Hypothetical particle found in supergravity
In
theoretical physics , the dual graviton is a hypothetical
elementary particle that is a dual of the
graviton under
electric-magnetic duality , as an
S-duality , predicted by some formulations of
supergravity in eleven dimensions.
[3]
The dual graviton was first
hypothesized in 1980.
[4] It was theoretically modeled in 2000s,
[1]
[2] which was then predicted in eleven-dimensional mathematics of SO(8)
supergravity in the framework of electric-magnetic duality.
[3] It again emerged in the E 11 generalized geometry in eleven dimensions,
[5] and the E 7 generalized vielbein-geometry in eleven dimensions.
[6] While there is no local coupling between graviton and dual graviton, the field introduced by dual graviton may be coupled to a
BF model as non-local gravitational fields in extra dimensions.
[7]
A massive dual gravity of Ogievetsky–Polubarinov model
[8] can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.
[9]
[10]
The previously mentioned theories of dual graviton are in flat space. In
de Sitter and
anti-de Sitter spaces (A)dS, the massless dual graviton exhibits less gauge symmetries dynamics compared with those of
Curtright field in flat space, hence the mixed-symmetry field propagates in more degrees of freedom.
[11] However, the dual graviton in (A)dS transforms under GL(D) representation, which is identical to that of massive dual graviton in flat space.
[12] This apparent paradox can be resolved using the unfolding technique in Brink, Metsaev, and Vasiliev conjecture.
[13]
[14] For the massive dual graviton in (A)dS, the flat limit is clarified after expressing dual field in terms of the
Stueckelberg coupling of a massless spin-2 field with a
Proca field.
[11]
Dual linearized gravity
The dual formulations of linearized gravity are described by a mixed Young symmetry tensor
T
λ
1
λ
2
⋯
λ
D
−
3
μ
{\displaystyle T_{\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu }}
, the so-called dual graviton, in any spacetime dimension D > 4 with the following characters:
[2]
[15]
T
λ
1
λ
2
⋯
λ
D
−
3
μ
=
T
λ
1
λ
2
⋯
λ
D
−
3
μ
,
{\displaystyle T_{\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu }=T_{[\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}]\mu },}
T
λ
1
λ
2
⋯
λ
D
−
3
μ
=
0.
{\displaystyle T_{[\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu ]}=0.}
where square brackets show antisymmetrization.
For 5-D spacetime, the spin-2 dual graviton is described by the
Curtright field
T
α
β
γ
{\displaystyle T_{\alpha \beta \gamma }}
. The symmetry properties imply that
T
α
β
γ
=
T
α
β
γ
,
{\displaystyle T_{\alpha \beta \gamma }=T_{[\alpha \beta ]\gamma },}
T
α
β
γ
+
T
β
γ
α
+
T
γ
α
β
=
0.
{\displaystyle T_{[\alpha \beta ]\gamma }+T_{[\beta \gamma ]\alpha }+T_{[\gamma \alpha ]\beta }=0.}
The Lagrangian action for the spin-2 dual graviton
T
λ
1
λ
2
μ
{\displaystyle T_{\lambda _{1}\lambda _{2}\mu }}
in 5-D spacetime, the
Curtright field , becomes
[2]
[15]
L
d
u
a
l
=
−
1
12
(
F
α
β
γ
δ
F
α
β
γ
δ
−
3
F
α
β
ξ
ξ
F
α
β
λ
λ
)
,
{\displaystyle {\cal {L}}_{\rm {dual}}=-{\frac {1}{12}}\left(F_{[\alpha \beta \gamma ]\delta }F^{[\alpha \beta \gamma ]\delta }-3F_{[\alpha \beta \xi ]}{}^{\xi }F^{[\alpha \beta \lambda ]}{}_{\lambda }\right),}
where
F
α
β
γ
δ
{\displaystyle F_{\alpha \beta \gamma \delta }}
is defined as
F
α
β
γ
δ
=
∂
α
T
β
γ
δ
+
∂
β
T
γ
α
δ
+
∂
γ
T
α
β
δ
,
{\displaystyle F_{[\alpha \beta \gamma ]\delta }=\partial _{\alpha }T_{[\beta \gamma ]\delta }+\partial _{\beta }T_{[\gamma \alpha ]\delta }+\partial _{\gamma }T_{[\alpha \beta ]\delta },}
and the gauge symmetry of the
Curtright field is
δ
σ
,
α
T
α
β
γ
=
2
(
∂
α
σ
β
γ
+
∂
α
α
β
γ
−
∂
γ
α
α
β
)
.
