# Dual graviton

In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality predicted by some formulations of supergravity in eleven dimensions.

Composition Elementary particle Gravitation Hypothetical Self 2000s 0 e 2

The dual graviton was first hypothesized in 1980. It was theoretically modeled in 2000s, which was then predicted in eleven-dimensional mathematics of SO(8) supergravity in the framework of electric-magnetic duality. It again emerged in the E11 generalized geometry in eleven dimensions, and the E7 generalized vielbeine-geometry in eleven dimensions. 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.

## Dual linearized gravity

The dual formulations of linearized gravity are described by a mixed Young symmetry tensor $T_{\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu }$ , the so-called dual graviton, in any spacetime dimension D > 4 with the following characters:

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

where square brackets show antisymmetrization.

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

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

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

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

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

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

and the gauge symmetry of the Curtright field is

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

The dual Riemann curvature tensor of the dual graviton is defined as follows:

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

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

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

They fulfill the following Bianchi identities

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

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

## Dual graviton coupling with BF theory

Dual gravitons have interaction with topological BF model in D = 5 through the following Lagrangian action

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

where

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

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

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

where $g=\det(g_{\mu \nu })$  is the determinant of the metric tensor matrix, and $R$  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. There are the following relation between the gravitoelectic field $E_{ab}[h_{ab}]$  and gravitomagnetic field $B_{ab}[h_{ab}]$  of the graviton $h_{ab}$  and the gravitoelectic field $E_{ab}[T_{abc}]$  and gravitomagnetic field $B_{ab}[T_{abc}]$  of the dual graviton $T_{abc}$ :

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

and scalar curvature $R$  with dual scalar curvature $E$ :

$E=\star R$
$R=-\star E$

where $\star$  denotes the Hodge dual.

## Dual graviton in conformal gravity

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

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

where $C_{ABCD}$  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.