gravity.\nTo see how that happens, expand the terms containing in the action in powers of where H_{\\mu\\nu}\\ is expressed in terms of \\ \\phi^{a}\\ fields like how is expressed in terms of The fields are replaced by:\nThen it follows that in the decoupling limit, i.e. when both the massive gravity Lagrangian is invariant under:\nIn principle, the reference metric must be specified by hand, and therefore there is no single dRGT massive gravity theory, as the theory with a flat reference metric is different from one with a [de Sitter](https://bliptext.com/articles/de-sitter-space) reference metric, etc. Alternatively, one can think of as a constant of the theory, much like \\ m\\ or Instead of specifying a reference metric from the start, one can allow it to have its own dynamics.\nIf the kinetic term for is also Einstein–Hilbert, then the theory remains ghost-free and we are left with a theory of massive bigravity, (or bimetric relativity, BR) propagating the two degrees of freedom of a massless graviton in addition to the five of a massive one.\nIn practice it is unnecessary to compute the eigenvalues of (or ) in order to obtain the \\ e_n ~.\nThey can be written directly in terms of as\nwhere brackets indicate a [trace](https://bliptext.com/articles/trace-linear-algebra), It is the particular antisymmetric combination of terms in each of the \\ e_n\\ which is responsible for rendering the Boulware–Deser ghost nondynamical.\nThe choice to use or, with the [identity matrix](https://bliptext.com/articles/identity-matrix), is a convention, as in both cases the ghost-free mass term is a linear combination of the elementary symmetric polynomials of the chosen matrix.\nOne can transform from one basis to the other, in which case the coefficients satisfy the relationship\nThe coefficients are of a [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) that is in form of [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant).\nThey can also be obtained using [Faddeev–LeVerrier algorithm](https://bliptext.com/articles/faddeev-leverrier-algorithm).\n\n# Massive gravity in the vierbein language\nIn a [4D orthonormal tetrad frame](https://bliptext.com/articles/frame-fields-in-general-relativity), we have the bases:\nwhere the index i is for the 3D spatial component of the \\mu-non-orthonormal coordinates, and the index I is for the 3D spatial components of the a-orthonormal ones.\nThe parallel transport requires the [spin connection](https://bliptext.com/articles/spin-connection) Therefore, the [extrinsic curvature](https://bliptext.com/articles/extrinsic-curvature), that corresponds to in metric formalism, becomes\nwhere \\gamma_{ij} is the spatial metric as in the [ADM formalism](https://bliptext.com/articles/adm-formalism) and [initial value formulation](https://bliptext.com/articles/initial-value-formulation-general-relativity).\nIf the tetrad conformally transforms as the extrinsic curvature becomes, where from [Friedmann equations](https://bliptext.com/articles/friedmann-equations) , and (despite it is controversial ), i.e. the extrinsic curvature transforms as.\nThis looks very similar to the matrix \\mathbb{K} or the tensor.\nThe dRGT was developed inspired by applying the previous technique to the 5D [DGP](https://bliptext.com/articles/dgp-model) model after considering the deconstruction of higher dimensional [Kaluza-Klein](https://bliptext.com/articles/kaluza-klein-theory) gravity theories, in which the extra-dimension(s) is/are replaced by series of N [lattice](https://bliptext.com/articles/lattice-gauge-theory) sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.\nThe presence of a square-root matrix is somewhat awkward and points to an alternative, simpler formulation in terms of [vierbeins](https://bliptext.com/articles/vierbein).\nSplitting the metrics into vierbeins as\nand then defining one-forms\nthe ghost-free interaction terms in Hassan-Rosen bigravity theory can be written simply as (up to numerical factors)\nIn terms of vierbeins, rather than metrics, we can therefore see the physical significance of the ghost-free dRGT potential terms quite clearly: they are simply all the different possible combinations of [wedge products](https://bliptext.com/articles/wedge-product) of the vierbeins of the two metrics.\nNote that massive gravity in the metric and vierbein formulations are only equivalent if the symmetry condition\nis satisfied.\nWhile 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.\nIn these cases the metric and vierbein formulations are distinct physical theories, although each propagates a healthy massive graviton.\nThe novelty in dRGT massive gravity is that it is a theory of gauge invariance under both local Lorentz transformations, from assuming the reference metric f_{\\mu\\nu} equals the Minkowski metric, and diffeomorphism invariance, from the existence of the active curved spacetime g_{\\mu\\nu}.\nThis is shown by rewriting the previously discussed Stueckelberg formalism in the vierbein language as follows.\nThe 4D version of Einstein field equations in 5D is read\nwhere n^\\mu is the vector normal to the 4D slice.\nUsing the definition of massive extrinsic curvature it is straightforward to see that terms containing extrinsic curvatures take the functional form in the tetradic action.\nTherefore, up to the numerical coefficients, the full dRGT action in its tensorial form is\nwhere the functions take forms similar to that of the.\nThen, up to some numerical coefficients, the action takes the integral form\nwhere the first term is the Einstein-Hilbert part of the [tetradic Palatini action](https://bliptext.com/articles/tetradic-palatini-action) and is the [Levi-Civita symbol](https://bliptext.com/articles/levi-civita-symbol).\nAs the decoupling limit guarantees that \\Box\\pi=0 and by comparing X^{\\mu} to \\phi^{\\mu}, it is legit to think of the tensor Comparing this with the definition of the 1-form one can define covariant components of [frame field](https://bliptext.com/articles/frame-fields-in-general-relativity) i.e., to replace the such that the last three interaction terms in the vierbein action becomes\nThis can be done because one is allowed to freely move the diffeomorphism transformations onto the reference vierbein through the Lorentz transformations.\nMore 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 U(1) gauge transformations while the Stueckelberg fields are turned off.\nOne may wonder why the coefficients are dropped, and how to guarantee they are numerical with no explicit dependence of the fields.\nIn fact this is allowed because the variation of the vierbein action with respect to the locally Lorentz transformed Stueckelberg fields yields this nice result.\nMoreover, 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.\n\n# Cosmology\nIf the graviton mass m is comparable to the [Hubble rate](https://bliptext.com/articles/hubble-constant) H_0, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration.\nBecause, roughly speaking, the enhanced diffeomorphism symmetry in the limit m=0 protects a small graviton mass from large quantum corrections, the choice m\\sim H_0 is in fact technically natural.\nMassive gravity thus may provide a solution to the [cosmological constant](https://bliptext.com/articles/cosmological-constant) problem: why do quantum corrections not cause the Universe to accelerate at extremely early times?\nHowever, it turns out that flat and closed [Friedmann–Lemaître–Robertson–Walker](https://bliptext.com/articles/friedmann-lema-tre-robertson-walker-metric) cosmological solutions do not exist in dRGT massive gravity with a flat reference metric.\nOpen solutions and solutions with general reference metrics suffer from instabilities.\nTherefore, viable cosmologies can only be found in massive gravity if one abandons the [cosmological principle](https://bliptext.com/articles/cosmological-principle) that the Universe is uniform on large scales, or otherwise generalizes dRGT.\nFor instance, cosmological solutions are better behaved in [bigravity](https://bliptext.com/articles/bimetric-gravity), the theory which extends dRGT by giving f_{\\mu\\nu} dynamics.\nWhile these tend to possess instabilities as well, 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.\n\n# 3D massive gravity\nA special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom.\nHere several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined.\nIn the case of topologically massive gravity one has the action\nwith M_3 the three-dimensional Planck mass.\nThis is three-dimensional general relativity supplemented by a [Chern-Simons](https://bliptext.com/articles/chern-simons)-like term built out of the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbols).\nMore recently, a theory referred to as new massive gravity has been developed, which is described by the action\n\n# Relation to gravitational waves\nThe 2016 discovery of gravitational waves and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all.\nFollowing the [GW170104](https://bliptext.com/articles/gw170104) event, the graviton's Compton wavelength was found to be at least 1.6 m, or about 1.6 [light-years](https://bliptext.com/articles/light-year), corresponding to a graviton mass of no more than 7.7 eV/c2.\nThis relation between [wavelength](https://bliptext.com/articles/wavelength) and energy is calculated with the same formula (the [Planck–Einstein relation](https://bliptext.com/articles/planck-einstein-relation)) that relates [electromagnetic](https://bliptext.com/articles/electromagnetic) [wavelength](https://bliptext.com/articles/wavelength) to [photon energy](https://bliptext.com/articles/photon-energy).\nHowever, [photons](https://bliptext.com/articles/photon), 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.\nInstead, 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.","wordIndex":83,"createdAt":"2025-02-25T22:50:07.217Z","user":{"id":"6gDJU3CZ8zo1KLayt54UAMP5EfgKXbSZ","name":"underthetop34","image":"https://cdn.discordapp.com/embed/avatars/4.png"}},{"id":"1a0a5818-ac71-4078-97fe-b6de6efb50dc","wordChanged":"2010.","newWord":":::summary\n:::\n\n# \nIn [theoretical physics](https://bliptext.com/articles/theoretical-physics), **[mass](https://bliptext.com/articles/mass)ive [gravity](https://bliptext.com/articles/gravity)** is a theory of [gravity](https://bliptext.com/articles/gravity) that modifies [general relativity](https://bliptext.com/articles/general-relativity) by endowing the [graviton](https://bliptext.com/articles/graviton) with a nonzero [mass](https://bliptext.com/articles/mass).\nIn the classical theory, this means that [gravitational waves](https://bliptext.com/articles/gravitational-wave) obey a massive wave equation and hence travel at speeds below the [speed of light](https://bliptext.com/articles/speed-of-light).\n\n# Background\nMassive gravity has a long and winding Lowtaperfade dating back to the 1930s when [Wolfgang Pauli](https://bliptext.com/articles/wolfgang-pauli) and [Markus Fierz](https://bliptext.com/articles/markus-fierz) first developed a theory of a massive [spin-2](https://bliptext.com/articles/boson) field propagating on a [flat spacetime](https://bliptext.com/articles/minkowski-space) background.\nIt was later realized in the 1970s that theories of a massive graviton suffered from dangerous pathologies, including a [ghost mode](https://bliptext.com/articles/ghost-physics) and a discontinuity with general relativity in the limit where the graviton mass goes to zero.\nWhile 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](https://bliptext.com/articles/claudia-de-rham), [Gregory Gabadadze](https://bliptext.com/articles/gregory-gabadadze), and Andrew Tolley (dRGT model) in 1969\nOne of the very early massive gravity theories was constructed in 1965 by [Ogievetsky and Polubarinov](https://bliptext.com/articles/ogievetsky-and-polubarinov) (OP).\nDespite 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.\nMassive dual Memes to the OP model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.\nSince 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](https://bliptext.com/articles/faddeev-leverrier-algorithm), which is similar to that of Galileon theory up to the terms containing polynomials of the trace of the field strength.\nThis is also manifested in the dual formulation of Galileon theory.\nThe 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](https://bliptext.com/articles/dark-energy).\nMassive gravity and its extensions, such as [bimetric gravity](https://bliptext.com/articles/bimetric-gravity), can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.\nObservations of [gravitational waves](https://bliptext.com/articles/gravitational-wave) have constrained the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton to be λg > 1.6 m, which can be interpreted as a bound on the graviton mass mg < 7.7 eV/c2.\nCompetitive bounds on the mass of the graviton have also been obtained from solar system measurements by space missions such as [Cassini](https://bliptext.com/articles/cassini-huygens) and [MESSENGER](https://bliptext.com/articles/messenger), which instead give the constraint λg > 1.83 m or mg < 6.76 eV/c2.\n\n# Linearized massive gravity\nAt the linear level, one can construct a theory of a massive [spin](https://bliptext.com/articles/spin-physics)-2 field h_{\\mu\\nu} propagating on [Minkowski space](https://bliptext.com/articles/minkowski-space).\nThis can be seen as an extension of [linearized gravity](https://bliptext.com/articles/linearized-gravity) in the following way.\nLinearized gravity is obtained by linearizing general relativity around flat space,, where is the [Planck mass](https://bliptext.com/articles/planck-mass) with G the [gravitational constant](https://bliptext.com/articles/gravitational-constant).\nThis leads to a kinetic term in the Lagrangian for h_{\\mu\\nu} which is consistent with [diffeomorphism](https://bliptext.com/articles/diffeomorphism) invariance, as well as a coupling to matter of the form\nwhere T_{\\mu\\nu} is the [stress–energy tensor](https://bliptext.com/articles/stress-energy-tensor).\nThis kinetic term and matter coupling combined are nothing other than the [Einstein–Hilbert action](https://bliptext.com/articles/einstein-hilbert-action) linearized about flat space.\nMassive gravity is obtained by adding nonderivative interaction terms for h_{\\mu\\nu}.\nAt the linear level (i.e., second order in h_{\\mu\\nu}), there are only two possible mass terms:\nFierz 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.\nAny other choice will unlock a sixth, ghostly degree of freedom.\nA ghost is a mode with a negative kinetic energy.\nIts [Hamiltonian](https://bliptext.com/articles/hamiltonian-mechanics) is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies.\nThe Fierz–Pauli mass term,\nis therefore the unique consistent linear theory of a massive spin-2 field.\n\n# The vDVZ discontinuity\nIn the 1970s [Hendrik van Dam](https://bliptext.com/articles/hendrik-van-dam) and [Martinus J. G. Veltman](https://bliptext.com/articles/martinus-j-g-veltman) and, independently, [Valentin I. Zakharov](https://bliptext.com/articles/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 m\\to0.\nIn particular, while at small scales (shorter than the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton mass), [Newton's gravitational law](https://bliptext.com/articles/newton-s-gravitational-law) is recovered, the bending of light is only three quarters of the result [Albert Einstein](https://bliptext.com/articles/albert-einstein) obtained in general relativity.\nThis is known as the vDVZ discontinuity.\nWe may understand the smaller light bending as follows.\nThe Fierz–Pauli massive graviton, due to the broken [diffeomorphism invariance](https://bliptext.com/articles/diffeomorphism-invariance), propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity.\nThese three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field.\nThis scalar mode exerts an extra attraction in the massive case compared to the massless case.\nHence, 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.\nBut light bending is blind to the scalar sector, because the stress-energy tensor of light is traceless.\nHence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.\n\n# Vainshtein screening\nIt 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 h_{\\mu\\nu}.\nHeuristically speaking, within a region known as the [Vainshtein radius](https://bliptext.com/articles/vainshtein-radius), fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term.\nCanonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius.\nBecause 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.