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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 E₁₁ generalized geometry in eleven dimensions,^[5] and the E₇ 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 ${\displaystyle T_{\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu ))$, the so-called dual graviton, in any spacetime dimension D > 4 with the following characters:^[2][15]

${\displaystyle T_{\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu }=T_{[\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}]\mu },}$

${\displaystyle T_{[\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu ]}=0.}$

where square brackets show antisymmetrization.

For 5-D spacetime, the spin-2 dual graviton is described by the Curtright field ${\displaystyle T_{\alpha \beta \gamma ))$. The symmetry properties imply that

${\displaystyle T_{\alpha \beta \gamma }=T_{[\alpha \beta ]\gamma },}$

${\displaystyle T_{[\alpha \beta ]\gamma }+T_{[\beta \gamma ]\alpha }+T_{[\gamma \alpha ]\beta }=0.}$

The Lagrangian action for the spin-2 dual graviton ${\displaystyle T_{\lambda _{1}\lambda _{2}\mu ))$ in 5-D spacetime, the Curtright field, becomes^[2][15]

${\displaystyle {\cal {L))_{\rm {dual))=-{\frac {1}{12))\left(F_{[\alpha \beta \gamma ]\delta }F^{[\alpha \beta \gamma ]\delta }-3F_{[\alpha \beta \xi ]}{}^{\xi }F^{[\alpha \beta \lambda ]}{}_{\lambda }\right),}$

where ${\displaystyle F_{\alpha \beta \gamma \delta ))$ is defined as

${\displaystyle F_{[\alpha \beta \gamma ]\delta }=\partial _{\alpha }T_{[\beta \gamma ]\delta }+\partial _{\beta }T_{[\gamma \alpha ]\delta }+\partial _{\gamma }T_{[\alpha \beta ]\delta },}$

and the gauge symmetry of the Curtright field is

${\displaystyle \delta _{\sigma ,\alpha }T_{[\alpha \beta ]\gamma }=2(\partial _{[\alpha }\sigma _{\beta ]\gamma }+\partial _{[\alpha }\alpha _{\beta ]\gamma }-\partial _{\gamma }\alpha _{\alpha \beta }).}$

The dual Riemann curvature tensor of the dual graviton is defined as follows:^[2]

${\displaystyle E_{[\alpha \beta \delta ][\varepsilon \gamma ]}\equiv {\frac {1}{2))(\partial _{\varepsilon }F_{[\alpha \beta \delta ]\gamma }-\partial _{\gamma }F_{[\alpha \beta \delta ]\varepsilon }),}$

and the dual Ricci curvature tensor and scalar curvature of the dual graviton become, respectively

${\displaystyle E_{[\alpha \beta ]\gamma }=g^{\varepsilon \delta }E_{[\alpha \beta \delta ][\varepsilon \gamma ]},}$

${\displaystyle E_{\alpha }=g^{\beta \gamma }E_{[\alpha \beta ]\gamma }.}$

They fulfill the following Bianchi identities

${\displaystyle \partial _{\alpha }(E^{[\alpha \beta ]\gamma }+g^{\gamma [\alpha }E^{\beta ]})=0,}$

where ${\displaystyle g^{\alpha \beta ))$ is the 5-D spacetime metric.

Massive dual gravity

In 4-D, the Lagrangian of the spinless massive version of the dual gravity is

${\displaystyle {\mathcal {L_{\rm {dual,massive))^{\rm {spinless))))=-{\frac {1}{2))u+{\frac {1}{2))(v-gu)^{2}+{\frac {1}{3))g(v-gu)^{3}\sideset {_{3)){_{2))F(1,{\frac {1}{2)),{\frac {3}{2));2,{\frac {5}{2));-4g^{2}(v-gu)^{2}),}$

where ${\displaystyle V^{\mu }={\frac {1}{6))\epsilon ^{\mu \alpha \beta \gamma }V_{\alpha \beta \gamma }~,v=V_{\mu }V^{\mu }{\text{and))~u=\partial _{\mu }V^{\mu }.}$^[16] The coupling constant ${\displaystyle g/m}$ appears in the equation of motion to couple the trace of the conformally improved energy momentum tensor ${\displaystyle \theta }$ to the field as in the following equation

