Massive gravity

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

Massive gravity has a long and winding history, dating back to the 1930s when Wolfgang Pauli and Markus Fierz first developed a theory of a massive spin-2 field propagating on a flat spacetime background. It was later realized in the 1970s that theories of a massive graviton suffered from dangerous pathologies, including a ghost mode and a discontinuity with general relativity in the limit where the graviton mass goes to zero. While solutions to these problems had existed for some time in three spacetime dimensions, they were not solved in four dimensions and higher until the work of Claudia de Rham, Gregory Gabadadze, and Andrew Tolley in 2010.

There are several motivations, theoretical and observational, for studying massive gravity. On the observational side, the fact that general relativity is modified at large distances in massive gravity provides a possible explanation for the accelerated expansion of the Universe that does not require any dark energy. Massive gravity and its extensions, such as bimetric gravity, can yield cosmological solutions which do in fact display late-time acceleration in agreement with observations.

Linearized massive gravity

At the linear level, one can construct a theory of a massive spin-2 field $h_{\mu \nu }$ propagating on Minkowski space. This can be seen as an extension of linearized gravity in the following way. Linearized gravity is obtained by linearizing general relativity around flat space, $g_{\mu \nu }=\eta _{\mu \nu }+M_{\mathrm {Pl} }^{-1}h_{\mu \nu }$, where $M_{\mathrm {Pl} }=(8\pi G)^{-1/2}$ is the Planck mass with $G$ the gravitational constant. This leads to a kinetic term in the Lagrangian for $h_{\mu \nu }$ which is consistent with diffeomorphism invariance, as well as a coupling to matter of the form

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

where T is the stress–energy tensor.

Massive gravity is obtained by adding nonderivative interaction terms for $h_{\mu \nu }$. At the linear level (i.e., second order in $h_{\mu \nu }$, there are only two possible mass terms:

${\mathcal {L}}_{\mathrm {int} }=ah^{\mu \nu }h_{\mu \nu }+b\left(\eta ^{\mu \nu }h_{\mu \nu }\right)^{2}.$

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

${\mathcal {L}}_{\mathrm {FP} }=m^{2}\left(h^{\mu \nu }h_{\mu \nu }-\left(\eta ^{\mu \nu }h_{\mu \nu }\right)^{2}\right)$

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

The vDVZ discontinuity and Vainshtein screening

In the 1970s Hendrik van Dam and Martinus J. G. Veltman and, independently, Vladimir E. Zakharov discovered a peculiar property of Fierz-Pauli massive gravity: its predictions did not uniformly reduce to those of general relativity in the limit $m\to 0$. In particular, while at small scales (shorter than the Compton wavelength of the graviton mass), the Newton's gravitational law is recovered, the bending of light is only three quarters of the result Albert Einstein obtained in general relativity. This is known as the vDVZ discontinuity.

We may understand the smaller light bending as follows. The Fierz-Pauli massive graviton, due to the absence of diffeomorphism invariance, propagates three extra degrees of freedom compared to the massless graviton of linearized general relativity. These three degrees of freedom package themselves into a vector field, which is irrelevant for our purposes, and a scalar field. This scalar mode exerts an extra attraction in the massive case compared to the massless case. Hence, if one wants measurements of the force exerted between nonrelativistic masses to agree, the coupling constant of the massive theory should be smaller than that of the massless theory. But light bending is blind to the scalar sector, because the light energy momentum tensor is traceless. Hence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one.

It was argued by Vainshtein two years later that the vDVZ discontinuity is an artifact of the linear theory, and that the predictions of general relativity are in fact recovered at small scales when one takes into account nonlinear effects, i.e., higher than quadratic terms in $h_{\mu \nu }$. Heuristically speaking, within a region known as the Vainshtein radius, fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term. Canonically normalizing the scalar around this background therefore leads to a heavily-suppressed kinetic term, damping fluctuations of the scalar within the Vainshtein radius. Because the extra force mediated by the scalar is proportional to (minus) its gradient, this leads to a much smaller extra force than we would have calculated just using the linear Fierz-Pauli theory.

This phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as DGP and certain scalar-tensor theories, where it is crucial for hiding the effects of modified gravity in the solar system. This allows these theories to match terrestrial and solar-system tests of gravity as well as general relativity does, while maintaining large deviations at longer distances. In this way these theories can lead to cosmic acceleration and have observable imprints on the large-scale structure of the Universe without running afoul of other, much more stringent constraints from observations closer to home.

The Boulware-Deser ghost

At the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and Stanley Deser found in 1972 that generic nonlinear extensions of the Fierz-Pauli theory reintroduced the dangerous ghost mode; 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. As a result, this Boulware-Deser ghost would be present around, for example, highly inhomogeneous backgrounds.

This 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 $h^{\mu\nu}T_{\mu\nu}$ breaks diffeomorphism invariance. This must be remedied by adding new terms at higher and higher order, ad infinitum. For a massless graviton, this process converges and the end result is well-known: one simply arrives at general relativity. This is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless graviton.

In 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. The Boulware-Deser ghost presents a serious obstacle to such an endeavor. The vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable. In fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware-Deser ghost.

Ghost-free massive gravity

In 2010 a breakthrough was achieved when de Rham, Gabadadze, and Tolley constructed, order by order, a theory of massive gravity with coefficients tuned to avoid the Boulware-Deser ghost by packaging all ghostly (i.e., higher-derivative) operators into total derivatives which do not contribute to the equations of motion. The complete absence of the Boulware-Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by Fawad Hassan and Rachel Rosen.

The action for the ghost-free de Rham-Gabadadze-Tolley (dRGT) massive gravity is given by

$S=\int d^{4}x{\sqrt {-g}}\left(-{\frac {M_{\mathrm {Pl} }^{2}}{2}}R+m^{2}M_{\mathrm {Pl} }^{2}\displaystyle \sum _{n=0}^{4}\alpha _{n}e_{n}(\mathbb {K} )+{\mathcal {L}}_{\mathrm {m} }(g,\Phi _{i})\right),$

or, equivalently,

$S=\int d^{4}x{\sqrt {-g}}\left(-{\frac {M_{\mathrm {Pl} }^{2}}{2}}R+m^{2}M_{\mathrm {Pl} }^{2}\displaystyle \sum _{n=0}^{4}\beta _{n}e_{n}(\mathbb {X} )+{\mathcal {L}}_{\mathrm {m} }(g,\Phi _{i})\right).$

The ingredients require some explanation. As in standard general relativity, there is an Einstein-Hilbert kinetic term proportional to the Ricci scalar $R$ and a minimal coupling to the matter Lagrangian ${\mathcal {L}}_{\mathrm {m} }$. The 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 ${\mathcal {O}}(1)$) closely related to the mass of the graviton.

The interaction potential is built out of elementary symmetric polynomials $e_{n}$ of the eigenvalues of the matrices $\mathbb {K} =\mathbb {I} -{\sqrt {g^{-1}f}}$ or $\mathbb {X} ={\sqrt {g^{-1}f}}$, parametrized by dimensionless coupling constants $\alpha _{i}$ or $\beta _{i}$. Here ${\sqrt {g^{-1}f}}$ is the matrix square root of the matrix $g^{-1}f$. Written in index notation, $\mathbb {X}$ is defined by the relation

$X^{\mu }{}_{\alpha }X^{\alpha }{}_{\nu }=g^{\mu \alpha }f_{\nu \alpha }.$

We have introduced a reference metric $f_{\mu \nu }$ in order to construct the interaction term. There is a simple reason for this: it is impossible to construct a nontrivial interaction (i.e., nonderivative) term from $g_{\mu \nu }$ alone. The only possibilities are $g^{\mu \alpha }g_{\alpha \nu }=\delta _{\nu }^{\mu }$ and $\operatorname {det} g$, both of which lead to a cosmological constant term rather than a bona fide interaction. Physically, $f_{\mu \nu }$ corresponds to the background metric around which fluctuations take the Fierz-Pauli form. This means that, for instance, nonlinearly completing the Fierz-Pauli theory around Minkowski space given above will lead to dRGT massive gravity with $f_{\mu \nu }=\eta _{\mu \nu }$, although the proof of the absence of the Boulware-Deser ghost holds around general reference metrics.