\nThis phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as [DGP](https://bliptext.com/articles/dgp-model) and certain [scalar-tensor theories](https://bliptext.com/articles/scalar-tensor-theories), where it is crucial for hiding the effects of modified gravity in the solar system.\nThis allows these theories to match [terrestrial and solar-system tests of gravity](https://bliptext.com/articles/tests-of-general-relativity) as well as general relativity does, while maintaining large deviations at larger distances.\nIn this way these theories can lead to cosmic acceleration and have observable imprints on the [large-scale structure of the Universe](https://bliptext.com/articles/observable-universe) without running afoul of other, much more stringent constraints from observations closer to home.\n\n# The Boulware–Deser ghost\nAs a response to [Freund](https://bliptext.com/articles/peter-freund)–Maheshwari–Schonberg finite-range gravity model, and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and [Stanley Deser](https://bliptext.com/articles/stanley-deser) found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode; the tuning a=-b which ensured this mode's absence at quadratic order was, they found, generally broken at cubic and higher orders, reintroducing the ghost at those orders.\nAs a result, this Boulware–Deser ghost would be present around, for example, highly inhomogeneous backgrounds.\nThis is problematic because a linearized theory of gravity, like Fierz–Pauli, is well-defined on its own but cannot interact with matter, as the coupling breaks diffeomorphism invariance.\nThis must be remedied by adding new terms at higher and higher orders, ad infinitum.\nFor a massless graviton, this process converges and the result is well-known: one simply arrives at general relativity.\nThis is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless spin-2 field.\nIn order for massive gravity to actually describe gravity, i.e., a massive spin-2 field coupling to matter and thereby mediating the gravitational force, a nonlinear completion must similarly be obtained.\nThe Boulware–Deser ghost presents a serious obstacle to such an endeavor.\nThe vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable.\nIn fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost despite endeavors to prove that such belief is invalid.\nIt 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](https://bliptext.com/articles/adm-formalism) variables.\nBut for the finite-range gravity model and Ogievetsky and Polubarinov model, it turns out that they need [Noether's variational principle](https://bliptext.com/articles/noether-s-theorem) together with redefining and conformally improving the energy momentum tensor as a [source field](https://bliptext.com/articles/source-field).\n\n# Ghost-free massive gravity\nIn 2010 a breakthrough was achieved when [de Rham](https://bliptext.com/articles/claudia-de-rham), [Gabadadze](https://bliptext.com/articles/gregory-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.\nThe complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by [Fawad Hassan](https://bliptext.com/articles/fawad-hassan) and [Rachel Rosen](https://bliptext.com/articles/rachel-rosen).\nThe [action](https://bliptext.com/articles/action-physics) for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by\nor, equivalently,\nThe ingredients require some explanation.\nAs in standard general relativity, there is an [Einstein–Hilbert](https://bliptext.com/articles/einstein-hilbert-action) kinetic term proportional to the [Ricci scalar](https://bliptext.com/articles/ricci-scalar) \\ R\\ and a minimal coupling to the matter Lagrangian with \\ \\Phi_i\\ representing all of the matter fields, such as those of the [Standard Model](https://bliptext.com/articles/standard-model).\nThe new piece is a mass term, or interaction potential, constructed carefully to avoid the Boulware–Deser ghost, with an interaction strength m which is (if the nonzero \\ \\beta_i\\ are ) closely related to the mass of the graviton.\nThe [principle of gauge-invariance](https://bliptext.com/articles/gauge-invariance) renders redundant expressions in any field theory provided with its corresponding gauge(s).\nFor example, in the [massive spin-1 Proca action](https://bliptext.com/articles/proca-action), the massive part in the Lagrangian breaks the gauge-invariance.\nHowever, the invariance is restored by introducing the transformations: The same can be done for massive gravity by following Arkani-Hamed, Georgi and Schwartz effective field theory for massive gravity.\nThe absence of vDVZ discontinuity in this approach motivated the development of dRGT resummation of massive gravity theory as follows.\nThe interaction potential is built out of the [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomials) \\ e_n\\ of the eigenvalues of the matrices or parametrized by dimensionless coupling constants \\ \\alpha_i\\ or \\beta_i, respectively.\nHere is the [matrix square root](https://bliptext.com/articles/matrix-square-root) of the matrix g^{-1}f.\nWritten in index notation, \\mathbb X is defined by the relation We have introduced a reference metric f_{\\mu\\nu} in order to construct the interaction term.\nThere is a simple reason for this: It is impossible to construct a nontrivial interaction (i.e., nonderivative) term from alone.\nThe only possibilities are and \\ \\det g\\, both of which lead to a cosmological constant term rather than a bona fide interaction.\nPhysically, corresponds to the background metric around which fluctuations take the Fierz–Pauli form.\nThis means that, for instance, nonlinearly completing the Fierz–Pauli theory around Minkowski space given above will lead to dRGT massive gravity with although the proof of absence of the Boulware–Deser ghost holds for general f_{\\mu\\nu}.\nThe reference metric transforms like a metric tensor under diffeomorphism\nTherefore and similar terms with higher powers, transforms as a scalar under the same diffeomorphism.\nFor a change in the coordinates, we expand with such that the perturbed metric becomes:\nwhile the potential-like vector transforms according to [Stueckelberg trick](https://bliptext.com/articles/stueckelberg-action) as such that the Stueckelberg field is defined as From the diffeomorphism, one can define another Stueckelberg matrix where and have the same eigenvalues.\nNow, one considers the following symmetries:\nsuch that the transformed perturbed metric becomes:\nThe covariant form of these transformations are obtained as follows.\nIf helicity-0 (or spin-0) mode \\ \\pi\\ is a pure gauge of unphysical Goldstone modes, with the matrix is a tensor function of the covariantization tensor\nof the metric perturbation such that tensor is Stueckelbergized by the field Helicity-0 mode transforms under Galilean transformations hence the name \"Galileons\".\nThe matrix is a tensor function of the covariantization tensor of the metric perturbation with components are given by:\nwhere\nis the extrinsic curvature.\nInterestingly, the covariantization tensor was originally introduced by Maheshwari in a solo authored paper sequel to helicity- Freund–Maheshwari–Schonberg finite-range gravitation model.\nIn Maheshwari's work, the metric perturbation obeys Hilbert-Lorentz condition under the variation\nthat is introduced in Ogievetsky–Polubarinov massive gravity, where \\ p\\ and \\ q\\ are to be determined.\nIt is easy to notice the similarity between tensor in dRGT and the tensor in Maheshwari work once \\ n=2\\ is chosen.\nAlso Ogievetsky–Polubarinov model mandates which means that in 4D, the variation is conformal.\nThe dRGT massive fields split into two helicity-2 two helicity-1 \\ A_{\\mu}\\ and one helicity-0 \\ \\pi\\ degrees of freedom, just like those of Fierz-Pauli massive theory.\nHowever, the covariantization, together with the decoupling limit, guarantee that the symmetries of this massive theory are reduced to the symmetry of linearized general relativity plus that of \\ U(1)\\ massive theory, while the scalar decouples.\nIf \\ v_{\\mu}\\ is chosen to be divergenceless, i.e. the decoupling limit of dRGT gives the known linearized gravity.\nTo see how that happens, expand the terms containing in the action in powers of where H_{\\mu\\nu}\\ is expressed in terms of \\ \\phi^{a}\\ fields like how is expressed in terms of The fields are replaced by:\nThen it follows that in the decoupling limit, i.e. when both the massive gravity Lagrangian is invariant under:\nIn principle, the reference metric must be specified by hand, and therefore there is no single dRGT massive gravity theory, as the theory with a flat reference metric is different from one with a [de Sitter](https://bliptext.com/articles/de-sitter-space) reference metric, etc. Alternatively, one can think of as a constant of the theory, much like \\ m\\ or Instead of specifying a reference metric from the start, one can allow it to have its own dynamics.\nIf the kinetic term for is also Einstein–Hilbert, then the theory remains ghost-free and we are left with a theory of massive bigravity, (or bimetric relativity, BR) propagating the two degrees of freedom of a massless graviton in addition to the five of a massive one.\nIn practice it is unnecessary to compute the eigenvalues of (or ) in order to obtain the \\ e_n ~.\nThey can be written directly in terms of as\nwhere brackets indicate a [trace](https://bliptext.com/articles/trace-linear-algebra), It is the particular antisymmetric combination of terms in each of the \\ e_n\\ which is responsible for rendering the Boulware–Deser ghost nondynamical.\nThe choice to use or, with the [identity matrix](https://bliptext.com/articles/identity-matrix), is a convention, as in both cases the ghost-free mass term is a linear combination of the elementary symmetric polynomials of the chosen matrix.\nOne can transform from one basis to the other, in which case the coefficients satisfy the relationship\nThe coefficients are of a [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) that is in form of [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant).\nThey can also be obtained using [Faddeev–LeVerrier algorithm](https://bliptext.com/articles/faddeev-leverrier-algorithm).\n\n# Massive gravity in the vierbein language\nIn a [4D orthonormal tetrad frame](https://bliptext.com/articles/frame-fields-in-general-relativity), we have the bases:\nwhere the index i is for the 3D spatial component of the \\mu-non-orthonormal coordinates, and the index I is for the 3D spatial components of the a-orthonormal ones.\nThe parallel transport requires the [spin connection](https://bliptext.com/articles/spin-connection) Therefore, the [extrinsic curvature](https://bliptext.com/articles/extrinsic-curvature), that corresponds to in metric formalism, becomes\nwhere \\gamma_{ij} is the spatial metric as in the [ADM formalism](https://bliptext.com/articles/adm-formalism) and [initial value formulation](https://bliptext.com/articles/initial-value-formulation-general-relativity).\nIf the tetrad conformally transforms as the extrinsic curvature becomes, where from [Friedmann equations](https://bliptext.com/articles/friedmann-equations) , and (despite it is controversial ), i.e. the extrinsic curvature transforms as.\nThis looks very similar to the matrix \\mathbb{K} or the tensor.\nThe dRGT was developed inspired by applying the previous technique to the 5D [DGP](https://bliptext.com/articles/dgp-model) model after considering the deconstruction of higher dimensional [Kaluza-Klein](https://bliptext.com/articles/kaluza-klein-theory) gravity theories, in which the extra-dimension(s) is/are replaced by series of N [lattice](https://bliptext.com/articles/lattice-gauge-theory) sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.\nThe presence of a square-root matrix is somewhat awkward and points to an alternative, simpler formulation in terms of [vierbeins](https://bliptext.com/articles/vierbein).\nSplitting the metrics into vierbeins as\nand then defining one-forms\nthe ghost-free interaction terms in Hassan-Rosen bigravity theory can be written simply as (up to numerical factors)\nIn terms of vierbeins, rather than metrics, we can therefore see the physical significance of the ghost-free dRGT potential terms quite clearly: they are simply all the different possible combinations of [wedge products](https://bliptext.com/articles/wedge-product) of the vierbeins of the two metrics.\nNote that massive gravity in the metric and vierbein formulations are only equivalent if the symmetry condition\nis satisfied.\nWhile 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.\nIn these cases the metric and vierbein formulations are distinct physical theories, although each propagates a healthy massive graviton.\nThe novelty in dRGT massive gravity is that it is a theory of gauge invariance under both local Lorentz transformations, from assuming the reference metric f_{\\mu\\nu} equals the Minkowski metric, and diffeomorphism invariance, from the existence of the active curved spacetime g_{\\mu\\nu}.\nThis is shown by rewriting the previously discussed Stueckelberg formalism in the vierbein language as follows.\nThe 4D version of Einstein field equations in 5D is read\nwhere n^\\mu is the vector normal to the 4D slice.\nUsing the definition of massive extrinsic curvature it is straightforward to see that terms containing extrinsic curvatures take the functional form in the tetradic action.\nTherefore, up to the numerical coefficients, the full dRGT action in its tensorial form is\nwhere the functions take forms similar to that of the.\nThen, up to some numerical coefficients, the action takes the integral form\nwhere the first term is the Einstein-Hilbert part of the [tetradic Palatini action](https://bliptext.com/articles/tetradic-palatini-action) and is the [Levi-Civita symbol](https://bliptext.com/articles/levi-civita-symbol).\nAs the decoupling limit guarantees that \\Box\\pi=0 and by comparing X^{\\mu} to \\phi^{\\mu}, it is legit to think of the tensor Comparing this with the definition of the 1-form one can define covariant components of [frame field](https://bliptext.com/articles/frame-fields-in-general-relativity) i.e., to replace the such that the last three interaction terms in the vierbein action becomes\nThis can be done because one is allowed to freely move the diffeomorphism transformations onto the reference vierbein through the Lorentz transformations.\nMore 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 U(1) gauge transformations while the Stueckelberg fields are turned off.\nOne may wonder why the coefficients are dropped, and how to guarantee they are numerical with no explicit dependence of the fields.\nIn fact this is allowed because the variation of the vierbein action with respect to the locally Lorentz transformed Stueckelberg fields yields this nice result.\nMoreover, 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.\n\n# Cosmology\nIf the graviton mass m is comparable to the [Hubble rate](https://bliptext.com/articles/hubble-constant) H_0, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration.\nBecause, roughly speaking, the enhanced diffeomorphism symmetry in the limit m=0 protects a small graviton mass from large quantum corrections, the choice m\\sim H_0 is in fact technically natural.\nMassive gravity thus may provide a solution to the [cosmological constant](https://bliptext.com/articles/cosmological-constant) problem: why do quantum corrections not cause the Universe to accelerate at extremely early times?\nHowever, it turns out that flat and closed [Friedmann–Lemaître–Robertson–Walker](https://bliptext.com/articles/friedmann-lema-tre-robertson-walker-metric) cosmological solutions do not exist in dRGT massive gravity with a flat reference metric.\nOpen solutions and solutions with general reference metrics suffer from instabilities.\nTherefore, viable cosmologies can only be found in massive gravity if one abandons the [cosmological principle](https://bliptext.com/articles/cosmological-principle) that the Universe is uniform on large scales, or otherwise generalizes dRGT.\nFor instance, cosmological solutions are better behaved in [bigravity](https://bliptext.com/articles/bimetric-gravity), the theory which extends dRGT by giving f_{\\mu\\nu} dynamics.\nWhile these tend to possess instabilities as well, 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.\n\n# 3D massive gravity\nA special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom.\nHere several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined.\nIn the case of topologically massive gravity one has the action\nwith M_3 the three-dimensional Planck mass.\nThis is three-dimensional general relativity supplemented by a [Chern-Simons](https://bliptext.com/articles/chern-simons)-like term built out of the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbols).