# Dual graviton

Composition Elementary particle Gravitation Hypothetical Self 2000s[1][2] 0 e 2

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.[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 vielbeine-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]

## 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]

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

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]

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

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.

## 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.[8] 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 ${\displaystyle T_{abc}}$:[9]

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

and scalar curvature ${\displaystyle R}$ with dual scalar curvature ${\displaystyle E}$:[9]

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

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.[10]