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Intheoretical physics, thedual graviton is a hypotheticalelementary particle that is a dual of thegraviton underelectric-magnetic duality, as anS-duality, predicted by some formulations ofeleven-dimensional supergravity.^[3]

The dual graviton was firsthypothesized 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 theE₁₁ generalized geometry in eleven dimensions,^[5] and theE₇ 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 aBF model as non-local gravitational fields in extra dimensions.^[7]

Amassive 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. Inde Sitter andanti-de Sitter spaces (A)dS, the massless dual graviton exhibits less gauge symmetries dynamics compared with those ofCurtright 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 theStueckelberg coupling of a massless spin-2 field with aProca field.^[11]

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 dimensionD > 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 theCurtright 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, theCurtright field, becomes^[2][15]

${\displaystyle {\mathcal{L}}_{\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}}=-\frac{1}{12}\left(F_{[\alpha \beta \gamma ]\delta }{F}^{[\alpha \beta \gamma ]\delta }-3F_{[\alpha \beta \xi ]}{}^{\xi }{F}^{[\alpha \beta \lambda ]}{}_{\lambda }\right),}$

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

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

and the gauge symmetry of theCurtright field is

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

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

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

and the dualRicci curvature tensor andscalar curvature of the dual graviton become, respectively

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

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

They fulfill the following Bianchi identities

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

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

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

${\displaystyle {\mathcal{L}}_{\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e}}^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{s}}=-\frac{1}{2}u+\frac{1}{2}(v-gu{)}^2+\frac{1}{3}g(v-gu{)}^3{\phantom{F}}_3{F}_2(1,\frac{1}{2},\frac{3}{2};2,\frac{5}{2};-4g^2(v-gu{)}^2),}$

where${\displaystyle V^{\mu }=\frac{1}{6}{ϵ}^{\mu \alpha \beta \gamma }{V}_{\alpha \beta \gamma },v=V_{\mu }{V}^{\mu }\text{and}u={\mathrm{\partial }}_{\mu }{V}^{\mu }.}$^[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

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

And for the spin-2 massive dual gravity in 4-D,^[10] the Lagrangian is formulated in terms of theHessian matrix that also constitutesHorndeski 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${\displaystyle K_{\mu }^{\nu }=3{\mathrm{\partial }}_{\alpha }{T}_{[\beta \gamma ]\mu }{ϵ}^{\alpha \beta \gamma \nu }}$.

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

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

where the${\displaystyle P_{\alpha \beta \gamma ,\lambda \mu \nu }=2{ϵ}_{\alpha \beta \lambda \mu }{\eta }_{\gamma \nu }+{ϵ}_{\alpha \gamma \lambda \mu }{\eta }_{\beta \nu }-{ϵ}_{\beta \gamma \lambda \mu }{\eta }_{\alpha \nu }}$ isYoung 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 gravitons have interaction with topologicalBF model inD = 5 through the following Lagrangian action^[7]

${\displaystyle S_{\mathrm{L}}=\int d^{5}x({\mathcal{L}}_{\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}}+{\mathcal{L}}_{\mathrm{B}\mathrm{F}}).}$

where

${\displaystyle {\mathcal{L}}_{\mathrm{B}\mathrm{F}}=Tr[\mathbf{B}\wedge \mathbf{F}]}$

Here,${\displaystyle \mathbf{F}\equiv d\mathbf{A}\sim R_{ab}}$ is thecurvature 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 inD > 4:

${\displaystyle S_{\mathrm{B}\mathrm{F}}=\int d^{5}x{\mathcal{L}}_{\mathrm{B}\mathrm{F}}\sim S_{\mathrm{E}\mathrm{H}}=\frac{1}{2}\int {\mathrm{d}}^{5}xR\sqrt{-g}.}$

where${\displaystyle g=det(g_{\mu \nu })}$ is the determinant of themetric tensor matrix, and${\displaystyle R}$ is theRicci scalar.

In similar manner while we definegravitoelectromagnetism for the graviton, we can define electric and magnetic fields for the dual graviton.^[17] There are the following relation between the gravitoelectric field${\displaystyle E_{ab}[h_{ab}]}$ and gravitomagnetic field${\displaystyle B_{ab}[h_{ab}]}$ of the graviton${\displaystyle h_{ab}}$ and the gravitoelectric field${\displaystyle E_{ab}[T_{abc}]}$ and gravitomagnetic field${\displaystyle B_{ab}[T_{abc}]}$ of the dual graviton${\displaystyle T_{abc}}$:^[18][15]

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

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

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

${\displaystyle E=\star R}$

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

where${\displaystyle \star }$ denotes theHodge dual.

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

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

where${\displaystyle C_{ABCD}}$ is theWeyl tensor inD = 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 inD = 4.^[19]

It is easy to notice the similarity between theLanczos 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.

• Graviton
• Gravitino
• Gravity
• Gravitoelectromagnetism
• Curtright field
• Taub–NUT space
• Nielsen–Olesen vortex
• 't Hooft loop
• Dual photon
• Massive gravity
• Horndeski's theory
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