\nMore recently, a theory referred to as new massive gravity has been developed, which is described by the action\n\n# Relation to gravitational waves\nThe 2016 discovery of gravitational waves and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all.\nFollowing the [GW170104](https://bliptext.com/articles/gw170104) event, the graviton's Compton wavelength was found to be at least 1.6 m, or about 1.6 [light-years](https://bliptext.com/articles/light-year), corresponding to a graviton mass of no more than 7.7 eV/c2.\nThis relation between [wavelength](https://bliptext.com/articles/wavelength) and energy is calculated with the same formula (the [Planck–Einstein relation](https://bliptext.com/articles/planck-einstein-relation)) that relates [electromagnetic](https://bliptext.com/articles/electromagnetic) [wavelength](https://bliptext.com/articles/wavelength) to [photon energy](https://bliptext.com/articles/photon-energy).\nHowever, [photons](https://bliptext.com/articles/photon), 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.\nInstead, 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.","wordIndex":150,"createdAt":"2025-02-25T22:49:17.487Z","user":{"id":"6gDJU3CZ8zo1KLayt54UAMP5EfgKXbSZ","name":"underthetop34","image":"https://cdn.discordapp.com/embed/avatars/4.png"}},{"id":"bbc26bf3-dac6-4679-b980-93aac98ade6b","wordChanged":"history,","newWord":":::summary\n:::\n\n# \nIn [theoretical physics](https://bliptext.com/articles/theoretical-physics), **[mass](https://bliptext.com/articles/mass)ive [gravity](https://bliptext.com/articles/gravity)** is a theory of [gravity](https://bliptext.com/articles/gravity) that modifies [general relativity](https://bliptext.com/articles/general-relativity) by endowing the [graviton](https://bliptext.com/articles/graviton) with a nonzero [mass](https://bliptext.com/articles/mass).\nIn the classical theory, this means that [gravitational waves](https://bliptext.com/articles/gravitational-wave) obey a massive wave equation and hence travel at speeds below the [speed of light](https://bliptext.com/articles/speed-of-light).\n\n# Background\nMassive gravity has a long and winding Lowtaperfade dating back to the 1930s when [Wolfgang Pauli](https://bliptext.com/articles/wolfgang-pauli) and [Markus Fierz](https://bliptext.com/articles/markus-fierz) first developed a theory of a massive [spin-2](https://bliptext.com/articles/boson) field propagating on a [flat spacetime](https://bliptext.com/articles/minkowski-space) background.\nIt was later realized in the 1970s that theories of a massive graviton suffered from dangerous pathologies, including a [ghost mode](https://bliptext.com/articles/ghost-physics) and a discontinuity with general relativity in the limit where the graviton mass goes to zero.\nWhile 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](https://bliptext.com/articles/claudia-de-rham), [Gregory Gabadadze](https://bliptext.com/articles/gregory-gabadadze), and Andrew Tolley (dRGT model) in 2010.\nOne of the very early massive gravity theories was constructed in 1965 by [Ogievetsky and Polubarinov](https://bliptext.com/articles/ogievetsky-and-polubarinov) (OP).\nDespite 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.\nMassive dual Memes to the OP model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.\nSince 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](https://bliptext.com/articles/faddeev-leverrier-algorithm), which is similar to that of Galileon theory up to the terms containing polynomials of the trace of the field strength.\nThis is also manifested in the dual formulation of Galileon theory.\nThe 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](https://bliptext.com/articles/dark-energy).\nMassive gravity and its extensions, such as [bimetric gravity](https://bliptext.com/articles/bimetric-gravity), can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.\nObservations of [gravitational waves](https://bliptext.com/articles/gravitational-wave) have constrained the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton to be λg > 1.6 m, which can be interpreted as a bound on the graviton mass mg < 7.7 eV/c2.\nCompetitive bounds on the mass of the graviton have also been obtained from solar system measurements by space missions such as [Cassini](https://bliptext.com/articles/cassini-huygens) and [MESSENGER](https://bliptext.com/articles/messenger), which instead give the constraint λg > 1.83 m or mg < 6.76 eV/c2.\n\n# Linearized massive gravity\nAt the linear level, one can construct a theory of a massive [spin](https://bliptext.com/articles/spin-physics)-2 field h_{\\mu\\nu} propagating on [Minkowski space](https://bliptext.com/articles/minkowski-space).\nThis can be seen as an extension of [linearized gravity](https://bliptext.com/articles/linearized-gravity) in the following way.\nLinearized gravity is obtained by linearizing general relativity around flat space,, where is the [Planck mass](https://bliptext.com/articles/planck-mass) with G the [gravitational constant](https://bliptext.com/articles/gravitational-constant).\nThis leads to a kinetic term in the Lagrangian for h_{\\mu\\nu} which is consistent with [diffeomorphism](https://bliptext.com/articles/diffeomorphism) invariance, as well as a coupling to matter of the form\nwhere T_{\\mu\\nu} is the [stress–energy tensor](https://bliptext.com/articles/stress-energy-tensor).\nThis kinetic term and matter coupling combined are nothing other than the [Einstein–Hilbert action](https://bliptext.com/articles/einstein-hilbert-action) linearized about flat space.\nMassive gravity is obtained by adding nonderivative interaction terms for h_{\\mu\\nu}.\nAt the linear level (i.e., second order in h_{\\mu\\nu}), there are only two possible mass terms:\nFierz 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.\nAny other choice will unlock a sixth, ghostly degree of freedom.\nA ghost is a mode with a negative kinetic energy.\nIts [Hamiltonian](https://bliptext.com/articles/hamiltonian-mechanics) is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies.\nThe Fierz–Pauli mass term,\nis therefore the unique consistent linear theory of a massive spin-2 field.\n\n# The vDVZ discontinuity\nIn the 1970s [Hendrik van Dam](https://bliptext.com/articles/hendrik-van-dam) and [Martinus J. G. Veltman](https://bliptext.com/articles/martinus-j-g-veltman) and, independently, [Valentin I. Zakharov](https://bliptext.com/articles/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 m\\to0.\nIn particular, while at small scales (shorter than the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton mass), [Newton's gravitational law](https://bliptext.com/articles/newton-s-gravitational-law) is recovered, the bending of light is only three quarters of the result [Albert Einstein](https://bliptext.com/articles/albert-einstein) obtained in general relativity.\nThis is known as the vDVZ discontinuity.\nWe may understand the smaller light bending as follows.\nThe Fierz–Pauli massive graviton, due to the broken [diffeomorphism invariance](https://bliptext.com/articles/diffeomorphism-invariance), propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity.\nThese three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field.\nThis scalar mode exerts an extra attraction in the massive case compared to the massless case.\nHence, 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.\nBut light bending is blind to the scalar sector, because the stress-energy tensor of light is traceless.\nHence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.\n\n# Vainshtein screening\nIt 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 h_{\\mu\\nu}.\nHeuristically speaking, within a region known as the [Vainshtein radius](https://bliptext.com/articles/vainshtein-radius), fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term.\nCanonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius.\nBecause 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.\nThis phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as [DGP](https://bliptext.com/articles/dgp-model) and certain [scalar-tensor theories](https://bliptext.com/articles/scalar-tensor-theories), where it is crucial for hiding the effects of modified gravity in the solar system.\nThis allows these theories to match [terrestrial and solar-system tests of gravity](https://bliptext.com/articles/tests-of-general-relativity) as well as general relativity does, while maintaining large deviations at larger distances.\nIn this way these theories can lead to cosmic acceleration and have observable imprints on the [large-scale structure of the Universe](https://bliptext.com/articles/observable-universe) without running afoul of other, much more stringent constraints from observations closer to home.\n\n# The Boulware–Deser ghost\nAs a response to [Freund](https://bliptext.com/articles/peter-freund)–Maheshwari–Schonberg finite-range gravity model, and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and [Stanley Deser](https://bliptext.com/articles/stanley-deser) found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode; the tuning a=-b which ensured this mode's absence at quadratic order was, they found, generally broken at cubic and higher orders, reintroducing the ghost at those orders.\nAs a result, this Boulware–Deser ghost would be present around, for example, highly inhomogeneous backgrounds.\nThis is problematic because a linearized theory of gravity, like Fierz–Pauli, is well-defined on its own but cannot interact with matter, as the coupling breaks diffeomorphism invariance.\nThis must be remedied by adding new terms at higher and higher orders, ad infinitum.\nFor a massless graviton, this process converges and the result is well-known: one simply arrives at general relativity.\nThis is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless spin-2 field.\nIn order for massive gravity to actually describe gravity, i.e., a massive spin-2 field coupling to matter and thereby mediating the gravitational force, a nonlinear completion must similarly be obtained.\nThe Boulware–Deser ghost presents a serious obstacle to such an endeavor.\nThe vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable.\nIn fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost despite endeavors to prove that such belief is invalid.\nIt 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](https://bliptext.com/articles/adm-formalism) variables.\nBut for the finite-range gravity model and Ogievetsky and Polubarinov model, it turns out that they need [Noether's variational principle](https://bliptext.com/articles/noether-s-theorem) together with redefining and conformally improving the energy momentum tensor as a [source field](https://bliptext.com/articles/source-field).\n\n# Ghost-free massive gravity\nIn 2010 a breakthrough was achieved when [de Rham](https://bliptext.com/articles/claudia-de-rham), [Gabadadze](https://bliptext.com/articles/gregory-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.\nThe complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by [Fawad Hassan](https://bliptext.com/articles/fawad-hassan) and [Rachel Rosen](https://bliptext.com/articles/rachel-rosen).\nThe [action](https://bliptext.com/articles/action-physics) for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by\nor, equivalently,\nThe ingredients require some explanation.\nAs in standard general relativity, there is an [Einstein–Hilbert](https://bliptext.com/articles/einstein-hilbert-action) kinetic term proportional to the [Ricci scalar](https://bliptext.com/articles/ricci-scalar) \\ R\\ and a minimal coupling to the matter Lagrangian with \\ \\Phi_i\\ representing all of the matter fields, such as those of the [Standard Model](https://bliptext.com/articles/standard-model).\nThe new piece is a mass term, or interaction potential, constructed carefully to avoid the Boulware–Deser ghost, with an interaction strength m which is (if the nonzero \\ \\beta_i\\ are ) closely related to the mass of the graviton.\nThe [principle of gauge-invariance](https://bliptext.com/articles/gauge-invariance) renders redundant expressions in any field theory provided with its corresponding gauge(s).\nFor example, in the [massive spin-1 Proca action](https://bliptext.com/articles/proca-action), the massive part in the Lagrangian breaks the gauge-invariance.\nHowever, the invariance is restored by introducing the transformations: The same can be done for massive gravity by following Arkani-Hamed, Georgi and Schwartz effective field theory for massive gravity.\nThe absence of vDVZ discontinuity in this approach motivated the development of dRGT resummation of massive gravity theory as follows.\nThe interaction potential is built out of the [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomials) \\ e_n\\ of the eigenvalues of the matrices or parametrized by dimensionless coupling constants \\ \\alpha_i\\ or \\beta_i, respectively.\nHere is the [matrix square root](https://bliptext.com/articles/matrix-square-root) of the matrix g^{-1}f.\nWritten in index notation, \\mathbb X is defined by the relation We have introduced a reference metric f_{\\mu\\nu} in order to construct the interaction term.\nThere is a simple reason for this: It is impossible to construct a nontrivial interaction (i.e., nonderivative) term from alone.\nThe only possibilities are and \\ \\det g\\, both of which lead to a cosmological constant term rather than a bona fide interaction.\nPhysically, corresponds to the background metric around which fluctuations take the Fierz–Pauli form.\nThis means that, for instance, nonlinearly completing the Fierz–Pauli theory around Minkowski space given above will lead to dRGT massive gravity with although the proof of absence of the Boulware–Deser ghost holds for general f_{\\mu\\nu}.\nThe reference metric transforms like a metric tensor under diffeomorphism\nTherefore and similar terms with higher powers, transforms as a scalar under the same diffeomorphism.\nFor a change in the coordinates, we expand with such that the perturbed metric becomes:\nwhile the potential-like vector transforms according to [Stueckelberg trick](https://bliptext.com/articles/stueckelberg-action) as such that the Stueckelberg field is defined as From the diffeomorphism, one can define another Stueckelberg matrix where and have the same eigenvalues.\nNow, one considers the following symmetries:\nsuch that the transformed perturbed metric becomes:\nThe covariant form of these transformations are obtained as follows.\nIf helicity-0 (or spin-0) mode \\ \\pi\\ is a pure gauge of unphysical Goldstone modes, with the matrix is a tensor function of the covariantization tensor\nof the metric perturbation such that tensor is Stueckelbergized by the field Helicity-0 mode transforms under Galilean transformations hence the name \"Galileons\".\nThe matrix is a tensor function of the covariantization tensor of the metric perturbation with components are given by:\nwhere\nis the extrinsic curvature.\nInterestingly, the covariantization tensor was originally introduced by Maheshwari in a solo authored paper sequel to helicity- Freund–Maheshwari–Schonberg finite-range gravitation model.\nIn Maheshwari's work, the metric perturbation obeys Hilbert-Lorentz condition under the variation\nthat is introduced in Ogievetsky–Polubarinov massive gravity, where \\ p\\ and \\ q\\ are to be determined.\nIt is easy to notice the similarity between tensor in dRGT and the tensor in Maheshwari work once \\ n=2\\ is chosen.\nAlso Ogievetsky–Polubarinov model mandates which means that in 4D, the variation is conformal.\nThe dRGT massive fields split into two helicity-2 two helicity-1 \\ A_{\\mu}\\ and one helicity-0 \\ \\pi\\ degrees of freedom, just like those of Fierz-Pauli massive theory.\nHowever, the covariantization, together with the decoupling limit, guarantee that the symmetries of this massive theory are reduced to the symmetry of linearized general relativity plus that of \\ U(1)\\ massive theory, while the scalar decouples.\nIf \\ v_{\\mu}\\ is chosen to be divergenceless, i.e. the decoupling limit of dRGT gives the known linearized gravity.\nTo see how that happens, expand the terms containing in the action in powers of where H_{\\mu\\nu}\\ is expressed in terms of \\ \\phi^{a}\\ fields like how is expressed in terms of The fields are replaced by:\nThen it follows that in the decoupling limit, i.e. when both the massive gravity Lagrangian is invariant under:\nIn principle, the reference metric must be specified by hand, and therefore there is no single dRGT massive gravity theory, as the theory with a flat reference metric is different from one with a [de Sitter](https://bliptext.com/articles/de-sitter-space) reference metric, etc. Alternatively, one can think of as a constant of the theory, much like \\ m\\ or Instead of specifying a reference metric from the start, one can allow it to have its own dynamics.