Generalized interaction in multigravity

Theoretical and Mathematical Physics, 177(1): 1400–1411 (2013)
GENERALIZED INTERACTION IN MULTIGRAVITY
S. A. Duplij∗
and A. T. Kotvytskiy∗
We consider a general approach to describing the interaction in multigravity models in a D-dimensional
space–time. We present various possibilities for generalizing the invariant volume. We derive the most
general form of the interaction potential, which becomes a Pauli–Fierz-type model in the bigravity case.
Analyzing this model in detail in the (3+1)-expansion formalism and also requiring the absence of ghosts
leads to this bigravity model being completely equivalent to the Pauli–Fierz model. We thus in a concrete
example show that introducing an interaction between metrics is equivalent to introducing the graviton
mass.
Keywords: multigravity, bigravity, massive gravity, invariant volume, interaction potential, Pauli–Fierz
model
1. Introduction
Multigravity together with conformal gravity [1] and scalar theories [2] is one possible extension of
general relativity [3], [4]. In early papers, a particular case of multigravity was called the f–g theory
or strong gravity [5]–[7]. This construction was later successfully applied in quantum gravity and brane
theory [8]–[10], in theories with discrete dimensions [11], [12], in renormalization theory [13], and in massive
gravity [14] and was used to explain such experimental facts as dark energy and dark matter [15]–[17] and
the accelerated expansion of the Universe [18], [19]. Considering nonlinear formulations of multigravity is
therefore important (this was done for bigravity in [20]).
On the other hand, progress in the theory of massive gravity was achieved in [21], where the Pauli–
Fierz mass term was extended in a linearized gravity and it was shown that such a model is free of ghost
modes [22]. The theory was further extended to the case of an arbitrary additional metric [23]. The main
properties of such theories were considered in [24], [25], and the absence of ghost terms in nonlinear models
was proved in [26]. In theories of gravity with nonzero mass, we encounter a singularity because of which
a theory does not tend to general relativity as the graviton mass tends to zero [27], [28]. The Vainshtein
mechanism [29] allows avoiding such an inhomogeneity in the parameter space [23], [30]; moreover, such an
inhomogeneity can be eliminated in the case of a nonﬂat background metric [31], [32].
Here, we describe the general approach for describing interactions in multigravity models in a D-
dimensional space–time (D > 3). In Sec. 2, we study various possibilities of generalizing the invariant
volume of interaction dΩ
(N)
int , which is restricted by the conditions that the invariant volume dΩ
(N)
int must be
a scalar that passes to the standard volume
√
g dD
x in the limit in which all metrics coincide. The function
dΩ
(N)
int must also be monotonic and uniform in all the metrics. In Sec. 3, we derive the most general form
of the interaction potential and show that in the simplest case of two metrics (bigravity), it is given by
a Pauli–Fierz-type model. A detailed analysis of this model in the formalism of (3+1)-expansion under
the condition that ghosts are absent leads to this bigravity model in the weak-ﬁeld limit being completely
∗
Karazin Kharkov State University, Kharkov, Ukraine, e-mail: sduplij@gmail.com.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 177, No. 1, pp. 137–150, October, 2013.
Original article submitted February 26, 2013; revised May 21, 2013.
1400 0040-5779/13/1771-1400 c 2013 Springer Science+Business Media, Inc.

equivalent to the Pauli–Fierz model. In fact, this means that introducing an interaction between the tensor
fields g
(1)
μν and g
(2)
μν is equivalent to introducing a graviton mass. In the appendix, we present a new method
for calculating
√
g for small excitations, which can be used with any background metric. In the case of a
flat Minkowski background space–time, we obtain the standard expression.
2. Multigravity and the generalization of the invariant volume of
interaction
We consider the union of N different universes each of which is described by its metric g
(i)
μν, where
i = 1, . . . , N. We use the signature (+,
D−1
−, . . . , − ) in the D-dimensional space–time. We write the action
for the ith universe in the form
SG(i) = dΩ(i)
[L(i)
gr (g(i)
) + Lmat(g(i)
, Φ(i)
)], (1)
where dΩ(i)
= d4
x g(i) is the invariant volume, g(i)
= | det(g
(i)
μν)| is the scalar density with weight two,
g
(i)
μν is the metric tensor of the ith universe, L
(i)
gr (g(i)
) is the Lagrangian describing the gravitational field,
and the Lagrangian L
(i)
mat(g(i)
, Φ(i)
) describes the coupling between the gravity and matter fields Φ(i)
. The
integral in (1) is taken over the total manifold of N universes.