{\displaystyle \delta _{\sigma ,\alpha }T_{[\alpha \beta ]\gamma }=2(\partial _{[\alpha }\sigma _{\beta ]\gamma }+\partial _{[\alpha }\alpha _{\beta ]\gamma }-\partial _{\gamma }\alpha _{\alpha \beta }).}
The dual
Riemann curvature tensor of the dual graviton is defined as follows:
[2]
E
α
β
δ
ε
γ
≡
1
2
(
∂
ε
F
α
β
δ
γ
−
∂
γ
F
α
β
δ
ε
)
,
{\displaystyle E_{[\alpha \beta \delta ][\varepsilon \gamma ]}\equiv {\frac {1}{2}}(\partial _{\varepsilon }F_{[\alpha \beta \delta ]\gamma }-\partial _{\gamma }F_{[\alpha \beta \delta ]\varepsilon }),}
and the dual
Ricci curvature tensor and
scalar curvature of the dual graviton become, respectively
E
α
β
γ
=
g
ε
δ
E
α
β
δ
ε
γ
,
{\displaystyle E_{[\alpha \beta ]\gamma }=g^{\varepsilon \delta }E_{[\alpha \beta \delta ][\varepsilon \gamma ]},}
E
α
=
g
β
γ
E
α
β
γ
.
{\displaystyle E_{\alpha }=g^{\beta \gamma }E_{[\alpha \beta ]\gamma }.}
They fulfill the following Bianchi identities
∂
α
(
E
α
β
γ
+
g
γ
α
E
β
)
=
0
,
{\displaystyle \partial _{\alpha }(E^{[\alpha \beta ]\gamma }+g^{\gamma [\alpha }E^{\beta ]})=0,}
where
g
α
β
{\displaystyle g^{\alpha \beta }}
is the 5-D spacetime metric.
Massive dual gravity
In 4-D, the Lagrangian of the spinless massive version of the dual gravity is
L
d
u
a
l
,
m
a
s
s
i
v
e
s
p
i
n
l
e
s
s
=
−
1
2
u
+
1
2
(
v
−
g
u
)
2
+
1
3
g
(
v
−
g
u
)
3
F
3
F
2
(
1
,
1
2
,
3
2
;
2
,
5
2
;
−
4
g
2
(
v
−
g
u
)
2
)
,
{\displaystyle {\mathcal {L_{\rm {dual,massive}}^{\rm {spinless}}}}=-{\frac {1}{2}}u+{\frac {1}{2}}(v-gu)^{2}+{\frac {1}{3}}g(v-gu)^{3}\sideset {_{3}}{_{2}}F(1,{\frac {1}{2}},{\frac {3}{2}};2,{\frac {5}{2}};-4g^{2}(v-gu)^{2}),}
where
V
μ
=
1
6
ϵ
μ
α
β
γ
V
α
β
γ
,
v
=
V
μ
V
μ
and
u
=
∂
μ
V
μ
.
{\displaystyle V^{\mu }={\frac {1}{6}}\epsilon ^{\mu \alpha \beta \gamma }V_{\alpha \beta \gamma }~,v=V_{\mu }V^{\mu }{\text{and}}~u=\partial _{\mu }V^{\mu }.}
[16] The coupling constant
g
/
m
{\displaystyle g/m}
appears in the equation of motion to couple the trace of the conformally improved energy momentum tensor
θ
{\displaystyle \theta }
to the field as in the following equation
(
◻
+
m
2
)
V
μ
=
g
m
∂
μ
θ
.