\nIf the kinetic term for is also Einstein–Hilbert, then the theory remains ghost-free and we are left with a theory of massive bigravity, (or bimetric relativity, BR) propagating the two degrees of freedom of a massless graviton in addition to the five of a massive one.\nIn practice it is unnecessary to compute the eigenvalues of (or ) in order to obtain the \\ e_n ~.\nThey can be written directly in terms of as\nwhere brackets indicate a [trace](https://bliptext.com/articles/trace-linear-algebra), It is the particular antisymmetric combination of terms in each of the \\ e_n\\ which is responsible for rendering the Boulware–Deser ghost nondynamical.\nThe choice to use or, with the [identity matrix](https://bliptext.com/articles/identity-matrix), is a convention, as in both cases the ghost-free mass term is a linear combination of the elementary symmetric polynomials of the chosen matrix.\nOne can transform from one basis to the other, in which case the coefficients satisfy the relationship\nThe coefficients are of a [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) that is in form of [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant).\nThey can also be obtained using [Faddeev–LeVerrier algorithm](https://bliptext.com/articles/faddeev-leverrier-algorithm).\n\n# Massive gravity in the vierbein language\nIn a [4D orthonormal tetrad frame](https://bliptext.com/articles/frame-fields-in-general-relativity), we have the bases:\nwhere the index i is for the 3D spatial component of the \\mu-non-orthonormal coordinates, and the index I is for the 3D spatial components of the a-orthonormal ones.\nThe parallel transport requires the [spin connection](https://bliptext.com/articles/spin-connection) Therefore, the [extrinsic curvature](https://bliptext.com/articles/extrinsic-curvature), that corresponds to in metric formalism, becomes\nwhere \\gamma_{ij} is the spatial metric as in the [ADM formalism](https://bliptext.com/articles/adm-formalism) and [initial value formulation](https://bliptext.com/articles/initial-value-formulation-general-relativity).\nIf the tetrad conformally transforms as the extrinsic curvature becomes, where from [Friedmann equations](https://bliptext.com/articles/friedmann-equations) , and (despite it is controversial ), i.e. the extrinsic curvature transforms as.\nThis looks very similar to the matrix \\mathbb{K} or the tensor.\nThe dRGT was developed inspired by applying the previous technique to the 5D [DGP](https://bliptext.com/articles/dgp-model) model after considering the deconstruction of higher dimensional [Kaluza-Klein](https://bliptext.com/articles/kaluza-klein-theory) gravity theories, in which the extra-dimension(s) is/are replaced by series of N [lattice](https://bliptext.com/articles/lattice-gauge-theory) sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.\nThe presence of a square-root matrix is somewhat awkward and points to an alternative, simpler formulation in terms of [vierbeins](https://bliptext.com/articles/vierbein).\nSplitting the metrics into vierbeins as\nand then defining one-forms\nthe ghost-free interaction terms in Hassan-Rosen bigravity theory can be written simply as (up to numerical factors)\nIn terms of vierbeins, rather than metrics, we can therefore see the physical significance of the ghost-free dRGT potential terms quite clearly: they are simply all the different possible combinations of [wedge products](https://bliptext.com/articles/wedge-product) of the vierbeins of the two metrics.\nNote that massive gravity in the metric and vierbein formulations are only equivalent if the symmetry condition\nis satisfied.\nWhile 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.\nIn these cases the metric and vierbein formulations are distinct physical theories, although each propagates a healthy massive graviton.\nThe novelty in dRGT massive gravity is that it is a theory of gauge invariance under both local Lorentz transformations, from assuming the reference metric f_{\\mu\\nu} equals the Minkowski metric, and diffeomorphism invariance, from the existence of the active curved spacetime g_{\\mu\\nu}.\nThis is shown by rewriting the previously discussed Stueckelberg formalism in the vierbein language as follows.\nThe 4D version of Einstein field equations in 5D is read\nwhere n^\\mu is the vector normal to the 4D slice.\nUsing the definition of massive extrinsic curvature it is straightforward to see that terms containing extrinsic curvatures take the functional form in the tetradic action.\nTherefore, up to the numerical coefficients, the full dRGT action in its tensorial form is\nwhere the functions take forms similar to that of the.\nThen, up to some numerical coefficients, the action takes the integral form\nwhere the first term is the Einstein-Hilbert part of the [tetradic Palatini action](https://bliptext.com/articles/tetradic-palatini-action) and is the [Levi-Civita symbol](https://bliptext.com/articles/levi-civita-symbol).\nAs the decoupling limit guarantees that \\Box\\pi=0 and by comparing X^{\\mu} to \\phi^{\\mu}, it is legit to think of the tensor Comparing this with the definition of the 1-form one can define covariant components of [frame field](https://bliptext.com/articles/frame-fields-in-general-relativity) i.e., to replace the such that the last three interaction terms in the vierbein action becomes\nThis can be done because one is allowed to freely move the diffeomorphism transformations onto the reference vierbein through the Lorentz transformations.\nMore 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 U(1) gauge transformations while the Stueckelberg fields are turned off.\nOne may wonder why the coefficients are dropped, and how to guarantee they are numerical with no explicit dependence of the fields.\nIn fact this is allowed because the variation of the vierbein action with respect to the locally Lorentz transformed Stueckelberg fields yields this nice result.\nMoreover, 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.\n\n# Cosmology\nIf the graviton mass m is comparable to the [Hubble rate](https://bliptext.com/articles/hubble-constant) H_0, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration.\nBecause, roughly speaking, the enhanced diffeomorphism symmetry in the limit m=0 protects a small graviton mass from large quantum corrections, the choice m\\sim H_0 is in fact technically natural.\nMassive gravity thus may provide a solution to the [cosmological constant](https://bliptext.com/articles/cosmological-constant) problem: why do quantum corrections not cause the Universe to accelerate at extremely early times?\nHowever, it turns out that flat and closed [Friedmann–Lemaître–Robertson–Walker](https://bliptext.com/articles/friedmann-lema-tre-robertson-walker-metric) cosmological solutions do not exist in dRGT massive gravity with a flat reference metric.\nOpen solutions and solutions with general reference metrics suffer from instabilities.\nTherefore, viable cosmologies can only be found in massive gravity if one abandons the [cosmological principle](https://bliptext.com/articles/cosmological-principle) that the Universe is uniform on large scales, or otherwise generalizes dRGT.\nFor instance, cosmological solutions are better behaved in [bigravity](https://bliptext.com/articles/bimetric-gravity), the theory which extends dRGT by giving f_{\\mu\\nu} dynamics.\nWhile these tend to possess instabilities as well, 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.\n\n# 3D massive gravity\nA special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom.\nHere several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined.\nIn the case of topologically massive gravity one has the action\nwith M_3 the three-dimensional Planck mass.\nThis is three-dimensional general relativity supplemented by a [Chern-Simons](https://bliptext.com/articles/chern-simons)-like term built out of the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbols).\nMore recently, a theory referred to as new massive gravity has been developed, which is described by the action\n\n# Relation to gravitational waves\nThe 2016 discovery of gravitational waves and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all.\nFollowing the [GW170104](https://bliptext.com/articles/gw170104) event, the graviton's Compton wavelength was found to be at least 1.6 m, or about 1.6 [light-years](https://bliptext.com/articles/light-year), corresponding to a graviton mass of no more than 7.7 eV/c2.\nThis relation between [wavelength](https://bliptext.com/articles/wavelength) and energy is calculated with the same formula (the [Planck–Einstein relation](https://bliptext.com/articles/planck-einstein-relation)) that relates [electromagnetic](https://bliptext.com/articles/electromagnetic) [wavelength](https://bliptext.com/articles/wavelength) to [photon energy](https://bliptext.com/articles/photon-energy).\nHowever, [photons](https://bliptext.com/articles/photon), 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.\nInstead, 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.","wordIndex":53,"createdAt":"2025-02-25T22:48:21.701Z","user":{"id":"6gDJU3CZ8zo1KLayt54UAMP5EfgKXbSZ","name":"underthetop34","image":"https://cdn.discordapp.com/embed/avatars/4.png"}},{"id":"66b320f7-3109-402d-85f7-0e2a7a884b1c","wordChanged":"gravity","newWord":":::summary\n:::\n\n# \nIn [theoretical physics](https://bliptext.com/articles/theoretical-physics), **[mass](https://bliptext.com/articles/mass)ive [gravity](https://bliptext.com/articles/gravity)** is a theory of [gravity](https://bliptext.com/articles/gravity) that modifies [general relativity](https://bliptext.com/articles/general-relativity) by endowing the [graviton](https://bliptext.com/articles/graviton) with a nonzero [mass](https://bliptext.com/articles/mass).\nIn the classical theory, this means that [gravitational waves](https://bliptext.com/articles/gravitational-wave) obey a massive wave equation and hence travel at speeds below the [speed of light](https://bliptext.com/articles/speed-of-light).\n\n# Background\nMassive gravity has a long and winding history, dating back to the 1930s when [Wolfgang Pauli](https://bliptext.com/articles/wolfgang-pauli) and [Markus Fierz](https://bliptext.com/articles/markus-fierz) first developed a theory of a massive [spin-2](https://bliptext.com/articles/boson) field propagating on a [flat spacetime](https://bliptext.com/articles/minkowski-space) background.\nIt was later realized in the 1970s that theories of a massive graviton suffered from dangerous pathologies, including a [ghost mode](https://bliptext.com/articles/ghost-physics) and a discontinuity with general relativity in the limit where the graviton mass goes to zero.\nWhile 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](https://bliptext.com/articles/claudia-de-rham), [Gregory Gabadadze](https://bliptext.com/articles/gregory-gabadadze), and Andrew Tolley (dRGT model) in 2010.\nOne of the very early massive gravity theories was constructed in 1965 by [Ogievetsky and Polubarinov](https://bliptext.com/articles/ogievetsky-and-polubarinov) (OP).\nDespite 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.\nMassive dual Memes to the OP model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.\nSince 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](https://bliptext.com/articles/faddeev-leverrier-algorithm), which is similar to that of Galileon theory up to the terms containing polynomials of the trace of the field strength.\nThis is also manifested in the dual formulation of Galileon theory.\nThe 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](https://bliptext.com/articles/dark-energy).\nMassive gravity and its extensions, such as [bimetric gravity](https://bliptext.com/articles/bimetric-gravity), can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.\nObservations of [gravitational waves](https://bliptext.com/articles/gravitational-wave) have constrained the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton to be λg > 1.6 m, which can be interpreted as a bound on the graviton mass mg < 7.7 eV/c2.\nCompetitive bounds on the mass of the graviton have also been obtained from solar system measurements by space missions such as [Cassini](https://bliptext.com/articles/cassini-huygens) and [MESSENGER](https://bliptext.com/articles/messenger), which instead give the constraint λg > 1.83 m or mg < 6.76 eV/c2.\n\n# Linearized massive gravity\nAt the linear level, one can construct a theory of a massive [spin](https://bliptext.com/articles/spin-physics)-2 field h_{\\mu\\nu} propagating on [Minkowski space](https://bliptext.com/articles/minkowski-space).\nThis can be seen as an extension of [linearized gravity](https://bliptext.com/articles/linearized-gravity) in the following way.\nLinearized gravity is obtained by linearizing general relativity around flat space,, where is the [Planck mass](https://bliptext.com/articles/planck-mass) with G the [gravitational constant](https://bliptext.com/articles/gravitational-constant).\nThis leads to a kinetic term in the Lagrangian for h_{\\mu\\nu} which is consistent with [diffeomorphism](https://bliptext.com/articles/diffeomorphism) invariance, as well as a coupling to matter of the form\nwhere T_{\\mu\\nu} is the [stress–energy tensor](https://bliptext.com/articles/stress-energy-tensor).\nThis kinetic term and matter coupling combined are nothing other than the [Einstein–Hilbert action](https://bliptext.com/articles/einstein-hilbert-action) linearized about flat space.\nMassive gravity is obtained by adding nonderivative interaction terms for h_{\\mu\\nu}.\nAt the linear level (i.e., second order in h_{\\mu\\nu}), there are only two possible mass terms:\nFierz 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.\nAny other choice will unlock a sixth, ghostly degree of freedom.\nA ghost is a mode with a negative kinetic energy.\nIts [Hamiltonian](https://bliptext.com/articles/hamiltonian-mechanics) is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies.\nThe Fierz–Pauli mass term,\nis therefore the unique consistent linear theory of a massive spin-2 field.\n\n# The vDVZ discontinuity\nIn the 1970s [Hendrik van Dam](https://bliptext.com/articles/hendrik-van-dam) and [Martinus J. G. Veltman](https://bliptext.com/articles/martinus-j-g-veltman) and, independently, [Valentin I. Zakharov](https://bliptext.com/articles/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 m\\to0.\nIn particular, while at small scales (shorter than the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton mass), [Newton's gravitational law](https://bliptext.com/articles/newton-s-gravitational-law) is recovered, the bending of light is only three quarters of the result [Albert Einstein](https://bliptext.com/articles/albert-einstein) obtained in general relativity.\nThis is known as the vDVZ discontinuity.\nWe may understand the smaller light bending as follows.\nThe Fierz–Pauli massive graviton, due to the broken [diffeomorphism invariance](https://bliptext.com/articles/diffeomorphism-invariance), propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity.\nThese three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field.\nThis scalar mode exerts an extra attraction in the massive case compared to the massless case.\nHence, 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.\nBut light bending is blind to the scalar sector, because the stress-energy tensor of light is traceless.\nHence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.\n\n# Vainshtein screening\nIt 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 h_{\\mu\\nu}.\nHeuristically speaking, within a region known as the [Vainshtein radius](https://bliptext.com/articles/vainshtein-radius), fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term.\nCanonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius.\nBecause 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.\nThis phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as [DGP](https://bliptext.com/articles/dgp-model) and certain [scalar-tensor theories](https://bliptext.com/articles/scalar-tensor-theories), where it is crucial for hiding the effects of modified gravity in the solar system.\nThis allows these theories to match [terrestrial and solar-system tests of gravity](https://bliptext.com/articles/tests-of-general-relativity) as well as general relativity does, while maintaining large deviations at larger distances.