Assuming “weakly coupled worlds” [20] and the “no-go” theorem [33], we can write the general action
for N massless gravitons as a sum of purely gravitational actions of form (1):
S0 =
N
i=1
SG(i).
Assuming that “weakly coupled worlds” mutually interact only through gravitational fields, we can write
the complete multigravity action in the form
Sfull =
N
i
SG(i) + Sint,
where the last term, Sint, describes the interaction between the universes. Choosing this term is crucial
when describing multigravity models [34].
In the general case of N-gravity in D dimensions, the action Sint is
Sint = dD
x W(g(i)
, . . . , g(N)
),
where dD
x and W(g(1)
, . . . , g(N)
) are scalar densities of opposite weights. By analogy with the standard
invariant volume dΩ = d4
x
√
g in general relativity [3], [4], we represent the expression dD
x W(g(i)
, . . . , g(N)
)
as
dD
x f(
√
g1, . . . ,
√
gN )V (g(i)
, . . . , g(N)
).
In this expression, V (g(i)
, . . . , g(N)
) ≡ V (g(i)
) is the scalar interaction potential, and f(
√
g1, . . . ,
√
gN ) is a
smooth positive function of N positive real arguments and has the weight +1. We introduce the invariant
volume of interaction
dΩ
(N)
int = dD
x f(
√
g1, . . . ,
√
gN ), (2)
which must be a scalar. Moreover, in the limit of coinciding arguments [34]
g(i)
μν = · · · = g(N)
μν ≡ gμν,
the invariant volume of interaction must transform into the standard invariant volume, dΩ
(N)
int → dΩ. To
satisfy all these requirements, the function f(
√
g1, . . . ,
√
gN ) must have the following properties:
1401

1. idempotency in the limit of coinciding arguments, f(
√
g, . . . ,
√
g ) =
√
g,
2. monotonicity,
3. homogeneity under a rescaling of all arguments, f(t
√
g1, . . . , t
√
gN ) = tα
f(
√
g1, . . . ,
√
gN ) (idempo-
tency implies that α = 1), and
4. total symmetricity in all arguments.
The homogeneity and symmetricity conditions for f(
√
g1, . . . ,
√
gN ) imply that the invariant volume
of interaction can be represented in the form [34]
dΩ
(N)
int = dD
x f(
√
g1, . . . ,
√
gN ) = dD
x 2N
√
g1 . . . gN f(y
(N)
1 , . . . , y
(N)
N ),
where
y
(N)
1 =
2N
gN−1
1 g−1
2 . . . g−1
N ,
y
(N)
2 = 2N
g−1
1 gN−1
2 . . . g−1
N ,
...
y
(N)
N = 2N
g−1
1 g−1
2 . . . g−1
N−1gN−1
N .
The variables y
(N)
i obviously satisfy the identity
y
(N)
1 y
(N)
2 · · · y
(N)
N = 1,
the function f is therefore in fact a function of N−1 arguments, and we can write the invariant volume of
interaction in the form
dΩ
(N)
int = d4
x f(
√
g1, . . . ,
√
gN ) = d4
x 2N
√
g1 · · · gN
ˆf(y
(N)
1 , . . . , y
(N)
N−1),
where
ˆf(y
(N)
1 , . . . , y
(N)
N−1)
def
= f y
(N)
1 , . . . , y
(N)
N−1,
1
y
(N)
1 y
(N)
2 · · · y
(N)
N−1
.
We note that y
(N)
i = 1 and f(1, . . . , 1) = 1 in the limit of coinciding metrics.
We choose the invariant volume of interaction as a sum of three means: the arithmetic mean, the
geometric mean, and the harmonic mean taken with arbitrary real coeﬃcients α, β, and γ. Then
dΩ
(N)
int = dD
x 2N
√
g1 · · · gN
1
α + β + γ
α
N
N
i=1
y
(N)
i + β + γ
N
N
i=1 1/y
(N)
i
, (3)
where α + β + γ = 0. For simplicity, we restrict ourself to this natural expression (3) for the invariant
volume of interaction in multigravity. We note that a particular case of (3) with α = γ = 0 and β = 1 for
bigravity (for N = 2) was considered in [20].