In 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 reference metric, etc. Alternatively, one can think of $f_{\mu \nu }$ as a constant of the theory, much like $m$ or $M_{\mathrm {Pl} }$. Instead of specifying a reference metric from the start, one can allow it to have its own dynamics. If the kinetic term for $f_{\mu \nu }$ is also Einstein-Hilbert, then the theory remains ghost-free and we are left with a theory of massive bigravity, propagating the two degrees of freedom of a massless graviton in addition to the five of a massive one.

In practice it is unnecessary to compute the eigenvalues of $\mathbb {X}$ (or $\mathbb {K}$) in order to obtain the $e_{n}$. They can be written directly in terms of $\mathbb {X}$ as

${\begin{aligned}e_{0}(\mathbb {X} )&=1,\\e_{1}(\mathbb {X} )&=[\mathbb {X} ],\\e_{2}(\mathbb {X} )&={\frac {1}{2}}\left([\mathbb {X} ]^{2}-[\mathbb {X} ^{2}]\right),\\e_{3}(\mathbb {X} )&={\frac {1}{6}}\left([\mathbb {X} ]^{3}-3[\mathbb {X} ][\mathbb {X} ^{2}]+2[\mathbb {X} ^{3}]\right),\\e_{4}(\mathbb {X} )&=\operatorname {det} \mathbb {X} ,\end{aligned}}$

where brackets indicate a trace, $[\mathbb {X} ]\equiv X^{\mu }{}_{\mu }$. It is the particular antisymmetric combination of terms in each of the $e_{n}$ which is responsible for rendering the Boulware-Deser ghost nondynamical.

The choice to use $\mathbb {X}$ or $\mathbb {K} =\mathbb {I} -\mathbb {X}$, with $\mathbb {I}$ the identity matrix, is convention, as in both cases the ghost-free mass term is a linear combination of the elementary symmetric polynomials of the matrix. One can transform from one basis to the other, in which case the coefficients satisfy the relationship

$\beta _{n}=(4-n)!\displaystyle \sum _{i=n}^{4}{\frac {(-1)^{i+n}}{(4-i)!(i-n)!}}\alpha _{i}.$

Massive gravity in the vielbein language

The presence of a square root matrix is somewhat awkward and points to an alternative, simpler formulation in terms of vielbeins. Splitting the metrics into vielbeins as

${\begin{aligned}g_{\mu \nu }=\eta _{ab}e^{a}{}_{\mu },\\f_{\mu \nu }=\eta _{ab}f^{a}{}_{\mu },\end{aligned}}$,

and then defining one-forms

${\begin{aligned}\mathbf {e} ^{a}=e^{a}{}_{\mu }dx^{\mu },\\\mathbf {f} ^{a}=f^{a}{}_{\mu }dx^{\mu },\end{aligned}}$

then the ghost-free interaction terms above can be written simply as (up to numerical factors)