${\displaystyle \left(\Box +m^{2}\right)V_{\mu }={\frac {g}{m))\partial _{\mu }\theta .}$

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

${\displaystyle {\text{det))(\delta _{\nu }^{\mu }+{\frac {g}{m))K_{\nu }^{\mu })=1-{\frac {1}{2))(g/m)^{2}K_{\alpha }^{\beta }K_{\beta }^{\alpha }+{\frac {1}{3))(g/m)^{3}K_{\alpha }^{\beta }K_{\beta }^{\gamma }K_{\gamma }^{\alpha }+{\frac {1}{8))(g/m)^{4}\left[(K_{\alpha }^{\beta }K_{\beta }^{\alpha })^{2}-2K_{\alpha }^{\beta }K_{\beta }^{\gamma }K_{\gamma }^{\delta }K_{\delta }^{\alpha }\right],}$

where ${\displaystyle K_{\mu }^{\nu }=3\partial _{\alpha }T_{[\beta \gamma ]\mu }\epsilon ^{\alpha \beta \gamma \nu ))$.

So the zeroth interaction part, i.e., the third term in the Lagrangian, can be read as ${\displaystyle K_{\alpha }^{\beta }\theta _{\beta }^{\alpha ))$ so the equation of motion becomes

${\displaystyle \left(\Box +m^{2}\right)T_{[\alpha \beta ]\gamma }={\frac {g}{m))P_{\alpha \beta \gamma ,\lambda \mu \nu }\partial ^{\lambda }\theta ^{\mu \nu },}$

where the ${\displaystyle P_{\alpha \beta \gamma ,\lambda \mu \nu }=2\epsilon _{\alpha \beta \lambda \mu }\eta _{\gamma \nu }+\epsilon _{\alpha \gamma \lambda \mu }\eta _{\beta \nu }-\epsilon _{\beta \gamma \lambda \mu }\eta _{\alpha \nu ))$ is Young symmetrizer of such SO(2) theory.

For solutions of the massive theory in arbitrary N-D, i.e., Curtright field ${\displaystyle T_{[\lambda _{1}\lambda _{2}...\lambda _{N-3}]\mu ))$, 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]

${\displaystyle S_{\rm {L))=\int d^{5}x({\cal {L))_{\rm {dual))+{\cal {L))_{\rm {BF))).}$

where

${\displaystyle {\cal {L))_{\rm {BF))=Tr[\mathbf {B} \wedge \mathbf {F} ]}$

Here, ${\displaystyle \mathbf {F} \equiv d\mathbf {A} \sim R_{ab))$ is the curvature form, and ${\displaystyle \mathbf {B} \equiv e^{a}\wedge e^{b))$ 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:

${\displaystyle S_{\rm {BF))=\int d^{5}x{\cal {L))_{\rm {BF))\sim S_{\rm {EH))={1 \over 2}\int \mathrm {d} ^{5}xR{\sqrt {-g)).}$

where ${\displaystyle g=\det(g_{\mu \nu })}$ is the determinant of the metric tensor matrix, and ${\displaystyle R}$ 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 ${\displaystyle E_{ab}[h_{ab}]}$ and gravitomagnetic field ${\displaystyle B_{ab}[h_{ab}]}$ of the graviton ${\displaystyle h_{ab))$ and the gravitoelectic field ${\displaystyle E_{ab}[T_{abc}]}$ and gravitomagnetic field ${\displaystyle B_{ab}[T_{abc}]}$ of the dual graviton ${\displaystyle T_{abc))$:^[18][15]

${\displaystyle B_{ab}[T_{abc}]=E_{ab}[h_{ab}]}$

${\displaystyle E_{ab}[T_{abc}]=-B_{ab}[h_{ab}]}$

and scalar curvature ${\displaystyle R}$ with dual scalar curvature ${\displaystyle E}$:^[18]

${\displaystyle E=\star R}$

${\displaystyle R=-\star E}$

where ${\displaystyle \star }$ denotes the Hodge dual.

Dual graviton in conformal gravity

The free (4,0) conformal gravity in D = 6 is defined as

${\displaystyle {\mathcal {S))=\int \mathrm {d} ^{6}x{\sqrt {-g))C_{ABCD}C^{ABCD},}$

where ${\displaystyle C_{ABCD))$ 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.