1402

3. The generalized interaction potential
We consider the general form of the multigravity interaction described by a scalar potential V (g(i)
, . . . ,
g(N)
) determined as a function of N metrics g
(i)
μν in a D-dimensional space–time. The symmetry group of
N universes is the direct product of groups of diffeomorphisms [20],
Gfull = Diff(ε(i)
μ ) × Diff(ε(2)
μ ) × · · · × Diff(ε(N)
μ ),
where a diffeomorphism Diff(ε
(i)
μ ) acts on the metric g
(i)
μν along the vector ε
(i)
μ (x). In accordance with
the known theorem [33], we can reduce the group Gfull to the diagonal subgroup when all vectors coincide,
ε
(i)
μ (x) = εμ(x). The infinitesimal transformations of each metric g
(i)
μν are then governed by the Lie derivative,
δg(i)
μν = Lεg(i)
μν = ερ
∂ρg(i)
μν + g(i)
μρ ∂νερ
+ g(i)
ρν ∂μερ
.
The scalar interaction potential must obviously be expressed via scalar functions of the metrics g
(i)
μν. We
can naturally choose these scalar functions as invariants of a tensor with one covariant and one contravariant
index constructed from the metrics, Hμ
ν = Hμ
ν(g(i)
, . . . , g(N)
). Eigenvalues of the matrix H corresponding to
the tensor Hμ
ν are then invariant under the action of general coordinate transformations xμ
→ ˜xμ
because
∂˜xα
∂xμ
Hμ
ν
∂xν
∂˜xβ
= Hα
β.
We parameterize the matrix H(g(i)
, . . . , g(N)
) using the following observation.
In most physically relevant models [4], the metric is diagonal,
g(i)
μν = diag(λ
(i)
0 , λ
(i)
1 , . . . , λ
(i)
D−1), (4)
where λ
(i)
a are eigenvalues of the ith metric. Hence, we can describe the structure of the matrix H(g(i)
, . . . ,
g(N)
) analogously to that of the invariant volume of interaction constructed in Sec. 2. Namely, we construct
N matrices H
(i)μ
ν as the product of diagonal matrices
H(i)μ
ν = g(i)μα1
g(i)
α1ρ1
g(i)ρ1β1
g
(2)
β1ρ2
· · · g(i)ρj−1αj
g(j)
αj ρj
g(i)ρj βj
g
(j+1)
βjρj+1
· · ·
· · · g(i)ρN−2αN−1
g(N−1)
αN−1ρN−1
g(i)ρN−1βN−1
g
(N)
βN−1ν.
The thus constructed matrices H(i)
satisfy the identity
H(1)
H(2)
· · · H(N)
= I, (5)
where I is the unit D×D matrix. As a result, we obtain N−1 independent matrices H(i)
. In the bigravity
case (for N = 2), we have two matrices
H(1)μ
ν = g(1)μβ1
g
(2)
β1ν, H(2)μ
ν = g(2)μα1
g(i)
α1ν,
which are mutually inverse, H(1)
H(2)
= I (see identity (5)), and it hence suffices to consider one of these
matrices (see, e.g., [20]). It is therefore reasonable to define the N2
matrices ˆp(i,j)
:
ˆp(i,j)μ
ν = g(i)μρ
g(j)
ρν , (6)
1403

where i, j = 1, 2, . . . , N. The matrices ˆp(i,j)
obviously satisfy the relations
ˆp(i,j)
ˆp(j,k)
= ˆp(i,k)
, (7)
ˆp(i,j)
ˆp(j,i)
= ˆp(i,i)
= I. (8)
Product (7) is associative and invertible (see equality (8)), but it is not defined for all elements, and the set
of p-variables is therefore a partial group [35]. We note that we have N(N−1)/2 independent p-matrices,
which commute in the case of diagonal metrics (4). In the bigravity case (for N = 2), we have
H(1)
= ˆp(1,2)
, H(2)
= ˆp(2,1)
.
