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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]

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

${\displaystyle T_{}=0.}$

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

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

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

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]

${\displaystyle E_{\mathrm{\alpha <!--\; \alpha \; -->}\mathrm{\beta <!--\; \beta \; -->}\mathrm{\delta <!--\; \delta \; -->}\mathrm{\epsilon <!--\; \epsilon \; -->}\mathrm{\gamma <!--\; \gamma \; -->}}\equiv <!--\; \equiv \; -->\frac{1}{2}({\mathrm{\partial <!--\; \partial \; -->}}_{\mathrm{\epsilon <!--\; \epsilon \; -->}}{F}_{\mathrm{\alpha <!--\; \alpha \; -->}\mathrm{\beta <!--\; \beta \; -->}\mathrm{\delta <!--\; \delta \; -->}\mathrm{\gamma <!--\; \gamma \; -->}}-<!--\; -\; -->{\mathrm{\partial <!--\; \partial \; -->}}_{\mathrm{\gamma <!--\; \gamma \; -->}}{F}_{\mathrm{\alpha <!--\; \alpha \; -->}}}$Zdroj:https://en.wikipedia.org?pojem=Dual_graviton
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