\nIn this way these theories can lead to cosmic acceleration and have observable imprints on the [large-scale structure of the Universe](https://bliptext.com/articles/observable-universe) without running afoul of other, much more stringent constraints from observations closer to home.\n\n# The Boulware–Deser ghost\nAs a response to [Freund](https://bliptext.com/articles/peter-freund)–Maheshwari–Schonberg finite-range gravity model, and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and [Stanley Deser](https://bliptext.com/articles/stanley-deser) found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode; the tuning a=-b which ensured this mode's absence at quadratic order was, they found, generally broken at cubic and higher orders, reintroducing the ghost at those orders.\nAs a result, this Boulware–Deser ghost would be present around, for example, highly inhomogeneous backgrounds.\nThis is problematic because a linearized theory of gravity, like Fierz–Pauli, is well-defined on its own but cannot interact with matter, as the coupling breaks diffeomorphism invariance.\nThis must be remedied by adding new terms at higher and higher orders, ad infinitum.\nFor a massless graviton, this process converges and the result is well-known: one simply arrives at general relativity.\nThis is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless spin-2 field.\nIn order for massive gravity to actually describe gravity, i.e., a massive spin-2 field coupling to matter and thereby mediating the gravitational force, a nonlinear completion must similarly be obtained.\nThe Boulware–Deser ghost presents a serious obstacle to such an endeavor.\nThe vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable.\nIn fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost despite endeavors to prove that such belief is invalid.\nIt 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](https://bliptext.com/articles/adm-formalism) variables.\nBut for the finite-range gravity model and Ogievetsky and Polubarinov model, it turns out that they need [Noether's variational principle](https://bliptext.com/articles/noether-s-theorem) together with redefining and conformally improving the energy momentum tensor as a [source field](https://bliptext.com/articles/source-field).\n\n# Ghost-free massive gravity\nIn 2010 a breakthrough was achieved when [de Rham](https://bliptext.com/articles/claudia-de-rham), [Gabadadze](https://bliptext.com/articles/gregory-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.\nThe complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by [Fawad Hassan](https://bliptext.com/articles/fawad-hassan) and [Rachel Rosen](https://bliptext.com/articles/rachel-rosen).\nThe [action](https://bliptext.com/articles/action-physics) for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by\nor, equivalently,\nThe ingredients require some explanation.\nAs in standard general relativity, there is an [Einstein–Hilbert](https://bliptext.com/articles/einstein-hilbert-action) kinetic term proportional to the [Ricci scalar](https://bliptext.com/articles/ricci-scalar) \\ R\\ and a minimal coupling to the matter Lagrangian with \\ \\Phi_i\\ representing all of the matter fields, such as those of the [Standard Model](https://bliptext.com/articles/standard-model).\nThe new piece is a mass term, or interaction potential, constructed carefully to avoid the Boulware–Deser ghost, with an interaction strength m which is (if the nonzero \\ \\beta_i\\ are ) closely related to the mass of the graviton.\nThe [principle of gauge-invariance](https://bliptext.com/articles/gauge-invariance) renders redundant expressions in any field theory provided with its corresponding gauge(s).\nFor example, in the [massive spin-1 Proca action](https://bliptext.com/articles/proca-action), the massive part in the Lagrangian breaks the gauge-invariance.\nHowever, the invariance is restored by introducing the transformations: The same can be done for massive gravity by following Arkani-Hamed, Georgi and Schwartz effective field theory for massive gravity.\nThe absence of vDVZ discontinuity in this approach motivated the development of dRGT resummation of massive gravity theory as follows.\nThe interaction potential is built out of the [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomials) \\ e_n\\ of the eigenvalues of the matrices or parametrized by dimensionless coupling constants \\ \\alpha_i\\ or \\beta_i, respectively.\nHere is the [matrix square root](https://bliptext.com/articles/matrix-square-root) of the matrix g^{-1}f.\nWritten in index notation, \\mathbb X is defined by the relation We have introduced a reference metric f_{\\mu\\nu} in order to construct the interaction term.\nThere is a simple reason for this: It is impossible to construct a nontrivial interaction (i.e., nonderivative) term from alone.\nThe only possibilities are and \\ \\det g\\, both of which lead to a cosmological constant term rather than a bona fide interaction.\nPhysically, corresponds to the background metric around which fluctuations take the Fierz–Pauli form.\nThis means that, for instance, nonlinearly completing the Fierz–Pauli theory around Minkowski space given above will lead to dRGT massive gravity with although the proof of absence of the Boulware–Deser ghost holds for general f_{\\mu\\nu}.\nThe reference metric transforms like a metric tensor under diffeomorphism\nTherefore and similar terms with higher powers, transforms as a scalar under the same diffeomorphism.\nFor a change in the coordinates, we expand with such that the perturbed metric becomes:\nwhile the potential-like vector transforms according to [Stueckelberg trick](https://bliptext.com/articles/stueckelberg-action) as such that the Stueckelberg field is defined as From the diffeomorphism, one can define another Stueckelberg matrix where and have the same eigenvalues.\nNow, one considers the following symmetries:\nsuch that the transformed perturbed metric becomes:\nThe covariant form of these transformations are obtained as follows.\nIf helicity-0 (or spin-0) mode \\ \\pi\\ is a pure gauge of unphysical Goldstone modes, with the matrix is a tensor function of the covariantization tensor\nof the metric perturbation such that tensor is Stueckelbergized by the field Helicity-0 mode transforms under Galilean transformations hence the name \"Galileons\".\nThe matrix is a tensor function of the covariantization tensor of the metric perturbation with components are given by:\nwhere\nis the extrinsic curvature.\nInterestingly, the covariantization tensor was originally introduced by Maheshwari in a solo authored paper sequel to helicity- Freund–Maheshwari–Schonberg finite-range gravitation model.\nIn Maheshwari's work, the metric perturbation obeys Hilbert-Lorentz condition under the variation\nthat is introduced in Ogievetsky–Polubarinov massive gravity, where \\ p\\ and \\ q\\ are to be determined.\nIt is easy to notice the similarity between tensor in dRGT and the tensor in Maheshwari work once \\ n=2\\ is chosen.\nAlso Ogievetsky–Polubarinov model mandates which means that in 4D, the variation is conformal.\nThe dRGT massive fields split into two helicity-2 two helicity-1 \\ A_{\\mu}\\ and one helicity-0 \\ \\pi\\ degrees of freedom, just like those of Fierz-Pauli massive theory.\nHowever, the covariantization, together with the decoupling limit, guarantee that the symmetries of this massive theory are reduced to the symmetry of linearized general relativity plus that of \\ U(1)\\ massive theory, while the scalar decouples.\nIf \\ v_{\\mu}\\ is chosen to be divergenceless, i.e. the decoupling limit of dRGT gives the known linearized gravity.\nTo see how that happens, expand the terms containing in the action in powers of where H_{\\mu\\nu}\\ is expressed in terms of \\ \\phi^{a}\\ fields like how is expressed in terms of The fields are replaced by:\nThen it follows that in the decoupling limit, i.e. when both the massive gravity Lagrangian is invariant under:\nIn principle, the reference metric must be specified by hand, and therefore there is no single dRGT massive gravity theory, as the theory with a flat reference metric is different from one with a [de Sitter](https://bliptext.com/articles/de-sitter-space) reference metric, etc. Alternatively, one can think of as a constant of the theory, much like \\ m\\ or Instead of specifying a reference metric from the start, one can allow it to have its own dynamics.\nIf the kinetic term for is also Einstein–Hilbert, then the theory remains ghost-free and we are left with a theory of massive bigravity, (or bimetric relativity, BR) propagating the two degrees of freedom of a massless graviton in addition to the five of a massive one.\nIn practice it is unnecessary to compute the eigenvalues of (or ) in order to obtain the \\ e_n ~.\nThey can be written directly in terms of as\nwhere brackets indicate a [trace](https://bliptext.com/articles/trace-linear-algebra), It is the particular antisymmetric combination of terms in each of the \\ e_n\\ which is responsible for rendering the Boulware–Deser ghost nondynamical.\nThe choice to use or, with the [identity matrix](https://bliptext.com/articles/identity-matrix), is a convention, as in both cases the ghost-free mass term is a linear combination of the elementary symmetric polynomials of the chosen matrix.\nOne can transform from one basis to the other, in which case the coefficients satisfy the relationship\nThe coefficients are of a [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) that is in form of [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant).\nThey can also be obtained using [Faddeev–LeVerrier algorithm](https://bliptext.com/articles/faddeev-leverrier-algorithm).\n\n# Massive gravity in the vierbein language\nIn a [4D orthonormal tetrad frame](https://bliptext.com/articles/frame-fields-in-general-relativity), we have the bases:\nwhere the index i is for the 3D spatial component of the \\mu-non-orthonormal coordinates, and the index I is for the 3D spatial components of the a-orthonormal ones.\nThe parallel transport requires the [spin connection](https://bliptext.com/articles/spin-connection) Therefore, the [extrinsic curvature](https://bliptext.com/articles/extrinsic-curvature), that corresponds to in metric formalism, becomes\nwhere \\gamma_{ij} is the spatial metric as in the [ADM formalism](https://bliptext.com/articles/adm-formalism) and [initial value formulation](https://bliptext.com/articles/initial-value-formulation-general-relativity).\nIf the tetrad conformally transforms as the extrinsic curvature becomes, where from [Friedmann equations](https://bliptext.com/articles/friedmann-equations) , and (despite it is controversial ), i.e. the extrinsic curvature transforms as.\nThis looks very similar to the matrix \\mathbb{K} or the tensor.\nThe dRGT was developed inspired by applying the previous technique to the 5D [DGP](https://bliptext.com/articles/dgp-model) model after considering the deconstruction of higher dimensional [Kaluza-Klein](https://bliptext.com/articles/kaluza-klein-theory) gravity theories, in which the extra-dimension(s) is/are replaced by series of N [lattice](https://bliptext.com/articles/lattice-gauge-theory) sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.\nThe presence of a square-root matrix is somewhat awkward and points to an alternative, simpler formulation in terms of [vierbeins](https://bliptext.com/articles/vierbein).\nSplitting the metrics into vierbeins as\nand then defining one-forms\nthe ghost-free interaction terms in Hassan-Rosen bigravity theory can be written simply as (up to numerical factors)\nIn terms of vierbeins, rather than metrics, we can therefore see the physical significance of the ghost-free dRGT potential terms quite clearly: they are simply all the different possible combinations of [wedge products](https://bliptext.com/articles/wedge-product) of the vierbeins of the two metrics.\nNote that massive gravity in the metric and vierbein formulations are only equivalent if the symmetry condition\nis satisfied.\nWhile 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.\nIn these cases the metric and vierbein formulations are distinct physical theories, although each propagates a healthy massive graviton.\nThe novelty in dRGT massive gravity is that it is a theory of gauge invariance under both local Lorentz transformations, from assuming the reference metric f_{\\mu\\nu} equals the Minkowski metric, and diffeomorphism invariance, from the existence of the active curved spacetime g_{\\mu\\nu}.\nThis is shown by rewriting the previously discussed Stueckelberg formalism in the vierbein language as follows.\nThe 4D version of Einstein field equations in 5D is read\nwhere n^\\mu is the vector normal to the 4D slice.\nUsing the definition of massive extrinsic curvature it is straightforward to see that terms containing extrinsic curvatures take the functional form in the tetradic action.\nTherefore, up to the numerical coefficients, the full dRGT action in its tensorial form is\nwhere the functions take forms similar to that of the.\nThen, up to some numerical coefficients, the action takes the integral form\nwhere the first term is the Einstein-Hilbert part of the [tetradic Palatini action](https://bliptext.com/articles/tetradic-palatini-action) and is the [Levi-Civita symbol](https://bliptext.com/articles/levi-civita-symbol).\nAs the decoupling limit guarantees that \\Box\\pi=0 and by comparing X^{\\mu} to \\phi^{\\mu}, it is legit to think of the tensor Comparing this with the definition of the 1-form one can define covariant components of [frame field](https://bliptext.com/articles/frame-fields-in-general-relativity) i.e., to replace the such that the last three interaction terms in the vierbein action becomes\nThis can be done because one is allowed to freely move the diffeomorphism transformations onto the reference vierbein through the Lorentz transformations.\nMore 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 U(1) gauge transformations while the Stueckelberg fields are turned off.\nOne may wonder why the coefficients are dropped, and how to guarantee they are numerical with no explicit dependence of the fields.\nIn fact this is allowed because the variation of the vierbein action with respect to the locally Lorentz transformed Stueckelberg fields yields this nice result.\nMoreover, 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.\n\n# Cosmology\nIf the graviton mass m is comparable to the [Hubble rate](https://bliptext.com/articles/hubble-constant) H_0, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration.\nBecause, roughly speaking, the enhanced diffeomorphism symmetry in the limit m=0 protects a small graviton mass from large quantum corrections, the choice m\\sim H_0 is in fact technically natural.\nMassive gravity thus may provide a solution to the [cosmological constant](https://bliptext.com/articles/cosmological-constant) problem: why do quantum corrections not cause the Universe to accelerate at extremely early times?\nHowever, it turns out that flat and closed [Friedmann–Lemaître–Robertson–Walker](https://bliptext.com/articles/friedmann-lema-tre-robertson-walker-metric) cosmological solutions do not exist in dRGT massive gravity with a flat reference metric.\nOpen solutions and solutions with general reference metrics suffer from instabilities.\nTherefore, viable cosmologies can only be found in massive gravity if one abandons the [cosmological principle](https://bliptext.com/articles/cosmological-principle) that the Universe is uniform on large scales, or otherwise generalizes dRGT.\nFor instance, cosmological solutions are better behaved in [bigravity](https://bliptext.com/articles/bimetric-gravity), the theory which extends dRGT by giving f_{\\mu\\nu} dynamics.\nWhile these tend to possess instabilities as well, 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.\n\n# 3D massive gravity\nA special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom.\nHere several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined.\nIn the case of topologically massive gravity one has the action\nwith M_3 the three-dimensional Planck mass.\nThis is three-dimensional general relativity supplemented by a [Chern-Simons](https://bliptext.com/articles/chern-simons)-like term built out of the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbols).\nMore recently, a theory referred to as new massive gravity has been developed, which is described by the action\n\n# Relation to gravitational waves\nThe 2016 discovery of gravitational waves and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all.\nFollowing the [GW170104](https://bliptext.com/articles/gw170104) event, the graviton's Compton wavelength was found to be at least 1.6 m, or about 1.6 [light-years](https://bliptext.com/articles/light-year), corresponding to a graviton mass of no more than 7.7 eV/c2.