Massive, Bi- and Multi-Gravity Formulation: A Summary

6 Massive, Bi- and Multi-Gravity Formulation: A Summary

The previous ‘deconstruction’ framework gave a intuitive argument for the emergence of a potential of the form (6.3 *) (or (6.1 *) in the vielbein language) and its bi- and multi-metric generalizations. In deconstruction or Kaluza–Klein decomposition a certain type of interaction arises naturally and we have seen that the whole spectrum of allowed potentials (or interactions) could be generated by extending the deconstruction procedure to a more general notion of derivative or by involving the mixing of more sites in the definition of the derivative along the extra dimensions. We here summarize the most general formulation for the theories of massive gravity about a generic reference metric, bi-gravity and multi-gravity and provide a dictionary between the different languages used in the literature.

The general action for ghost-free (or dRGT) massive gravity [144 *] in the vielbein language is [95 *, 314 *] (see however Footnote 13 with respect to Ref. [95 *], see also Refs. [502 , 410 ] for earlier work)

with

|------------------[--------------------------------------| ℒ (mass)(e,f) = 𝜀abcd c0ea ∧ eb ∧ ec ∧ ed + c1ea ∧ eb ∧ ec ∧ f d | a b c d a b c d | | +c2e ∧ e ∧ f ∧ f + c3e ∧ f ∧ f ∧ f | (6.2 ) | a b c d] | | +c4f ∧ f ∧ f ∧ f , | -----------------------------------------------------------

or in the metric language [144 *],

|-------------∫-----------(----------4--------------)---| | M-2Pl 4 √ --- m2- ∑ | |SmGR = 2 d x − g R + 2 αn ℒn[𝒦 [g, f]] . | (6.3 ) ------------------------------------n=0-----------------|

In what follows we will use the notation for the overall potential of massive gravity

so that

where $ℒGR [g ]$ is the standard GR Einstein–Hilbert Lagrangian for the dynamical metric $gμν$ and $fμν$ is the reference metric and for bi-gravity,

where both $gμν$ and $fμν$ are then dynamical metrics.

Both massive gravity and bi-gravity break one copy of diff invariance and so the Stückelberg fields can be introduced in exactly the same way in both cases $&tidle; 𝒰[g,f] → 𝒰[g,f]$ where the Stückelbergized metric $f&tidle;μν$ was introduced in (2.75 *) (or alternatively $𝒰 [g,f ] → 𝒰 [&tidle;g,f ]$ ). Thus bi-gravity is by no means an alternative to introducing the Stückelberg fields as is sometimes stated.

In these formulations, $ℒ 0$ (or the term proportional to $c 0$ ) correspond to a cosmological constant, $ℒ1$ to a tadpole, $ℒ2$ to the mass term and $ℒ3,4$ to allowed higher order interactions. The presence of the tadpole $ℒ1$ would imply a non-zero vev. The presence of the potentials $ℒ3,4$ without $ℒ2$ would lead to infinitely strongly coupled degrees of freedom and would thus be pathological. We recall that $𝒦 [g,f]$ is given in terms of the metrics $g$ and $f$ as

and the Lagrangians $ℒ n$ are defined as follows in arbitrary dimensions $d$ [144 *]

with $ℒ0 [Q ] = d!$ and $ℒ1[Q ] = (d − 1)![Q]$ or equivalently in four dimensions [292 *]

μναβ ℒ0 [Q ] = 𝜀 𝜀μναβ ′ (6.9 ) ℒ1 [Q ] = 𝜀μναβ𝜀μ′ναβQ μμ (6.10 ) μναβ μ′ ν′ ℒ2 [Q ] = 𝜀 𝜀μ′ν′αβQ ν Q ν (6.11 ) ℒ3 [Q ] = 𝜀μναβ𝜀μ′ν′α′βQ μν′Q ν′ν Q αα′ (6.12 ) μναβ μ′ ν′ α′ β′ ℒ4 [Q ] = 𝜀 𝜀μ′ν′α′β′Q ν Q ν Qα Q β . (6.13 )

We have introduced the constant $ℒ0$ ( $ℒ0 = 4!$ and $√ --- − gℒ0$ is nothing other than the cosmological constant) and the tadpole $ℒ1$ for completeness. Notice however that not all these five Lagrangians are independent and the tadpole can always be re-expressed in terms of a cosmological constant and the other potential terms.