${\begin{aligned}e_{0}(\mathbb {X} )\propto \epsilon _{abcd}\mathbf {e} ^{a}\wedge \mathbf {e} ^{b}\wedge \mathbf {e} ^{c}\wedge \mathbf {e} ^{d}\\e_{1}(\mathbb {X} )\propto \epsilon _{abcd}\mathbf {e} ^{a}\wedge \mathbf {e} ^{b}\wedge \mathbf {e} ^{c}\wedge \mathbf {f} ^{d}\\e_{2}(\mathbb {X} )\propto \epsilon _{abcd}\mathbf {e} ^{a}\wedge \mathbf {e} ^{b}\wedge \mathbf {f} ^{c}\wedge \mathbf {f} ^{d}\\e_{3}(\mathbb {X} )\propto \epsilon _{abcd}\mathbf {e} ^{a}\wedge \mathbf {f} ^{b}\wedge \mathbf {f} ^{c}\wedge \mathbf {f} ^{d}\\e_{4}(\mathbb {X} )\propto \epsilon _{abcd}\mathbf {f} ^{a}\wedge \mathbf {f} ^{b}\wedge \mathbf {f} ^{c}\wedge \mathbf {f} ^{d}\\\end{aligned}}$

In terms of vielbeins, 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 of the vielbeins of the two metrics.

Note that massive gravity in the metric and vielbein formulations are only equivalent if the symmetry condition

$(e^{-1})_{a}{}^{\mu }f_{b\nu }=(e^{-1})_{b}{}^{\mu }f_{a\nu }.$

is satisfied. While this is true for most physical situations, there may be cases, such as with matter coupling to both metrics or in multimetric theories with interaction cycles, in which it is not. In these cases the metric and vielbein formulations are distinct physical theories, although each propagates a healthy massive graviton.

Cosmology

If the graviton mass $m$ is comparable to the Hubble rate $H_{0}$, then at cosmological distances the mass term can produce a repulsive gravitational effect that leads to cosmic acceleration. Because, roughly speaking, the enhanced diffeomorphism symmetry in the limit $m=0$ protects a small graviton mass from large quantum corrections, the choice $m\sim H_{0}$ is in fact technically natural. Massive gravity thus may provide a solution to the cosmological constant problem: why do quantum corrections not cause the Universe to accelerate at extremely early times?

However, it turns out that flat and closed Friedmann–Lemaître–Robertson–Walker cosmological solutions do not exist in dRGT massive gravity with a flat reference metric. Open solutions and solutions with general reference metrics suffer from instabilities. Therefore viable cosmologies can only be found in massive gravity if one abandons the cosmological principle that the Universe is uniform on large scales, or otherwise generalizes dRGT. For instance, cosmological solutions are better behaved in bigravity, the theory which extends dRGT by giving $f_{\mu \nu }$ dynamics. While 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.

3D massive gravity

A special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom. Here several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined. In the case of topologically massive gravity one has the action

$S={\frac {M_{3}}{2}}\int d^{3}x{\sqrt {-g}}(R-2\Lambda )+{\frac {1}{4\mu }}\epsilon ^{\lambda \mu \nu }\Gamma _{\lambda \sigma }^{\rho }\left(\partial _{\mu }\Gamma _{\rho \nu }^{\sigma }+{\frac {2}{3}}\Gamma _{\mu \alpha }^{\sigma }\Gamma _{\nu \rho }^{\alpha }\right),$

with $M_{3}$ the three-dimensional Planck mass. This is three-dimensional general relativity supplemented by a Chern-Simons-like term built out of the Christoffel symbols.

More recently, a theory referred to as new massive gravity has been developed, which is described by the action

$S=M_{3}\int d^{3}x{\sqrt {-g}}\left[\pm R+{\frac {1}{m^{2}}}\left(R_{\mu \nu }R^{\mu \nu }-{\frac {3}{8}}R^{2}\right)\right].$

See also

• Accelerating universe
• Alternatives to general relativity
• Bimetric theory
• DGP model
• Scalar–tensor theory

Further reading

Review articles

• de Rham, Claudia (2014), "Massive Gravity", Living Reviews in Relativity, 17: 7, arXiv:1401.4173, doi:10.12942/lrr-2014-7{{citation}}: CS1 maint: unflagged free DOI (link)
• Hinterbichler, Kurt (2012), "Theoretical Aspects of Massive Gravity", Reviews of Modern Physics, 84: 671–710, arXiv:1105.3735, doi:10.1103/RevModPhys.84.671