\nThis relation between [wavelength](https://bliptext.com/articles/wavelength) and energy is calculated with the same formula (the [Planck–Einstein relation](https://bliptext.com/articles/planck-einstein-relation)) that relates [electromagnetic](https://bliptext.com/articles/electromagnetic) [wavelength](https://bliptext.com/articles/wavelength) to [photon energy](https://bliptext.com/articles/photon-energy).\nHowever, [photons](https://bliptext.com/articles/photon), 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.\nInstead, 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.","wordIndex":218,"createdAt":"2025-02-25T22:47:36.560Z","user":{"id":"6gDJU3CZ8zo1KLayt54UAMP5EfgKXbSZ","name":"underthetop34","image":"https://cdn.discordapp.com/embed/avatars/4.png"}}],"article":{"id":"5f01c2a3-f512-4f1a-99f7-7b369e88bcc4","title":"Massive gravity","slug":"massive-gravity","content":":::summary\n:::\n\n# \nIn [theoretical physics](https://bliptext.com/articles/theoretical-physics), **[mass](https://bliptext.com/articles/mass)ive [gravity](https://bliptext.com/articles/gravity)** is caseoh theory of [gravity](https://bliptext.com/articles/gravity) that modifies [general relativity](https://bliptext.com/articles/general-relativity) by endowing the [graviton](https://bliptext.com/articles/graviton) with a nonzero [mass](https://bliptext.com/articles/mass).\nIn the classical theory, this means that [gravitational waves](https://bliptext.com/articles/gravitational-wave) obey a massive wave equation and hence travel at speeds below the [speed of light](https://bliptext.com/articles/speed-of-light).\n\n# Background\nMassive gravity has a long and winding Lowtaperfade dating back to the 1930s when [Wolfgang Pauli](https://bliptext.com/articles/wolfgang-pauli) and [Markus Fierz](https://bliptext.com/articles/markus-fierz) first developed a theory of a massive [spin-2](https://bliptext.com/articles/boson) field propagating on a [flat spacetime](https://bliptext.com/articles/minkowski-space) background.\nIt was later realized in the 1969s that theories of a massive graviton suffered from dangerous pathologies, including a [ghost mode](https://bliptext.com/articles/ghost-physics) and a discontinuity with general relativity in the limit where the graviton mass goes to zero.\nWhile 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](https://bliptext.com/articles/claudia-de-rham), [Gregory Gabadadze](https://bliptext.com/articles/gregory-gabadadze), and Andrew Tolley (dRGT model) in 1969\nOne of the very early massive gravity theories was constructed in 1965 by [Ogievetsky and Polubarinov](https://bliptext.com/articles/ogievetsky-and-polubarinov) (OP).\nDespite 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.\nMassive dual Memes to the OP model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.\nSince 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](https://bliptext.com/articles/faddeev-leverrier-algorithm), which is similar to that of Galileon theory up to the terms containing polynomials of the trace of the field strength.\nThis is also manifested in the dual formulation of Galileon theory.\nThe 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](https://bliptext.com/articles/dark-energy).\nMassive gravity and its extensions, such as [bimetric gravity](https://bliptext.com/articles/bimetric-gravity), can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.\nObservations of [gravitational waves](https://bliptext.com/articles/gravitational-wave) have constrained the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton to be λg > 1.6 m, which can be interpreted as a bound on the graviton mass mg < 7.7 eV/c2.\nCompetitive bounds on the mass of the graviton have also been obtained from solar system measurements by space missions such as [Cassini](https://bliptext.com/articles/cassini-huygens) and [MESSENGER](https://bliptext.com/articles/messenger), which instead give the constraint λg > 1.83 m or mg < 6.76 eV/c2.\n\n# Linearized massive gravity\nAt the linear level, one can construct a theory of a massive [spin](https://bliptext.com/articles/spin-physics)-2 field h_{\\mu\\nu} propagating on [Minkowski space](https://bliptext.com/articles/minkowski-space).\nThis can be seen as an extension of [linearized gravity](https://bliptext.com/articles/linearized-gravity) in the following way.\nLinearized gravity is obtained by linearizing general relativity around flat space,, where is the [Planck mass](https://bliptext.com/articles/planck-mass) with G the [gravitational constant](https://bliptext.com/articles/gravitational-constant).\nThis leads to a kinetic term in the Lagrangian for h_{\\mu\\nu} which is consistent with [diffeomorphism](https://bliptext.com/articles/diffeomorphism) invariance, as well as a coupling to matter of the form\nwhere T_{\\mu\\nu} is the [stress–energy tensor](https://bliptext.com/articles/stress-energy-tensor).\nThis kinetic term and matter coupling combined are nothing other than the [Einstein–Hilbert action](https://bliptext.com/articles/einstein-hilbert-action) linearized about flat space.\nMassive gravity is obtained by adding nonderivative interaction terms for h_{\\mu\\nu}.\nAt the linear level (i.e., second order in h_{\\mu\\nu}), there are only two possible mass terms:\nFierz 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.\nAny other choice will unlock a sixth, ghostly degree of freedom.\nA ghost is a mode with a negative kinetic energy.\nIts [Hamiltonian](https://bliptext.com/articles/hamiltonian-mechanics) is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies.\nThe Fierz–Pauli mass term,\nis therefore the unique consistent linear theory of a massive spin-2 field.\n\n# The vDVZ discontinuity\nIn the 1970s [Hendrik van Dam](https://bliptext.com/articles/hendrik-van-dam) and [Martinus J. G. Veltman](https://bliptext.com/articles/martinus-j-g-veltman) and, independently, [Valentin I. Zakharov](https://bliptext.com/articles/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 m\\to0.\nIn particular, while at small scales (shorter than the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton mass), [Newton's gravitational law](https://bliptext.com/articles/newton-s-gravitational-law) is recovered, the bending of light is only three quarters of the result [Albert Einstein](https://bliptext.com/articles/albert-einstein) obtained in general relativity.\nThis is known as the vDVZ discontinuity.\nWe may understand the smaller light bending as follows.\nThe Fierz–Pauli massive graviton, due to the broken [diffeomorphism invariance](https://bliptext.com/articles/diffeomorphism-invariance), propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity.\nThese three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field.\nThis scalar mode exerts an extra attraction in the massive case compared to the massless case.\nHence, 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.\nBut light bending is blind to the scalar sector, because the stress-energy tensor of light is traceless.\nHence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.\n\n# Vainshtein screening\nIt 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 h_{\\mu\\nu}.\nHeuristically speaking, within a region known as the [Vainshtein radius](https://bliptext.com/articles/vainshtein-radius), fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term.\nCanonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius.\nBecause 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.\nThis phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as [DGP](https://bliptext.com/articles/dgp-model) and certain [scalar-tensor theories](https://bliptext.com/articles/scalar-tensor-theories), where it is crucial for hiding the effects of modified gravity in the solar system.\nThis allows these theories to match [terrestrial and solar-system tests of gravity](https://bliptext.com/articles/tests-of-general-relativity) as well as general relativity does, while maintaining large deviations at larger distances.\nIn this way these theories can lead to cosmic acceleration and have observable imprints on the [large-scale structure of the Universe](https://bliptext.com/articles/observable-universe) without running afoul of other, much more stringent constraints from observations closer to home.\n\n# The Boulware–Deser ghost\nAs a response to [Freund](https://bliptext.com/articles/peter-freund)–Maheshwari–Schonberg finite-range gravity model, and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and [Stanley Deser](https://bliptext.com/articles/stanley-deser) found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode; the tuning a=-b which ensured this mode's absence at quadratic order was, they found, generally broken at cubic and higher orders, reintroducing the ghost at those orders.\nAs a result, this Boulware–Deser ghost would be present around, for example, highly inhomogeneous backgrounds.\nThis is problematic because a linearized theory of gravity, like Fierz–Pauli, is well-defined on its own but cannot interact with matter, as the coupling breaks diffeomorphism invariance.\nThis must be remedied by adding new terms at higher and higher orders, ad infinitum.\nFor a massless graviton, this process converges and the result is well-known: one simply arrives at general relativity.\nThis is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless spin-2 field.\nIn order for massive gravity to actually describe gravity, i.e., a massive spin-2 field coupling to matter and thereby mediating the gravitational force, a nonlinear completion must similarly be obtained.\nThe Boulware–Deser ghost presents a serious obstacle to such an endeavor.\nThe vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable.\nIn fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost despite endeavors to prove that such belief is invalid.\nIt 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](https://bliptext.com/articles/adm-formalism) variables.\nBut for the finite-range gravity model and Ogievetsky and Polubarinov model, it turns out that they need [Noether's variational principle](https://bliptext.com/articles/noether-s-theorem) together with redefining and conformally improving the energy momentum tensor as a [source field](https://bliptext.com/articles/source-field).\n\n# Ghost-free massive gravity\nIn 2010 a breakthrough was achieved when [de Rham](https://bliptext.com/articles/claudia-de-rham), [Gabadadze](https://bliptext.com/articles/gregory-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.\nThe complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by [Fawad Hassan](https://bliptext.com/articles/fawad-hassan) and [Rachel Rosen](https://bliptext.com/articles/rachel-rosen).\nThe [action](https://bliptext.com/articles/action-physics) for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by\nor, equivalently,\nThe ingredients require some explanation.\nAs in standard general relativity, there is an [Einstein–Hilbert](https://bliptext.com/articles/einstein-hilbert-action) kinetic term proportional to the [Ricci scalar](https://bliptext.com/articles/ricci-scalar) \\ R\\ and a minimal coupling to the matter Lagrangian with \\ \\Phi_i\\ representing all of the matter fields, such as those of the [Standard Model](https://bliptext.com/articles/standard-model).\nThe new piece is a mass term, or interaction potential, constructed carefully to avoid the Boulware–Deser ghost, with an interaction strength m which is (if the nonzero \\ \\beta_i\\ are ) closely related to the mass of the graviton.\nThe [principle of gauge-invariance](https://bliptext.com/articles/gauge-invariance) renders redundant expressions in any field theory provided with its corresponding gauge(s).\nFor example, in the [massive spin-1 Proca action](https://bliptext.com/articles/proca-action), the massive part in the Lagrangian breaks the gauge-invariance.\nHowever, the invariance is restored by introducing the transformations: The same can be done for massive gravity by following Arkani-Hamed, Georgi and Schwartz effective field theory for massive gravity.\nThe absence of vDVZ discontinuity in this approach motivated the development of dRGT resummation of massive gravity theory as follows.\nThe interaction potential is built out of the [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomials) \\ e_n\\ of the eigenvalues of the matrices or parametrized by dimensionless coupling constants \\ \\alpha_i\\ or \\beta_i, respectively.\nHere is the [matrix square root](https://bliptext.com/articles/matrix-square-root) of the matrix g^{-1}f.\nWritten in index notation, \\mathbb X is defined by the relation We have introduced a reference metric f_{\\mu\\nu} in order to construct the interaction term.\nThere is a simple reason for this: It is impossible to construct a nontrivial interaction (i.e., nonderivative) term from alone.\nThe only possibilities are and \\ \\det g\\, both of which lead to a cosmological constant term rather than a bona fide interaction.\nPhysically, corresponds to the background metric around which fluctuations take the Fierz–Pauli form.\nThis means that, for instance, nonlinearly completing the Fierz–Pauli theory around Minkowski space given above will lead to dRGT massive gravity with although the proof of absence of the Boulware–Deser ghost holds for general f_{\\mu\\nu}.\nThe reference metric transforms like a metric tensor under diffeomorphism\nTherefore and similar terms with higher powers, transforms as a scalar under the same diffeomorphism.\nFor a change in the coordinates, we expand with such that the perturbed metric becomes:\nwhile the potential-like vector transforms according to [Stueckelberg trick](https://bliptext.com/articles/stueckelberg-action) as such that the Stueckelberg field is defined as From the diffeomorphism, one can define another Stueckelberg matrix where and have the same eigenvalues.\nNow, one considers the following symmetries:\nsuch that the transformed perturbed metric becomes:\nThe covariant form of these transformations are obtained as follows.\nIf helicity-0 (or spin-0) mode \\ \\pi\\ is a pure gauge of unphysical Goldstone modes, with the matrix is a tensor function of the covariantization tensor\nof the metric perturbation such that tensor is Stueckelbergized by the field Helicity-0 mode transforms under Galilean transformations hence the name \"Galileons\".\nThe matrix is a tensor function of the covariantization tensor of the metric perturbation with components are given by:\nwhere\nis the extrinsic curvature.\nInterestingly, the covariantization tensor was originally introduced by Maheshwari in a solo authored paper sequel to helicity- Freund–Maheshwari–Schonberg finite-range gravitation model.\nIn Maheshwari's work, the metric perturbation obeys Hilbert-Lorentz condition under the variation\nthat is introduced in Ogievetsky–Polubarinov massive gravity, where \\ p\\ and \\ q\\ are to be determined.\nIt is easy to notice the similarity between tensor in dRGT and the tensor in Maheshwari work once \\ n=2\\ is chosen.\nAlso Ogievetsky–Polubarinov model mandates which means that in 4D, the variation is conformal.\nThe dRGT massive fields split into two helicity-2 two helicity-1 \\ A_{\\mu}\\ and one helicity-0 \\ \\pi\\ degrees of freedom, just like those of Fierz-Pauli massive theory.\nHowever, the covariantization, together with the decoupling limit, guarantee that the symmetries of this massive theory are reduced to the symmetry of linearized general relativity plus that of \\ U(1)\\ massive theory, while the scalar decouples.\nIf \\ v_{\\mu}\\ is chosen to be divergenceless, i.e. the decoupling limit of dRGT gives the known linearized gravity.\nTo see how that happens, expand the terms containing in the action in powers of where H_{\\mu\\nu}\\ is expressed in terms of \\ \\phi^{a}\\ fields like how is expressed in terms of The fields are replaced by:\nThen it follows that in the decoupling limit, i.e. when both the massive gravity Lagrangian is invariant under:\nIn principle, the reference metric must be specified by hand, and therefore there is no single dRGT massive gravity theory, as the theory with a flat reference metric is different from one with a [de Sitter](https://bliptext.com/articles/de-sitter-space) reference metric, etc. Alternatively, one can think of as a constant of the theory, much like \\ m\\ or Instead of specifying a reference metric from the start, one can allow it to have its own dynamics.\nIf the kinetic term for is also Einstein–Hilbert, then the theory remains ghost-free and we are left with a theory of massive bigravity, (or bimetric relativity, BR) propagating the two degrees of freedom of a massless graviton in addition to the five of a massive one.