Dual graviton

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]

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

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

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][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]

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

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.

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

${\displaystyle {\mathcal {L_{\rm {dual,massive}}^{\rm {spinless}}}}=-{\frac {1}{2}}u+{\frac {1}{2}}(v-gu)^{2}+{\frac {1}{3}}g(v-gu)^{3}\sideset {_{3}}{_{2}}F(1,{\frac {1}{2}},{\frac {3}{2}};2,{\frac {5}{2}};-4g^{2}(v-gu)^{2}),}$

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

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

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${\displaystyle {\text{det}}(\delta _{\nu }^{\mu }+{\frac {g}{m}}K_{\nu }^{\mu })=1-{\frac {1}{2}}(g/m)^{2}K_{\alpha }^{\beta }K_{\beta }^{\alpha }+{\frac {1}{3}}(g/m)^{3}K_{\alpha }^{\beta }K_{\beta }^{\gamma }K_{\gamma }^{\alpha }+{\frac {1}{8}}(g/m)^{4}\left[(K_{\alpha }^{\beta }K_{\beta }^{\alpha })^{2}-2K_{\alpha }^{\beta }K_{\beta }^{\gamma }K_{\gamma }^{\delta }K_{\delta }^{\alpha }\right],}$

where ${\displaystyle K_{\mu }^{\nu }=3\partial _{\alpha }T_{[\beta \gamma ]\mu }\epsilon ^{\alpha \beta \gamma \nu }}$.

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

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

where the ${\displaystyle P_{\alpha \beta \gamma ,\lambda \mu \nu }=2\epsilon _{\alpha \beta \lambda \mu }\eta _{\gamma \nu }+\epsilon _{\alpha \gamma \lambda \mu }\eta _{\beta \nu }-\epsilon _{\beta \gamma \lambda \mu }\eta _{\alpha \nu }}$ is Young 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 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.

In similar manner while we define gravitomagnetic and gravitoelectic for the graviton, we can define electric and magnetic fields for the dual graviton.[17] 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 ${\displaystyle T_{abc}}$:[18][15]

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

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

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

${\displaystyle E=\star R}$

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

where ${\displaystyle \star }$ denotes the Hodge dual.

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.[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. However, Lanczos tensor is a tensor of geometry in D=4,[20] meanwhile Curtright tensor is a field tensor in arbitrary dimensions.