We construct the matrices H(i)
from six matrices ˆp(i,j)
(among which three are independent) in the
case of ternary gravity (for N = 3):
H(1)
= ˆp(1,3)
ˆp(1,2)
,
H(2)
= ˆp(2,1)
ˆp(2,3)
,
H(3)
= ˆp(3,2)
ˆp(3,1)
.
The matrices H(i)
satisfy the identity
H(1)
H(2)
H(3)
= I.
Taking equality (4) into account, we can write the eigenvalues of the matrices H
(i)μ
ν using metric eigenvalues,
H(i)
= diag
(λ
(i)
0 )N
R0
,
(λ
(i)
1 )N
R1
, . . . ,
(λ
(i)
D−1)N
RD−1
, (9)
where Ra =
N
i=1 λ
(i)
a . It then follows from (9) that
det H(i)
=
(det g(i)
)N
N
j=1 det g(j)
, (10)
and obviously N
j=1 det H(j)
= 1 (see identity (5)).
We note that for the metric g
(i)
μν with the signature (+,
D−1
−, . . . , − ), the signs of the eigenvalues are
λ
(i)
0 > 0, λ
(i)
1 < 0, . . . , λ
(i)
D−1 < 0
(see, e.g., [4]). By virtue of relations (9) and (5), we find that all eigenvalues of the matrices H(i)
are nonzero
positive. We can then define the new variables
μ(i)
a = log
(λ
(i)
a )N
Ra
, a = 0, 1, . . ., D − 1, i = 1, 2, . . . , N, (11)
satisfying D identities
N
i=1
μ(i)
a = 0, a = 0, 1, . . . , D − 1. (12)
1404

As a result, the number of independent μ-variables is D(N − 1). We can therefore take a smooth function
of μ-variables as the scalar interaction potential,
V (g(i)
, g(2)
, . . . , g(N)
) = ˜v({μ(i)
a }).
Following [20] (where the particular case N = 2 and D = 4 was considered), we choose a more convenient
basis in the form of symmetric polynomials,
σ
(i)
k =
D−1
a=0
(μ(i)
a )k
, k = 1, 2, . . ., D, (13)
connected by D relations following from identities (12). We can therefore write the scalar interaction
potential for multigravity in the form
V (g(i)
, g(2)
, . . . , g(N)
) = v({σ
(i)
k }), k = 1, 2, . . ., D, i = 1, 2, . . . , N, (14)
where v is a scalar function of D(N−1) independent polynomials σ
(i)
k .
Naturally assuming the absence of interaction in the case of ﬂat spaces, we obtain the “boundary
condition”
v(0, 0, . . . , 0) = 0. (15)
We explicitly express scalar interaction potential (14) as a combination of invariants of the matrices
H(i)
. From relations (10), (11), and (13), we have
σ
(i)
k = tr(log H(i)
)k
.
We parameterize the metrics as
g(i)
μν = ημν + h(i)
μν, (16)
where h
(i)
μν are excitations over a ﬂat background. Keeping only terms quadratic in the excitations h
(i)
μν, which
correspond to the massive case and the absence of self-action, for σ
(i)
1 and σ
(i)
2 , we obtain the expressions
σ
(i)
1 =
1≤j≤N,
j=i
(h(i)
− h(j)
) − (h(i)
μν)2
− (h(j)
μν )2
, (17)
σ
(i)
2 = (N − 1)2
(h(i)
μν)2
+
1≤j≤N,
j=i
(h(j)
μν )2
+
+ 2
1≤k,j≤N,
j=k, k=i, j=i
h(j)μ
νh(k)ν
μ − 2(N − 1)h(i)μ
ν
1≤j≤N,
j=i
h(j)ν
μ, (18)
where h(i) def
= h
(i)
μνημν
and (h
(i)
μν)2 def
= h
(i)
μνh(i)μν
. We note that σ
(i)
k ∼ O((h(i)
)k
). Hence, if we keep only
quadratic terms, then we cannot consider expressions with powers k ≥ 3.
We can therefore represent the scalar interaction potential in the multigravity in the quadratic approx-
imation in the form
V (g(i)
) =
N
i=1
[aiσ
(i)
1 + bi(σ
(i)
1 )2
+ ciσ
(i)
2 ], (19)
where ai, bi, and ci are arbitrary real constants. Formula (17) implies that
N
i=1
σ
(i)
1 = 0,
which must also follow from identities (12).
