![](https://csdnimg.cn/release/download_crawler_static/12311945/bg3.jpg)
In the last line, we lower the index of S
μ
ν
by using the
metric as g
μρ
S
ρ
ν
≡ S
μν
and define the higher rank tensor
g
μ
1
ν
1
μ
2
ν
2
μ
n
ν
n
as a totally antisymmetrization of the product
g
μ
1
ν
1
g
μ
2
ν
2
g
μ
n
ν
n
with respect to ν
1
; ν
2
; ν
n
, i.e.,
g
μ
1
ν
1
μ
2
ν
2
μ
n
ν
n
≡ n!δ
ν
1
½ρ
1
δ
ν
2
ρ
2
δ
ν
n
ρ
n
g
μ
1
ρ
1
g
μ
2
ρ
2
g
μ
n
ρ
n
: ð6Þ
Here, we eliminate the weight factor from the definition of
g
μ
1
ν
1
μ
n
ν
n
, e.g., g
μ
1
ν
1
μ
2
ν
2
≡ g
μ
1
ν
1
g
μ
2
ν
2
− g
μ
1
ν
2
g
μ
2
ν
1
. The fun-
damental properties of this tensor are summarized in
Appendix A. Furthermore, β
n
denote the free parameters.
Although the mass parameter m
2
is commonly introduced
by the replacement β
n
→ m
2
β
n
in (4), we omit this mass
parameter in order to regard β
n
as the free parameters.
The square root matrix S
μ
ν
≡
ffiffiffiffiffiffiffiffiffiffi
g
−1
f
p
μ
ν
is defined so that
it satisfies the relation,
S
μ
ρ
S
ρ
ν
¼ g
μρ
f
ρν
: ð7Þ
We assume Detðf
μν
Þ ≠ 0 and therefore, DetðS
μ
ν
Þ ≠ 0.
Furthermore, we should note the following important
relation that S
μν
≡ g
μρ
S
ρ
ν
is symmetric with respect to μν,
1
S
μν
¼ S
νμ
: ð9Þ
Our purpose is to derive the constraints of the dRGT
model described by the action (4). In the system where the
highest order time derivative terms in the EoM is second
order, the constraints can be defined as the linear combi-
nations of the EoM which do not include any second order
time derivatives of the dynamical variables, i.e., ∂
0
∂
0
g
μ
1
ν
1
.
2
Then, we now derive the EoM of the action (4).
A nontrivial point is how to calculate the variation of the
symmetric polynomial, δe
ðnÞ
ðSÞ. From the definition (5),
δe
ðnÞ
ðSÞ is given by,
δe
ðnÞ
ðSÞ¼Y
ðn−1Þ
μ
ν
ðSÞδS
μ
ν
; ð10Þ
where the matrix Y
μν
ðnÞ
ðSÞ is defined as follows,
Y
μν
ðnÞ
ðSÞ ≡
1
n!
g
μνμ
1
ν
1
μ
2
ν
2
μ
n
ν
n
S
μ
1
ν
1
S
μ
2
ν
2
S
μ
n
ν
n
: ð11Þ
We should note that the symmetric property Y
μν
ðnÞ
ðSÞ¼
Y
ðμνÞ
ðnÞ
ðSÞ follows the symmetry (9) and the symmetric
properties (A4) given in the Appendix A. In order to
calculate the right-hand side of (10), we use the following
algebraic relation,
δS
μ
ν
S
ν
ρ
þ S
μ
ν
δS
ν
ρ
¼ δg
μν
f
νρ
; ð12Þ
which is obtained by variating the relation (7).By
multiplying the matrix S
−1ρ
σ
Y
ðnÞ
σ
μ
ðSÞ on the both sides
of the above relation (12) and using the commutativity
½Y
ðnÞ
ðSÞ; S¼0, we obtain the relation, δS
μ
ν
Y
ðnÞ
ν
μ
ðSÞ¼
1
2
δg
μρ
S
ρν
Y
ðnÞ
ν
μ
ðSÞ. By substituting this relation,
δS
μ
ν
Y
ðnÞ
ν
μ
ðSÞ¼
1
2
δg
μρ
S
ρν
Y
ðnÞ
ν
μ
ðSÞ, into (10), we obtain,
δe
ðnÞ
ðSÞ¼
1
2
δg
μρ
S
ρν
Y
ðn−1Þ
ν
μ
ðSÞ: ð13Þ
Using Eq. (13) and the recursion relation, Y
μν
ðnÞ
ðSÞ¼
g
μν
e
ðnÞ
ðSÞ−S
μ
ρ
Y
ρν
ðn−1Þ
ðSÞ, given in Eq. (A12) of
Appendix A, we find,
δ½
ffiffiffiffiffiffi
−g
p
e
ðnÞ
ðSÞ ¼ −
ffiffiffiffiffiffi
−g
p
2
Y
ðnÞμν
ðSÞδg
μν
: ð14Þ
Hence, the EoM corresponding to the action (4) can be
determined as follows,
E
μν
≡ G
μν
þ
X
D−1
n¼0
β
n
Y
μν
ðnÞ
ðSÞ¼0: ð15Þ
Here, we define the Einstein tensor by G
μν
≡ R
μν
− Rg
μν
=2.
As we have mentioned, in the system where the highest
order time derivative terms in the EoM is second order, the
constraints can be defined as the linear combinations of
the EoM which does not depend on any second order time
derivatives of the dynamical metric, ∂
0
∂
0
g
μ
1
ν
1
. One of the
constraints is given by taking the divergence of the EoM
(15) and using the Bianchi identity ∇
μ
G
μν
¼ 0,
∇
μ
E
μν
¼
X
D−1
n¼1
β
n
∇
μ
Y
μν
ðnÞ
ðSÞ¼0: ð16Þ
Obviously, this equation is independent of ∂
0
∂
0
g
μ
1
ν
1
.
Hence, (16) can be regarded as a constraint corresponding
to ∂
μ
h
μν
− ∂
ν
h ¼ 0 in (1). We should note that the special
tuning of the potential term in the dRGT model (4) is not
essential for the existence of the constraint (16). In other
words, for a model whose potential term is different from
that of the dRGT model, such as Vðg; ηÞ in (2), the
constraint corresponding to (16) exists.
The special tuning of the potential terms in the dRGT
model (4) is essential for the existence of an additional
constraint corresponding to h ¼ 0 in (1). The covariant
expression of this constraint has not been obtained and it is
the purpose in this paper.
1
This symmetric property can be shown from the following
relation,
ffiffiffiffiffiffiffiffiffiffi
g
−1
f
q
μ
ν
¼ g
μρ
ffiffiffiffiffiffiffiffiffiffi
fg
−1
q
ρ
σ
g
σν
; ð8Þ
which can be confirmed by expanding the square root of the
quantity
ffiffiffiffiffiffiffiffiffiffi
g
−1
f
p
μ
ν
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þðg
−1
f − 1 Þ
p
μ
ν
(e.g., see [27]).
2
The validity of this definition of the constraints is explained
by the Lagrangian analysis (see [28–30] for the case of the FP
model).
COVARIANT CONSTRAINTS OF MASSIVE GRAVITY IN … PHYS. REV. D 99, 104054 (2019)
104054-3