\nIn practice it is unnecessary to compute the eigenvalues of (or ) in order to obtain the \\ e_n ~.\nThey can be written directly in terms of as\nwhere brackets indicate a [trace](https://bliptext.com/articles/trace-linear-algebra), It is the particular antisymmetric combination of terms in each of the \\ e_n\\ which is responsible for rendering the Boulware–Deser ghost nondynamical.\nThe choice to use or, with the [identity matrix](https://bliptext.com/articles/identity-matrix), is a convention, as in both cases the ghost-free mass term is a linear combination of the elementary symmetric polynomials of the chosen matrix.\nOne can transform from one basis to the other, in which case the coefficients satisfy the relationship\nThe coefficients are of a [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) that is in form of [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant).\nThey can also be obtained using [Faddeev–LeVerrier algorithm](https://bliptext.com/articles/faddeev-leverrier-algorithm).\n\n# Massive gravity in the vierbein language\nIn a [4D orthonormal tetrad frame](https://bliptext.com/articles/frame-fields-in-general-relativity), we have the bases:\nwhere the index i is for the 3D spatial component of the \\mu-non-orthonormal coordinates, and the index I is for the 3D spatial components of the a-orthonormal ones.\nThe parallel transport requires the [spin connection](https://bliptext.com/articles/spin-connection) Therefore, the [extrinsic curvature](https://bliptext.com/articles/extrinsic-curvature), that corresponds to in metric formalism, becomes\nwhere \\gamma_{ij} is the spatial metric as in the [ADM formalism](https://bliptext.com/articles/adm-formalism) and [initial value formulation](https://bliptext.com/articles/initial-value-formulation-general-relativity).\nIf the tetrad conformally transforms as the extrinsic curvature becomes, where from [Friedmann equations](https://bliptext.com/articles/friedmann-equations) , and (despite it is controversial ), i.e. the extrinsic curvature transforms as.\nThis looks very similar to the matrix \\mathbb{K} or the tensor.\nThe dRGT was developed inspired by applying the previous technique to the 5D [DGP](https://bliptext.com/articles/dgp-model) model after considering the deconstruction of higher dimensional [Kaluza-Klein](https://bliptext.com/articles/kaluza-klein-theory) gravity theories, in which the extra-dimension(s) is/are replaced by series of N [lattice](https://bliptext.com/articles/lattice-gauge-theory) sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.\nThe presence of a square-root matrix is somewhat awkward and points to an alternative, simpler formulation in terms of [vierbeins](https://bliptext.com/articles/vierbein).\nSplitting the metrics into vierbeins as\nand then defining one-forms\nthe ghost-free interaction terms in Hassan-Rosen bigravity theory can be written simply as (up to numerical factors)\nIn terms of vierbeins, rather than metrics, we can therefore see the physical significance of the ghost-free dRGT potential terms quite clearly: they are simply all the different possible combinations of [wedge products](https://bliptext.com/articles/wedge-product) of the vierbeins of the two metrics.\nNote that massive gravity in the metric and vierbein formulations are only equivalent if the symmetry condition\nis satisfied.\nWhile 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.\nIn these cases the metric and vierbein formulations are distinct physical theories, although each propagates a healthy massive graviton.\nThe novelty in dRGT massive gravity is that it is a theory of gauge invariance under both local Lorentz transformations, from assuming the reference metric f_{\\mu\\nu} equals the Minkowski metric, and diffeomorphism invariance, from the existence of the active curved spacetime g_{\\mu\\nu}.\nThis is shown by rewriting the previously discussed Stueckelberg formalism in the vierbein language as follows.\nThe 4D version of Einstein field equations in 5D is read\nwhere n^\\mu is the vector normal to the 4D slice.\nUsing the definition of massive extrinsic curvature it is straightforward to see that terms containing extrinsic curvatures take the functional form in the tetradic action.\nTherefore, up to the numerical coefficients, the full dRGT action in its tensorial form is\nwhere the functions take forms similar to that of the.\nThen, up to some numerical coefficients, the action takes the integral form\nwhere the first term is the Einstein-Hilbert part of the [tetradic Palatini action](https://bliptext.com/articles/tetradic-palatini-action) and is the [Levi-Civita symbol](https://bliptext.com/articles/levi-civita-symbol).\nAs the decoupling limit guarantees that \\Box\\pi=0 and by comparing X^{\\mu} to \\phi^{\\mu}, it is legit to think of the tensor Comparing this with the definition of the 1-form one can define covariant components of [frame field](https://bliptext.com/articles/frame-fields-in-general-relativity) i.e., to replace the such that the last three interaction terms in the vierbein action becomes\nThis can be done because one is allowed to freely move the diffeomorphism transformations onto the reference vierbein through the Lorentz transformations.\nMore 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 U(1) gauge transformations while the Stueckelberg fields are turned off.\nOne may wonder why the coefficients are dropped, and how to guarantee they are numerical with no explicit dependence of the fields.\nIn fact this is allowed because the variation of the vierbein action with respect to the locally Lorentz transformed Stueckelberg fields yields this nice result.\nMoreover, 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.\n\n# Cosmology\nIf the graviton mass m is comparable to the [Hubble rate](https://bliptext.com/articles/hubble-constant) H_0, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration.\nBecause, roughly speaking, the enhanced diffeomorphism symmetry in the limit m=0 protects a small graviton mass from large quantum corrections, the choice m\\sim H_0 is in fact technically natural.\nMassive gravity thus may provide a solution to the [cosmological constant](https://bliptext.com/articles/cosmological-constant) problem: why do quantum corrections not cause the Universe to accelerate at extremely early times?\nHowever, it turns out that flat and closed [Friedmann–Lemaître–Robertson–Walker](https://bliptext.com/articles/friedmann-lema-tre-robertson-walker-metric) cosmological solutions do not exist in dRGT massive gravity with a flat reference metric.\nOpen solutions and solutions with general reference metrics suffer from instabilities.\nTherefore, viable cosmologies can only be found in massive gravity if one abandons the [cosmological principle](https://bliptext.com/articles/cosmological-principle) that the Universe is uniform on large scales, or otherwise generalizes dRGT.\nFor instance, cosmological solutions are better behaved in [bigravity](https://bliptext.com/articles/bimetric-gravity), the theory which extends dRGT by giving f_{\\mu\\nu} dynamics.\nWhile these tend to possess instabilities as well, 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.\n\n# 3D massive gravity\nA special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom.\nHere several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined.\nIn the case of topologically massive gravity one has the action\nwith M_3 the three-dimensional Planck mass.\nThis is three-dimensional general relativity supplemented by a [Chern-Simons](https://bliptext.com/articles/chern-simons)-like term built out of the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbols).\nMore recently, a theory referred to as new massive gravity has been developed, which is described by the action\n\n# Relation to gravitational waves\nThe 2016 discovery of gravitational waves and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all.\nFollowing the [GW170104](https://bliptext.com/articles/gw170104) event, the graviton's Compton wavelength was found to be at least 1.6 m, or about 1.6 [light-years](https://bliptext.com/articles/light-year), corresponding to a graviton mass of no more than 7.7 eV/c2.\nThis relation between [wavelength](https://bliptext.com/articles/wavelength) and energy is calculated with the same formula (the [Planck–Einstein relation](https://bliptext.com/articles/planck-einstein-relation)) that relates [electromagnetic](https://bliptext.com/articles/electromagnetic) [wavelength](https://bliptext.com/articles/wavelength) to [photon energy](https://bliptext.com/articles/photon-energy).\nHowever, [photons](https://bliptext.com/articles/photon), 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.\nInstead, 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.","current_revision":"a11e975c-f3c7-455c-9171-996957c8d6ae","created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T22:44:41.675Z","updated_at":"2025-03-08T02:47:04.680Z","search_vector":"'graviti':2 'massiv':1","search_content":" \n \n\n# \nIn [theoretical physics](https://bliptext.com/articles/theoretical-physics), **[mass](https://bliptext.com/articles/mass)ive [gravity](https://bliptext.com/articles/gravity)** is a theory of [gravity](https://bliptext.com/articles/gravity) that modifies [general relativity](https://bliptext.com/articles/general-relativity) by endowing the [graviton](https://bliptext.com/articles/graviton) with a nonzero [mass](https://bliptext.com/articles/mass).\nIn the classical theory, this means that [gravitational waves](https://bliptext.com/articles/gravitational-wave) obey a massive wave equation and hence travel at speeds below the [speed of light](https://bliptext.com/articles/speed-of-light).\n\n# Background\nMassive gravity has a long and winding history, dating back to the 1930s when [Wolfgang Pauli](https://bliptext.com/articles/wolfgang-pauli) and [Markus Fierz](https://bliptext.com/articles/markus-fierz) first developed a theory of a massive [spin-2](https://bliptext.com/articles/boson) field propagating on a [flat spacetime](https://bliptext.com/articles/minkowski-space) background.\nIt was later realized in the 1970s that theories of a massive graviton suffered from dangerous pathologies, including a [ghost mode](https://bliptext.com/articles/ghost-physics) and a discontinuity with general relativity in the limit where the graviton mass goes to zero.\nWhile 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](https://bliptext.com/articles/claudia-de-rham), [Gregory Gabadadze](https://bliptext.com/articles/gregory-gabadadze), and Andrew Tolley (dRGT model) in 2010.\nOne of the very early massive gravity theories was constructed in 1965 by [Ogievetsky and Polubarinov](https://bliptext.com/articles/ogievetsky-and-polubarinov) (OP).\nDespite 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.\nMassive dual gravity to the OP model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.\nSince 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](https://bliptext.com/articles/faddeev-leverrier-algorithm), which is similar to that of Galileon theory up to the terms containing polynomials of the trace of the field strength.\nThis is also manifested in the dual formulation of Galileon theory.\nThe 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](https://bliptext.com/articles/dark-energy).\nMassive gravity and its extensions, such as [bimetric gravity](https://bliptext.com/articles/bimetric-gravity), can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.\nObservations of [gravitational waves](https://bliptext.com/articles/gravitational-wave) have constrained the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton to be λg > 1.6 m, which can be interpreted as a bound on the graviton mass mg 1.83 m or mg < 6.76 eV/c2.\n\n# Linearized massive gravity\nAt the linear level, one can construct a theory of a massive [spin](https://bliptext.com/articles/spin-physics)-2 field h_{\\mu\\nu} propagating on [Minkowski space](https://bliptext.com/articles/minkowski-space).\nThis can be seen as an extension of [linearized gravity](https://bliptext.com/articles/linearized-gravity) in the following way.\nLinearized gravity is obtained by linearizing general relativity around flat space,, where is the [Planck mass](https://bliptext.com/articles/planck-mass) with G the [gravitational constant](https://bliptext.com/articles/gravitational-constant).\nThis leads to a kinetic term in the Lagrangian for h_{\\mu\\nu} which is consistent with [diffeomorphism](https://bliptext.com/articles/diffeomorphism) invariance, as well as a coupling to matter of the form\nwhere T_{\\mu\\nu} is the [stress–energy tensor](https://bliptext.com/articles/stress-energy-tensor).\nThis kinetic term and matter coupling combined are nothing other than the [Einstein–Hilbert action](https://bliptext.com/articles/einstein-hilbert-action) linearized about flat space.\nMassive gravity is obtained by adding nonderivative interaction terms for h_{\\mu\\nu}.\nAt the linear level (i.e., second order in h_{\\mu\\nu}), there are only two possible mass terms:\nFierz 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.\nAny other choice will unlock a sixth, ghostly degree of freedom.\nA ghost is a mode with a negative kinetic energy.\nIts [Hamiltonian](https://bliptext.com/articles/hamiltonian-mechanics) is unbounded from below and it is therefore unstable to decay into particles of arbitrarily large positive and negative energies.\nThe Fierz–Pauli mass term,\nis therefore the unique consistent linear theory of a massive spin-2 field.\n\n# The vDVZ discontinuity\nIn the 1970s [Hendrik van Dam](https://bliptext.com/articles/hendrik-van-dam) and [Martinus J. G. Veltman](https://bliptext.com/articles/martinus-j-g-veltman) and, independently, [Valentin I. Zakharov](https://bliptext.com/articles/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 m\\to0.\nIn particular, while at small scales (shorter than the [Compton wavelength](https://bliptext.com/articles/compton-wavelength) of the graviton mass), [Newton's gravitational law](https://bliptext.com/articles/newton-s-gravitational-law) is recovered, the bending of light is only three quarters of the result [Albert Einstein](https://bliptext.com/articles/albert-einstein) obtained in general relativity.\nThis is known as the vDVZ discontinuity.\nWe may understand the smaller light bending as follows.\nThe Fierz–Pauli massive graviton, due to the broken [diffeomorphism invariance](https://bliptext.com/articles/diffeomorphism-invariance), propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity.\nThese three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field.\nThis scalar mode exerts an extra attraction in the massive case compared to the massless case.\nHence, 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.\nBut light bending is blind to the scalar sector, because the stress-energy tensor of light is traceless.\nHence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.\n\n# Vainshtein screening\nIt 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 h_{\\mu\\nu}.\nHeuristically speaking, within a region known as the [Vainshtein radius](https://bliptext.com/articles/vainshtein-radius), fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term.\nCanonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius.\nBecause 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.\nThis phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as [DGP](https://bliptext.com/articles/dgp-model) and certain [scalar-tensor theories](https://bliptext.com/articles/scalar-tensor-theories), where it is crucial for hiding the effects of modified gravity in the solar system.\nThis allows these theories to match [terrestrial and solar-system tests of gravity](https://bliptext.com/articles/tests-of-general-relativity) as well as general relativity does, while maintaining large deviations at larger distances.\nIn this way these theories can lead to cosmic acceleration and have observable imprints on the [large-scale structure of the Universe](https://bliptext.com/articles/observable-universe) without running afoul of other, much more stringent constraints from observations closer to home.\n\n# The Boulware–Deser ghost\nAs a response to [Freund](https://bliptext.com/articles/peter-freund)–Maheshwari–Schonberg finite-range gravity model, and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and [Stanley Deser](https://bliptext.com/articles/stanley-deser) found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode; the tuning a=-b which ensured this mode's absence at quadratic order was, they found, generally broken at cubic and higher orders, reintroducing the ghost at those orders.