{\displaystyle \left(\Box +m^{2}\right)V_{\mu }={\frac {g}{m}}\partial _{\mu }\theta .}
And for the spin-2 massive dual gravity in 4-D,
[10] the Lagrangian is formulated in terms of the
Hessian matrix that also constitutes
Horndeski theory (Galileons/
massive gravity ) through
det
(
δ
ν
μ
+
g
m
K
ν
μ
)
=
1
−
1
2
(
g
/
m
)
2
K
α
β
K
β
α
+
1
3
(
g
/
m
)
3
K
α
β
K
β
γ
K
γ
α
+
1
8
(
g
/
m
)
4
(
K
α
β
K
β
α
)
2
−
2
K
α
β
K
β
γ
K
γ
δ
K
δ
α
,
{\displaystyle {\text{det}}(\delta _{\nu }^{\mu }+{\frac {g}{m}}K_{\nu }^{\mu })=1-{\frac {1}{2}}(g/m)^{2}K_{\alpha }^{\beta }K_{\beta }^{\alpha }+{\frac {1}{3}}(g/m)^{3}K_{\alpha }^{\beta }K_{\beta }^{\gamma }K_{\gamma }^{\alpha }+{\frac {1}{8}}(g/m)^{4}\left[(K_{\alpha }^{\beta }K_{\beta }^{\alpha })^{2}-2K_{\alpha }^{\beta }K_{\beta }^{\gamma }K_{\gamma }^{\delta }K_{\delta }^{\alpha }\right],}
where
K
μ
ν
=
3
∂
α
T
β
γ
μ
ϵ
α
β
γ
ν
{\displaystyle K_{\mu }^{\nu }=3\partial _{\alpha }T_{[\beta \gamma ]\mu }\epsilon ^{\alpha \beta \gamma \nu }}
.
So the zeroth interaction part, i.e., the third term in the Lagrangian, can be read as
K
α
β
θ
β
α
{\displaystyle K_{\alpha }^{\beta }\theta _{\beta }^{\alpha }}
so the equation of motion becomes
(
◻
+
m
2
)
T
α
β
γ
=
g
m
P
α
β
γ
,
λ
μ
ν
∂
λ
θ
μ
ν
,
{\displaystyle \left(\Box +m^{2}\right)T_{[\alpha \beta ]\gamma }={\frac {g}{m}}P_{\alpha \beta \gamma ,\lambda \mu \nu }\partial ^{\lambda }\theta ^{\mu \nu },}
where the
P
α
β
γ
,
λ
μ
ν
=
2
ϵ
α
β
λ
μ
η
γ
ν
+
ϵ
α
γ
λ
μ
η
β
ν
−
ϵ
β
γ
λ
μ
η
α
ν
{\displaystyle P_{\alpha \beta \gamma ,\lambda \mu \nu }=2\epsilon _{\alpha \beta \lambda \mu }\eta _{\gamma \nu }+\epsilon _{\alpha \gamma \lambda \mu }\eta _{\beta \nu }-\epsilon _{\beta \gamma \lambda \mu }\eta _{\alpha \nu }}
is
Young symmetrizer of such SO(2) theory.
For solutions of the massive theory in arbitrary N-D, i.e., Curtright field
T
λ
1
λ
2
.
.
.
λ
N
−
3
μ
{\displaystyle T_{[\lambda _{1}\lambda _{2}...\lambda _{N-3}]\mu }}
, the symmetrizer becomes that of SO(N-2).
[9]
Dual graviton coupling with BF theory
Dual gravitons have interaction with topological
BF model in D = 5 through the following Lagrangian action
[7]
S
L
=
∫
d
5
x
(
L
d
u
a
l
+
L
B
F
)
.