1405

4. The Pauli–Fierz model in bigravity
As an example, we consider bigravity (N = 2) and obtain the Pauli–Fierz model from general principles.
Instead of relations (17) and (18), we have
σ
(1)
1 = −σ
(2)
1 = h(1)
− h(2)
− (h(1)
μν )2
− (h(2)
μν )2
≡ σ1,
σ
(1)
2 = σ
(2)
2 = (h(1)
μν )2
+ (h(2)
μν )2
− 2h(1)μ
νh(2)ν
μ ≡ σ2
(up to terms quadratic in the excitations h
(1)
μν and h
(2)
μν ). For the scalar interaction potential, sum (19) (with
condition (15) taken into account) becomes
V (g(1)
, g(2)
) = aσ1 + bσ2
1 + cσ2, (20)
where a, b, and c are arbitrary real constants with the dimension of the fourth power of mass. The total
bigravity action is then
S2 = −M2
1 d4
x R1
√
g1 − M2
2 d4
x R2
√
g2 + dΩ
(2)
int V (g(1)
, g(2)
), (21)
where M1,2 are constants with the dimension of mass and dΩ
(2)
int is invariant volume (2) of interaction for
bigravity, which in this case becomes
dΩ
(2)
int = d4
x 4
√
g1g2
1
α + β + γ
α
2
g1
g2
+
g2
g1
+ β + 2γ
g1
g2
+
g2
g1
−1
, (22)
where α, β, and γ are dimensionless parameters, α + β + γ = 0. We note that parameterization (16) of
equality (22) results in the expression
dΩ
(2)
int = d4
x 4
√
g1g2 + . . . ,
where the ellipsis denotes terms quadratic in the excitations h
(1)
μν and h
(2)
μν . These terms do not contribute
to (21), because we restrict ourself to the second order and scalar interaction potential (20) does not contain
terms without h
(1)
μν and h
(2)
μν . Using expansion (16) and applying it to action (21), we obtain
S2 = d4
x(Lkin + Lint), (23)
where
Lkin =
1
4
M2
1 [∂ρ
h(1)
μν ∂ρh(1)μν
− ∂μ
h(1)
∂μh(1)
+ 2∂μh(1)μν
∂νh(1)
− 2∂μh(1)μν
∂ρh(1)ρ
ν ] +
+
1
4
M2
2 [∂ρ
h(2)
μν ∂ρh(2)μν
− ∂μ
h(2)
∂μh(2)
+ 2∂μh(2)μν
∂νh(2)
− 2∂μh(2)μν
∂ρh(2)ρ
ν ],
Lint = a(h(1)
− h(2)
)2
+ b(h(1)
μν − h(2)
μν )(h(1)μν
− h(2)μν
) +
+ c(h(2)
μν h(2)μν
− h(1)
μν h(1)μν
) +
c
4
(h(1)
)2
− (h(2)
)2
.
1406

We further apply the (3+1)-expansion [31] to total action (23). Segregating the spatial and temporal
components in Lint, we obtain
Lint = a(h
(1)
00 − h
(2)
00 − h
(1)
ii + h
(2)
ii )2
+ b(h
(1)
00 − h
(2)
00 )(h
(1)
00 − h
(2)
00 ) −
− 2b(h
(1)
0i − h
(2)
0i )(h
(1)
0i − h
(2)
0i ) + b(h
(1)
ij − h
(2)
ij )(h
(1)
ij − h
(2)
ij ) +
+ c(h
(2)
00 h
(2)
00 − 2h
(2)
0i h
(2)
0i + h
(2)
ij h
(2)
ij − h
(1)
00 h
(1)
00 + 2h
(1)
0i h
(1)
0i − h
(1)
ij h
(1)
ij ) +
+
c
4
(h
(1)
00 − h
(1)
ii )2
− (h
(2)
00 − h
(2)
ii )2
.