Alternatively, we may express these scalars as follows [144 *]

ℒ0[Q ] = 4! (6.14 ) ℒ [Q ] = 3![Q] (6.15 ) 1 2 2 ℒ2[Q ] = 2!([Q ] − [Q ]) (6.16 ) ℒ3[Q ] = ([Q]3 − 3[Q][Q2] + 2[Q3 ]) (6.17 ) 4 2 2 2 2 3 4 ℒ4[Q ] = ([Q] − 6[Q] [Q ] + 3[Q ] + 8[Q ][Q ] − 6[Q ]). (6.18 )

These are easily generalizable to any number of dimensions, and in $d$ dimensions we find $d$ such independent scalars.

The multi-gravity action is a generalization to multiple interacting spin-2 fields with the same form for the interactions, and bi-gravity is the special case of two metrics ( $N = 2$ ), [314 *]

|-------------------------------------------------------------| | M 2 ∑N ∫ ( ) | SN = --Pl 𝜀abcdRab[ej] ∧ ecj ∧ edj + m2N ℒ(mass)(ej,ej+1) , (6.19 ) | 4 j=1 | ---------------------------------------------------------------

|---------------------------(----------------------------------)--| | M 2 ∑N ∫ ∘ ---- m2 ∑4 | |SN = --Pl d4x − gj R[gj] +--N- α (jn)ℒn[𝒦 [gj,gj+1]] .| (6.20 ) | 2 j=1 2 n=0 | ------------------------------------------------------------------

Inverse argument

We could have written this set of interactions in terms of $𝒦 [f, g]$ rather than $𝒦 [g,f]$ ,

∫ M-2Plm2-- 4 √ --- ∑4 𝒰 = 4 d x − g αn ℒn[𝒦 [g, f]] n=0 M 2m2 ∫ ∘ ---∑4 = --Pl--- d4x − f &tidle;αnℒn [𝒦[f,g]], (6.21 ) 4 n=0

with

( ) ( 1 0 0 0 0 ) ( ) &tidle;α0 | | α0 | &tidle;α1 | | − 4 − 1 0 0 0 | | α1 | || &tidle;α || = || 6 3 1 0 0 || || α || . (6.22 ) |( 2|) || − 4 − 3 − 2 − 1 0 || |( 2|) &tidle;α3 ( 1 1 1 1 1 ) α3 &tidle;α4 α4

Interestingly, the absence of tadpole and cosmological constant for say the metric $g$ implies $α0 = α1 = 0$ which in turn implies the absence of tadpole and cosmological constant for the other metric $f$ , $&tidle;α = &tidle;α = 0 0 1$ , and thus $&tidle;α = α = 1 2 2$ .

Alternative variables

Alternatively, another fully equivalent convention has also been used in the literature [292 ] in terms of $μ μα 𝕏 ν = g fαν$ defined in (2.76 *),

which is equivalent to (6.4 *) with $ℒ0 = 4!$ and

( ) ( ) ( ) β0 1 1 1 1 1 α0 || β1|| || 0 − 1 − 2 − 3 − 4 || || α1 || || β2|| = || 0 0 2 6 12 || || α2 || , (6.24 ) ( β3) ( 0 0 0 − 6 − 24) ( α3 ) β 0 0 0 0 24 α 4 4

or the inverse relation,

( ) ( ) ( ) α0 24 24 12 4 1 β0 || α1 || || 0 − 24 − 24 − 12 − 4|| || β1 || | α2 | = -1-| 0 0 12 12 6 | | β2 | , (6.25 ) |( α |) 24 |( 0 0 0 − 4 − 4|) |( β |) 3 3 α4 0 0 0 0 1 β4

so that in order to avoid a tadpole and a cosmological constant we need to set for instance $β4 = − (24β0 + 24β1 + 12 β2 + 4β3)$ and $β3 = − 6(4β0 + 3β1 + β2)$ .

Expansion about the reference metric

In the vielbein language the mass term is extremely simple, as can be seen in Eq. (6.1 *) with $𝒜$ defined in (2.60 *). Back to the metric language, this means that the mass term takes a remarkably simple form when writing the dynamical metric $gμν$ in terms of the reference metric $fμν$ and a difference $2 &tidle;h μν = 2hμν + hμν$ as

where $fαβ = (f− 1)αβ$ . The mass terms is then expressed as

where the $ℒn$ have the same expression as the $ℒn$ in (6.9 *) – (6.13 *) so $&tidle; ℒn$ is genuinely $th n$ order in $h μν$ . The expression (6.27 *) is thus at most quartic order in $hμν$ but is valid to all orders in $hμν$ , (there is no assumption that $hμν$ be small). In other words, the mass term (6.27 *) is not an expansion in $h μν$ truncated to a finite (quartic) order, but rather a fully equivalent way to rewrite the mass Lagrangian in terms of the variable $hμν$ rather than $gμν$ . Of course the kinetic term is intrinsically non-linear and includes a infinite expansion in $h μν$ . A generalization of such parameterizations are provided in [300 *].