{\displaystyle S_{\rm {L}}=\int d^{5}x({\cal {L}}_{\rm {dual}}+{\cal {L}}_{\rm {BF}}).}
where
L
B
F
=
T
r
B
∧
F
{\displaystyle {\cal {L}}_{\rm {BF}}=Tr[\mathbf {B} \wedge \mathbf {F} ]}
Here,
F
≡
d
A
∼
R
a
b
{\displaystyle \mathbf {F} \equiv d\mathbf {A} \sim R_{ab}}
is the
curvature form , and
B
≡
e
a
∧
e
b
{\displaystyle \mathbf {B} \equiv e^{a}\wedge e^{b}}
is the background field.
In principle, it should similarly be coupled to a BF model of gravity as the linearized Einstein–Hilbert action in D > 4:
S
B
F
=
∫
d
5
x
L
B
F
∼
S
E
H
=
1
2
∫
d
5
x
R
−
g
.
{\displaystyle S_{\rm {BF}}=\int d^{5}x{\cal {L}}_{\rm {BF}}\sim S_{\rm {EH}}={1 \over 2}\int \mathrm {d} ^{5}xR{\sqrt {-g}}.}
where
g
=
det
(
g
μ
ν
)
{\displaystyle g=\det(g_{\mu \nu })}
is the determinant of the
metric tensor matrix, and
R
{\displaystyle R}
is the
Ricci scalar .
Dual gravitoelectromagnetism
In similar manner while we define
gravitomagnetic and gravitoelectic for the graviton, we can define electric and magnetic fields for the dual graviton.
[17] There are the following relation between the gravitoelectic field
E
a
b
h
a
b
{\displaystyle E_{ab}[h_{ab}]}
and
gravitomagnetic field
B
a
b
h
a
b
{\displaystyle B_{ab}[h_{ab}]}
of the graviton
h
a
b
{\displaystyle h_{ab}}
and the gravitoelectic field
E
a
b
T
a
b
c
{\displaystyle E_{ab}[T_{abc}]}
and gravitomagnetic field
B
a
b
T
a
b
c
{\displaystyle B_{ab}[T_{abc}]}
of the dual graviton
T
a
b
c
{\displaystyle T_{abc}}
:
[18]
[15]
B
a
b
T
a
b
c
=
E
a
b
h
a
b
{\displaystyle B_{ab}[T_{abc}]=E_{ab}[h_{ab}]}
E
a
b
T
a
b
c
=
−
B
a
b
h
a
b
{\displaystyle E_{ab}[T_{abc}]=-B_{ab}[h_{ab}]}
and
scalar curvature
R
{\displaystyle R}
with dual scalar curvature
E
{\displaystyle E}
:
[18]
E
=
⋆
R
{\displaystyle E=\star R}
R
=
−
⋆
E
{\displaystyle R=-\star E}
where
⋆
{\displaystyle \star }
denotes the
Hodge dual .
Dual graviton in conformal gravity
The free (4,0)
conformal gravity in D = 6 is defined as
S
=
∫
d
6
x
−
g
C
A
B
C
D
C
A
B
C
D
,
{\displaystyle {\mathcal {S}}=\int \mathrm {d} ^{6}x{\sqrt {-g}}C_{ABCD}C^{ABCD},}
where
C
A
B
C
D
{\displaystyle C_{ABCD}}
is the
Weyl tensor in D = 6. The free (4,0) conformal gravity can be reduced to the graviton in the ordinary space, and the dual graviton in the dual space in D = 4.
[19]
It is easy to notice the similarity between the
Lanczos tensor , that generates the Weyl tensor in geometric theories of gravity, and Curtright tensor, particularly their shared symmetry properties of the linearized spin connection in Einstein's theory. However, Lanczos tensor is a tensor of geometry in D=4,
[20] meanwhile Curtright tensor is a field tensor in arbitrary dimensions.
See also
References
^
a
b
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a
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d
e
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^
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^
a
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^
a
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^
a
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^
a
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a
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^
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