We restrict our consideration to only the scalar sector because it suffices for eliminating ghost modes
from the spectrum (see [31] for the case of standard gravity). We write the (3+1)-expansion using the
parameterization
h
(r)
00 = 2ϕr, h
(r)
0i = ∂iBr, h
(r)
ij = −2(ψrδij − ∂i∂jEr),
where ϕr, ψr, Br, and Er are the scalar fields for the perturbed metric h
(r)
μν , r = 1, 2. From formula (23),
we obtain expressions for the kinetic and interaction terms:
Lkin = M2
1 [−2ψ1 ∂k∂kψ1 − 6 ˙ψ2
1 − 4ϕ1 ∂k∂kψ1 − 4 ˙ψ1 ∂k∂kB1 + 4 ˙ψ1 ∂k∂k
˙E1] +
+ M2
2 [−2ψ2 ∂k∂kψ2 − 6 ˙ψ2
2 − 4ϕ2 ∂k∂kψ2 − 4 ˙ψ2 ∂k∂kB2 + 4 ˙ψ2 ∂k∂k
˙E2], (24)
Lint = a 2(ϕ1 − ϕ2) + 6(ψ1 − ψ2) − 2Δ(E1 − E2)
2
+
+ b 4(ϕ1 − ϕ2)2
+ 2(B1 − B2)(ΔB1 − ΔB2) + 12(ψ1 − ψ2)2
+ 4(ΔE1 − ΔE2)2
−
− 8(ψ1 − ψ2)(ΔE1 − ΔE2) +
+ c 4(ϕ2
2 − ϕ2
1) + 12(ψ2
2 − ψ2
1) + B2ΔB2 − B1ΔB1 + 4((ΔE2)2
− (ΔE1)2
) +
+ 8(ψ1ΔE1 − ψ2ΔE2) + c (ϕ1 + 3ψ1 − ΔE1)2
− (ϕ2 + 3ψ2 − ΔE2)2
. (25)
We consider the part of the total Lagrangian that contains the scalar fields ϕ1 and ϕ2:
L(ϕ) = − 4M2
1 ϕ1Δψ1 − 4M2
2 ϕ2Δψ2 + ϕ2
1(4a + 4b − 3c) + ϕ2
2(4a + 4b + 3c) +
+ ϕ1(24a(ψ1 − ψ2) − 8a(ΔE1 − ΔE2) + 6cψ1 − 2cΔE1) +
+ ϕ2(−24a(ψ1 − ψ2) + 8a(ΔE1 − ΔE2) − 6cψ2 + 2cΔE2) − 8ϕ1ϕ2(a + b).
Obviously, the Lagrangian does not contain terms quadratic in the fields ϕ1 and ϕ2 if
4a + 4b − 3c = 0, 4a + 4b + 3c = 0, a + b = 0, (26)
i.e., scalar fields become nondynamical (see the details in [31]). System (26) is equivalent to the equations
a + b = 0, c = 0. (27)
We note that we can express the Lagrangian in terms of the differences of the corresponding fields only if
the above relations for the parameters are satisfied. Introducing the variables
ϕ = ϕ1 − ϕ2, B = B1 − B2, (28)
ψ = ψ1 − ψ2 , E = E1 − E2, (29)
1407

we can write interaction Lagrangian (25) in the form
L
(2)
int = 4a 6ψ2
+ 6ϕψ − 2ϕΔE − 4ψΔE −
1
2
BΔB . (30)
Expression (30) coincides with the massive Pauli–Fierz Lagrangian in the (3+1)-expansion of standard
gravity [31]. To prove the equivalence of bigravity (21) and the Pauli–Fierz theory, we must also consider
the kinetic part. We note that we can represent kinetic term (24) in terms of fields (28) and (29) only if we