\nAs a result, this Boulware–Deser ghost would be present around, for example, highly inhomogeneous backgrounds.\nThis is problematic because a linearized theory of gravity, like Fierz–Pauli, is well-defined on its own but cannot interact with matter, as the coupling breaks diffeomorphism invariance.\nThis must be remedied by adding new terms at higher and higher orders, ad infinitum.\nFor a massless graviton, this process converges and the result is well-known: one simply arrives at general relativity.\nThis is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless spin-2 field.\nIn order for massive gravity to actually describe gravity, i.e., a massive spin-2 field coupling to matter and thereby mediating the gravitational force, a nonlinear completion must similarly be obtained.\nThe Boulware–Deser ghost presents a serious obstacle to such an endeavor.\nThe vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable.\nIn fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost despite endeavors to prove that such belief is invalid.\nIt 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](https://bliptext.com/articles/adm-formalism) variables.\nBut for the finite-range gravity model and Ogievetsky and Polubarinov model, it turns out that they need [Noether's variational principle](https://bliptext.com/articles/noether-s-theorem) together with redefining and conformally improving the energy momentum tensor as a [source field](https://bliptext.com/articles/source-field).\n\n# Ghost-free massive gravity\nIn 2010 a breakthrough was achieved when [de Rham](https://bliptext.com/articles/claudia-de-rham), [Gabadadze](https://bliptext.com/articles/gregory-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.\nThe complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by [Fawad Hassan](https://bliptext.com/articles/fawad-hassan) and [Rachel Rosen](https://bliptext.com/articles/rachel-rosen).\nThe [action](https://bliptext.com/articles/action-physics) for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by\nor, equivalently,\nThe ingredients require some explanation.\nAs in standard general relativity, there is an [Einstein–Hilbert](https://bliptext.com/articles/einstein-hilbert-action) kinetic term proportional to the [Ricci scalar](https://bliptext.com/articles/ricci-scalar) \\ R\\ and a minimal coupling to the matter Lagrangian with \\ \\Phi_i\\ representing all of the matter fields, such as those of the [Standard Model](https://bliptext.com/articles/standard-model).\nThe new piece is a mass term, or interaction potential, constructed carefully to avoid the Boulware–Deser ghost, with an interaction strength m which is (if the nonzero \\ \\beta_i\\ are ) closely related to the mass of the graviton.\nThe [principle of gauge-invariance](https://bliptext.com/articles/gauge-invariance) renders redundant expressions in any field theory provided with its corresponding gauge(s).\nFor example, in the [massive spin-1 Proca action](https://bliptext.com/articles/proca-action), the massive part in the Lagrangian breaks the gauge-invariance.\nHowever, the invariance is restored by introducing the transformations: The same can be done for massive gravity by following Arkani-Hamed, Georgi and Schwartz effective field theory for massive gravity.\nThe absence of vDVZ discontinuity in this approach motivated the development of dRGT resummation of massive gravity theory as follows.\nThe interaction potential is built out of the [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomials) \\ e_n\\ of the eigenvalues of the matrices or parametrized by dimensionless coupling constants \\ \\alpha_i\\ or \\beta_i, respectively.\nHere is the [matrix square root](https://bliptext.com/articles/matrix-square-root) of the matrix g^{-1}f.\nWritten in index notation, \\mathbb X is defined by the relation We have introduced a reference metric f_{\\mu\\nu} in order to construct the interaction term.\nThere is a simple reason for this: It is impossible to construct a nontrivial interaction (i.e., nonderivative) term from alone.\nThe only possibilities are and \\ \\det g\\, both of which lead to a cosmological constant term rather than a bona fide interaction.\nPhysically, corresponds to the background metric around which fluctuations take the Fierz–Pauli form.\nThis means that, for instance, nonlinearly completing the Fierz–Pauli theory around Minkowski space given above will lead to dRGT massive gravity with although the proof of absence of the Boulware–Deser ghost holds for general f_{\\mu\\nu}.\nThe reference metric transforms like a metric tensor under diffeomorphism\nTherefore and similar terms with higher powers, transforms as a scalar under the same diffeomorphism.\nFor a change in the coordinates, we expand with such that the perturbed metric becomes:\nwhile the potential-like vector transforms according to [Stueckelberg trick](https://bliptext.com/articles/stueckelberg-action) as such that the Stueckelberg field is defined as From the diffeomorphism, one can define another Stueckelberg matrix where and have the same eigenvalues.\nNow, one considers the following symmetries:\nsuch that the transformed perturbed metric becomes:\nThe covariant form of these transformations are obtained as follows.\nIf helicity-0 (or spin-0) mode \\ \\pi\\ is a pure gauge of unphysical Goldstone modes, with the matrix is a tensor function of the covariantization tensor\nof the metric perturbation such that tensor is Stueckelbergized by the field Helicity-0 mode transforms under Galilean transformations hence the name \"Galileons\".\nThe matrix is a tensor function of the covariantization tensor of the metric perturbation with components are given by:\nwhere\nis the extrinsic curvature.\nInterestingly, the covariantization tensor was originally introduced by Maheshwari in a solo authored paper sequel to helicity- Freund–Maheshwari–Schonberg finite-range gravitation model.\nIn Maheshwari's work, the metric perturbation obeys Hilbert-Lorentz condition under the variation\nthat is introduced in Ogievetsky–Polubarinov massive gravity, where \\ p\\ and \\ q\\ are to be determined.\nIt is easy to notice the similarity between tensor in dRGT and the tensor in Maheshwari work once \\ n=2\\ is chosen.\nAlso Ogievetsky–Polubarinov model mandates which means that in 4D, the variation is conformal.\nThe dRGT massive fields split into two helicity-2 two helicity-1 \\ A_{\\mu}\\ and one helicity-0 \\ \\pi\\ degrees of freedom, just like those of Fierz-Pauli massive theory.\nHowever, the covariantization, together with the decoupling limit, guarantee that the symmetries of this massive theory are reduced to the symmetry of linearized general relativity plus that of \\ U(1)\\ massive theory, while the scalar decouples.\nIf \\ v_{\\mu}\\ is chosen to be divergenceless, i.e. the decoupling limit of dRGT gives the known linearized gravity.\nTo see how that happens, expand the terms containing in the action in powers of where H_{\\mu\\nu}\\ is expressed in terms of \\ \\phi^{a}\\ fields like how is expressed in terms of The fields are replaced by:\nThen it follows that in the decoupling limit, i.e. when both the massive gravity Lagrangian is invariant under:\nIn principle, the reference metric must be specified by hand, and therefore there is no single dRGT massive gravity theory, as the theory with a flat reference metric is different from one with a [de Sitter](https://bliptext.com/articles/de-sitter-space) reference metric, etc. Alternatively, one can think of as a constant of the theory, much like \\ m\\ or Instead of specifying a reference metric from the start, one can allow it to have its own dynamics.\nIf the kinetic term for is also Einstein–Hilbert, then the theory remains ghost-free and we are left with a theory of massive bigravity, (or bimetric relativity, BR) propagating the two degrees of freedom of a massless graviton in addition to the five of a massive one.\nIn practice it is unnecessary to compute the eigenvalues of (or ) in order to obtain the \\ e_n ~.\nThey can be written directly in terms of as\nwhere brackets indicate a [trace](https://bliptext.com/articles/trace-linear-algebra), It is the particular antisymmetric combination of terms in each of the \\ e_n\\ which is responsible for rendering the Boulware–Deser ghost nondynamical.\nThe choice to use or, with the [identity matrix](https://bliptext.com/articles/identity-matrix), is a convention, as in both cases the ghost-free mass term is a linear combination of the elementary symmetric polynomials of the chosen matrix.\nOne can transform from one basis to the other, in which case the coefficients satisfy the relationship\nThe coefficients are of a [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) that is in form of [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant).\nThey can also be obtained using [Faddeev–LeVerrier algorithm](https://bliptext.com/articles/faddeev-leverrier-algorithm).\n\n# Massive gravity in the vierbein language\nIn a [4D orthonormal tetrad frame](https://bliptext.com/articles/frame-fields-in-general-relativity), we have the bases:\nwhere the index i is for the 3D spatial component of the \\mu-non-orthonormal coordinates, and the index I is for the 3D spatial components of the a-orthonormal ones.\nThe parallel transport requires the [spin connection](https://bliptext.com/articles/spin-connection) Therefore, the [extrinsic curvature](https://bliptext.com/articles/extrinsic-curvature), that corresponds to in metric formalism, becomes\nwhere \\gamma_{ij} is the spatial metric as in the [ADM formalism](https://bliptext.com/articles/adm-formalism) and [initial value formulation](https://bliptext.com/articles/initial-value-formulation-general-relativity).\nIf the tetrad conformally transforms as the extrinsic curvature becomes, where from [Friedmann equations](https://bliptext.com/articles/friedmann-equations) , and (despite it is controversial ), i.e. the extrinsic curvature transforms as.\nThis looks very similar to the matrix \\mathbb{K} or the tensor.\nThe dRGT was developed inspired by applying the previous technique to the 5D [DGP](https://bliptext.com/articles/dgp-model) model after considering the deconstruction of higher dimensional [Kaluza-Klein](https://bliptext.com/articles/kaluza-klein-theory) gravity theories, in which the extra-dimension(s) is/are replaced by series of N [lattice](https://bliptext.com/articles/lattice-gauge-theory) sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components.\nThe presence of a square-root matrix is somewhat awkward and points to an alternative, simpler formulation in terms of [vierbeins](https://bliptext.com/articles/vierbein).\nSplitting the metrics into vierbeins as\nand then defining one-forms\nthe ghost-free interaction terms in Hassan-Rosen bigravity theory can be written simply as (up to numerical factors)\nIn terms of vierbeins, rather than metrics, we can therefore see the physical significance of the ghost-free dRGT potential terms quite clearly: they are simply all the different possible combinations of [wedge products](https://bliptext.com/articles/wedge-product) of the vierbeins of the two metrics.\nNote that massive gravity in the metric and vierbein formulations are only equivalent if the symmetry condition\nis satisfied.\nWhile 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.\nIn these cases the metric and vierbein formulations are distinct physical theories, although each propagates a healthy massive graviton.\nThe novelty in dRGT massive gravity is that it is a theory of gauge invariance under both local Lorentz transformations, from assuming the reference metric f_{\\mu\\nu} equals the Minkowski metric, and diffeomorphism invariance, from the existence of the active curved spacetime g_{\\mu\\nu}.\nThis is shown by rewriting the previously discussed Stueckelberg formalism in the vierbein language as follows.\nThe 4D version of Einstein field equations in 5D is read\nwhere n^\\mu is the vector normal to the 4D slice.\nUsing the definition of massive extrinsic curvature it is straightforward to see that terms containing extrinsic curvatures take the functional form in the tetradic action.\nTherefore, up to the numerical coefficients, the full dRGT action in its tensorial form is\nwhere the functions take forms similar to that of the.\nThen, up to some numerical coefficients, the action takes the integral form\nwhere the first term is the Einstein-Hilbert part of the [tetradic Palatini action](https://bliptext.com/articles/tetradic-palatini-action) and is the [Levi-Civita symbol](https://bliptext.com/articles/levi-civita-symbol).\nAs the decoupling limit guarantees that \\Box\\pi=0 and by comparing X^{\\mu} to \\phi^{\\mu}, it is legit to think of the tensor Comparing this with the definition of the 1-form one can define covariant components of [frame field](https://bliptext.com/articles/frame-fields-in-general-relativity) i.e., to replace the such that the last three interaction terms in the vierbein action becomes\nThis can be done because one is allowed to freely move the diffeomorphism transformations onto the reference vierbein through the Lorentz transformations.\nMore 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 U(1) gauge transformations while the Stueckelberg fields are turned off.\nOne may wonder why the coefficients are dropped, and how to guarantee they are numerical with no explicit dependence of the fields.\nIn fact this is allowed because the variation of the vierbein action with respect to the locally Lorentz transformed Stueckelberg fields yields this nice result.\nMoreover, 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.\n\n# Cosmology\nIf the graviton mass m is comparable to the [Hubble rate](https://bliptext.com/articles/hubble-constant) H_0, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration.\nBecause, roughly speaking, the enhanced diffeomorphism symmetry in the limit m=0 protects a small graviton mass from large quantum corrections, the choice m\\sim H_0 is in fact technically natural.\nMassive gravity thus may provide a solution to the [cosmological constant](https://bliptext.com/articles/cosmological-constant) problem: why do quantum corrections not cause the Universe to accelerate at extremely early times?\nHowever, it turns out that flat and closed [Friedmann–Lemaître–Robertson–Walker](https://bliptext.com/articles/friedmann-lema-tre-robertson-walker-metric) cosmological solutions do not exist in dRGT massive gravity with a flat reference metric.\nOpen solutions and solutions with general reference metrics suffer from instabilities.\nTherefore, viable cosmologies can only be found in massive gravity if one abandons the [cosmological principle](https://bliptext.com/articles/cosmological-principle) that the Universe is uniform on large scales, or otherwise generalizes dRGT.\nFor instance, cosmological solutions are better behaved in [bigravity](https://bliptext.com/articles/bimetric-gravity), the theory which extends dRGT by giving f_{\\mu\\nu} dynamics.\nWhile these tend to possess instabilities as well, 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.\n\n# 3D massive gravity\nA special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom.\nHere several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined.\nIn the case of topologically massive gravity one has the action\nwith M_3 the three-dimensional Planck mass.\nThis is three-dimensional general relativity supplemented by a [Chern-Simons](https://bliptext.com/articles/chern-simons)-like term built out of the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbols).\nMore recently, a theory referred to as new massive gravity has been developed, which is described by the action\n\n# Relation to gravitational waves\nThe 2016 discovery of gravitational waves and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all.\nFollowing the [GW170104](https://bliptext.com/articles/gw170104) event, the graviton's Compton wavelength was found to be at least 1.6 m, or about 1.6 [light-years](https://bliptext.com/articles/light-year), corresponding to a graviton mass of no more than 7.7 eV/c2.\nThis relation between [wavelength](https://bliptext.com/articles/wavelength) and energy is calculated with the same formula (the [Planck–Einstein relation](https://bliptext.com/articles/planck-einstein-relation)) that relates [electromagnetic](https://bliptext.com/articles/electromagnetic) [wavelength](https://bliptext.com/articles/wavelength) to [photon energy](https://bliptext.com/articles/photon-energy).\nHowever, [photons](https://bliptext.com/articles/photon), 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.\nInstead, 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.","revisionCount":5},"hasMore":false,"nextCursor":null}

History