The relation between the coefficients $κn$ and $αn$ is given by

( κ ) ( 1 0 0 0 0) ( α ) | 0 | | | | 0 | | κ1 | | 4 1 0 0 0| | α1 | || κ2 || = || 6 3 1 0 0|| || α2 || . (6.28 ) ( κ3 ) ( 4 3 3 1 0) ( α3 ) κ4 1 1 1 1 1 α4

The quadratic expansion about a background different from the reference metric was derived in Ref. [278 *]. Notice however that even though the mass term may not appear as having an exact Fierz–Pauli structure as shown in [278 ], it still has the correct structure to avoid any BD ghost, about any background [295 *, 294 *, 300 , 297 *].

	Abstract
1	Introduction
2	Massive and Interacting Fields
	2.1	Proca field
	2.2	Spin-2 field
	2.3	From linearized diffeomorphism to full diffeomorphism invariance
	2.4	Non-linear Stückelberg decomposition
	2.5	Boulware–Deser ghost
I	Massive Gravity from Extra Dimensions
3	Higher-Dimensional Scenarios
4	The Dvali–Gabadadze–Porrati Model
	4.1	Gravity induced on a brane
	4.2	Brane-bending mode
	4.3	Phenomenology of DGP
	4.4	Self-acceleration branch
	4.5	Degravitation
5	Deconstruction
	5.1	Formalism
	5.2	Ghost-free massive gravity
	5.3	Multi-gravity
	5.4	Bi-gravity
	5.5	Coupling to matter
	5.6	No new kinetic interactions
II	Ghost-free Massive Gravity
6	Massive, Bi- and Multi-Gravity Formulation: A Summary
7	Evading the BD Ghost in Massive Gravity
	7.1	ADM formulation
	7.2	Absence of ghost in the Stückelberg language
	7.3	Absence of ghost in the vielbein formulation
	7.4	Absence of ghosts in multi-gravity
8	Decoupling Limits
	8.1	Scaling versus decoupling
	8.2	Massive gravity as a decoupling limit of bi-gravity
	8.3	Decoupling limit of massive gravity
	8.4	$Λ3$ -decoupling limit of bi-gravity
9	Extensions of Ghost-free Massive Gravity
	9.1	Mass-varying
	9.2	Quasi-dilaton
	9.3	Partially massless
10	Massive Gravity Field Theory
	10.1	Vainshtein mechanism
	10.2	Validity of the EFT
	10.3	Non-renormalization
	10.4	Quantum corrections beyond the decoupling limit
	10.5	Strong coupling scale vs cutoff
	10.6	Superluminalities and (a)causality
	10.7	Galileon duality
III	Phenomenological Aspects of Ghost-free Massive Gravity
11	Phenomenology
	11.1	Gravitational waves
	11.2	Solar system
	11.3	Lensing
	11.4	Pulsars
	11.5	Black holes
12	Cosmology
	12.1	Cosmology in the decoupling limit
	12.2	FLRW solutions in the full theory
	12.3	Inhomogenous/anisotropic cosmological solutions
	12.4	Massive gravity on FLRW and bi-gravity
	12.5	Other proposals for cosmological solutions
IV	Other Theories of Massive Gravity
13	New Massive Gravity
	13.1	Formulation
	13.2	Absence of Boulware–Deser ghost
	13.3	Decoupling limit of new massive gravity
	13.4	Connection with bi-gravity
	13.5	3D massive gravity extensions
	13.6	Other 3D theories
	13.7	Black holes and other exact solutions
	13.8	New massive gravity holography
	13.9	Zwei-dreibein gravity
14	Lorentz-Violating Massive Gravity
	14.1	SO(3)-invariant mass terms
	14.2	Phase $m1 = 0$
	14.3	General massive gravity ( $m0 = 0$ )
15	Non-local massive gravity
16	Outlook
	Acknowledgments
	References
	Footnotes
	Figures