use the equations of motion. For this, we write total Lagrangian (24), (25) taking (28) into account. We
have
L
(2)
kin + L
(2)
int = M2
1 [−2ψ1 ∂k∂kψ1 − 6 ˙ψ2
1 − 4ϕ1 ∂k∂kψ1 − 4 ˙ψ1 ∂k∂kB1 + 4 ˙ψ1 ∂k∂k
˙E1] +
+ M2
2 [−2ψ2 ∂k∂kψ2 − 6 ˙ψ2
2 − 4ϕ2 ∂k∂kψ2 − 4 ˙ψ2 ∂k∂kB2 + 4 ˙ψ2 ∂k∂k
˙E2] +
+ 24a(ψ1 − ψ2)2
+ 4a[6(ϕ1 − ϕ2)(ψ1 − ψ2) − 2(ϕ1 − ϕ2)Δ(E1 − E2)] −
− 16a(ψ1 − ψ2)Δ(E1 − E2) − 2a(B1 − B2)Δ(B1 − B2). (31)
The Euler–Lagrange system of equations for the fields B1 and B2 is then
4M2
1 Δ ˙ψ1 + 4a(ΔB1 − ΔB2) = 0,
4M2
2 Δ ˙ψ2 + 4a(ΔB2 − ΔB1) = 0,
(32)
where we represent the relevant part of the Lagrangian in the form
L(B) = 4M2
1 ∂k
˙ψ1 ∂kB1 + 4M2
2 ∂k
˙ψ2 ∂kB2 + 2a(∂kB1 − ∂kB2)(∂kB1 − ∂kB2). (33)
By virtue of (28), system (32) transforms into
M2
1 Δ ˙ψ1 = −aΔB, M2
2 Δ ˙ψ2 = aΔB, (34)
which implies the equality
M2
1 ψ1 = −M2
2 ψ2. (35)
For the field ψ (see definitions (29)), we obtain
ψ = ψ1 − ψ2 = ψ1 +
M2
1
M2
2
ψ1 =
M2
1 + M2
2
M2
2
ψ1 = −
M2
2
M2
1
ψ2 − ψ2 = −
M2
1 + M2
2
M2
1
ψ2.
Taking Eqs. (34) into account, we obtain
B =
M2
1
−a
˙ψ1 =
M2
1 M2
2
−a(M2
1 + M2
2 )
˙ψ.
The part L(B) of the Lagrangian given by (33) then becomes
L(B) = 2
M4
1 M4
2
a(M2
1 + M2
2 )2
˙ψΔ ˙ψ.
1408

Varying expression (31) in the fields ϕ1 and ϕ2, we obtain the system
− M2
1 Δψ1 + 6a(ψ1 − ψ2) − 2a(ΔE1 − ΔE2) = 0,
− M2
2 Δψ2 − 6a(ψ1 − ψ2) + 2a(ΔE1 − ΔE2) = 0,
which by virtue of (29) and (35) is equivalent to the equation
ΔE = −
M2
1 M2
2
2a(M2
1 + M2
2 )
Δψ + 3ψ.
As a result, we can rewrite the part of Lagrangian that contains the fields E1 and E2 in the form
L(E) = 4M2
1
˙ψ1Δ ˙E1 + 4M2
2
˙ψ2Δ ˙E2 − 8aϕΔE − 16aψΔE =
= 4
M2
1 M2
2
M2
1 + M2
2
˙ψ −
M2
1 M2
2
2a(M2
1 + M2
2 )
Δ ˙ψ + 3 ˙ψ −
− 8a(ϕ + 2ψ) −
M2
1 M2
2
2a(M2
1 + M2
2 )
Δψ + 3ψ .
We also express the remaining terms in the kinetic term of total Lagrangian (31) in terms of the field ψ:
Lk(ψ) = − 2M2
1 ψ1Δψ1 − 2M2
2 ψ2Δψ2 −
− 6M2
1
˙ψ2
1 − 6M2
2
˙ψ2
2 − 4M2
1 ϕ1Δψ1 − 4M2
2 ϕ2Δψ2 =
= − 2
M2
1 M2
2
M2
1 + M2
2
(ψΔψ + 3 ˙ψ2
+ 2ϕΔψ).
Total Lagrangian (31) is
L
(2)
kin + L
(2)
int = Lk(ψ) + L(B) + L(E) + 24aψ2
+ 24aϕψ =
= 6
M2
1 M2
2
M2
1 + M2
2
( ˙ψ2
+ ψΔψ) − 24aψ2
.
We represent the constant a in terms of the new constant m2
g:
a =
1
4
M2
1 M2
2
M2
1 + M2
2
m2
g.
