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 eleven-dimensional supergravity.
[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] The dual formulations of linearized gravity are described by a mixed Young symmetry tensor
, the so-called dual graviton, in any spacetime dimension D > 4 with the following characters:[2][15] where square brackets show antisymmetrization.
For 5-D spacetime, the spin-2 dual graviton is described by the Curtright field
The symmetry properties imply that The Lagrangian action for the spin-2 dual graviton
α β γ δ
is defined as and the gauge symmetry of the Curtright field is The dual Riemann curvature tensor of the dual graviton is defined as follows:[2] and the dual Ricci curvature tensor and scalar curvature of the dual graviton become, respectively They fulfill the following Bianchi identities where
In 4-D, the Lagrangian of the spinless massive version of the dual gravity is
appears in the equation of motion to couple the trace of the conformally improved energy momentum tensor
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
So the zeroth interaction part, i.e., the third term in the Lagrangian, can be read as
α β γ , λ μ ν
α β γ , λ μ ν
For solutions of the massive theory in arbitrary N-D, i.e., Curtright field
[9] Dual gravitons have interaction with topological BF model in D = 5 through the following Lagrangian action[7] where Here,
In principle, it should similarly be coupled to a BF model of gravity as the linearized Einstein–Hilbert action in D > 4: where
is the determinant of the metric tensor matrix, and
In similar manner while we define gravitoelectromagnetism for the graviton, we can define electric and magnetic fields for the dual graviton.
The free (4,0) conformal gravity in D = 6 is defined as where
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.