The scalar sector of bigravity then becomes
L = 6
M2
1 M2
2
M2
1 + M2
2
( ˙ψ2
+ ψΔψ − m2
gψ2
) = 6
M2
1 M2
2
M2
1 + M2
2
(∂μψ ∂μ
ψ − m2
gψ2
),
where mg is the graviton mass. Taking conditions (27) into account, we can then write action (21) as
Sg = −M2
1 d4
x R1
√
−g1 − M2
2 d4
x R2
√
−g2 −
1
4
M2
1 M2
2
M2
1 + M2
2
d4
x (g1g2)1/4
(σ2 − σ2
1).
Hence, only the total action of bigravity results in the Pauli–Fierz theory. We note that the interaction
term was obtained in [20] based on semiheuristic reasonings, while we have obtained it in the quadratic
approximation framework using the (3+1)-expansion.
1409

5. Conclusions
We have constructed the invariant volume of interaction of multigravity in the general form. We used a
particular case of the volume taken as the sum of three different means (only the geometric mean was used
in [20]) to analyze the bigravity model. In the framework of the (3+1)-expansion formalism, we rigorously
(in the quadratic approximation) proved that the total bigravity Lagrangian (with kinetic terms of the
Einstein type taken into account) is equivalent to the massive Pauli–Fierz theory.
Appendix: Expansion of
√
g in small excitations
Standardly expanding
√
g in the small excitations hμν, we use the expression log(det gμν) = tr(log gμν)
and obtain
√
g = exp
1
2
tr(log gμν) .
Over the flat background metric gμν = ημν + hμν, we obtain
√
g = 1 +
1
2
h −
1
4
hμαhμα
+
1
8
h2
(A.1)
up to O(h2
), where h = hμνημν
.
We present the method for calculating the expansion of
√
g, which can be used for any background
metric (0)
gμν. The general formulas for expanding
√
g were presented in [34] up to the first order and in [36]
up to the second order. We have
gμν = (0)
gμν + hμν.
Hence (in the case D = 4), we obtain
det((0)
gμν + hμν) = εαβρσ
((0)
g0α + h0α)((0)
g1β + h1β)((0)
g2ρ + h2ρ)((0)
g3σ + h3σ),
where ε0123
= +1. Up to O(h2
), we have
det((0)
gμν + hμν) = det((0)
gμν) + hμνKμν
((0)
g) + hμνhαβFμναβ
((0)
g),
where
Kμν
= εαβρσ
(δμ
0 δν
α
(0)
g1β
(0)
g2ρ
(0)
g3σ + δμ
1 δν
β
(0)
g0α
(0)
g2ρ
(0)
g3σ +
+ δμ
2 δν
ρ
(0)
g0α
(0)
g1β
(0)
g3σ + δμ
3 δν
σ
(0)
g0α
(0)
g1β
(0)
g2ρ),
Fμναβ
= εχωρσ
(δμ
0 δν
χδα
1 δβ
ω
(0)
g2ρ
(0)
g3σ + δμ
0 δν
χδα
2 δβ
ρ
(0)
g1ω
(0)
g3σ +
+ δμ
0 δν
χδα
3 δβ
σ
(0)
g1ω
(0)
g2ρ + δμ
1 δν
ωδα
2 δβ
ρ
(0)
g0χ
(0)
g3σ +
+ δμ
1 δν
ωδα
3 δβ
σ
(0)
g0χ
(0)
g2ρ + δμ
2 δν
ρ δα
3 δβ
σ
(0)
g0χ
(0)
g1ω).
The general expression for the expansion of
√
g then becomes
√
g = (0)g −
hμνKμν
((0)
g) + hμνhαβFμναβ
((0)
g)
2 (0)g
−
hμνKμν
((0)
g)
2
8 ((0)g)3
, (A.2)
where (0)
g = | det (0)
gμν|.
1410

In the standard case of expansion over a flat metric (0)
gμν = ημν considered in the paper, the expressions
for (0)
g Kμν
and Fμναβ
become
det((0)
gμν) = (0)
g = −1, Kμν
= −ημν
, Fμναβ
=
1
2
(ηαμ
ηβν
− ημν
ηαβ
),
and we have
√
g = 1 −
−h + hμνhαβ(ηαμ
ηβν
− ημν
ηαβ
)/2
2
−
(−h)2
8
= 1 +
1
2
h −
1
4
hμνhμν
+
1
8
h2
for (A.2). It is important that this expression coincides with (A.1).
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Generalized interaction in multigravity

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