| Dual graviton | |
| Composition: | Elementary particle |
| Group: | Gauge boson |
| Interaction: | Gravitation |
| Antiparticle: | Self |
| Status: | Hypothetical |
| Theorized: | 2000s[1] [2] |
| Electric Charge: | 0 e |
| Spin: | 2 |
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.[3]
The dual graviton was first hypothesized in 1980.[4] 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.[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.
The dual formulations of linearized gravity are described by a mixed Young symmetry tensor
| T | |
| λ1λ2 … λD-3\mu |
| T | |
| λ1λ2 … λD-3\mu |
=
| T | |
| [λ1λ2 … λD-3]\mu |
,
| T | |
| [λ1λ2 … λD-3\mu] |
=0.
T\alpha\beta\gamma
T\alpha\beta\gamma=T[\alpha\beta]\gamma,
T[\alpha\beta]\gamma+T[\beta\gamma]\alpha+T[\gamma\alpha=0.
The Lagrangian action for the spin-2 dual graviton
| T | |
| λ1λ2\mu |
{\calL}\rm=-
| 1 | |
| 12 |
\left(F[\alpha\beta\gamma]\deltaF[\alpha\beta\gamma]\delta-3F[\alpha\beta\xi]{}\xiF[\alpha\betaλ]{}λ\right),
F\alpha\beta\gamma\delta
F[\alpha\beta\gamma]\delta=\partial\alphaT[\beta\gamma+\partial\betaT[\gamma\alpha]\delta+\partial\gammaT[\alpha\beta]\delta,
\delta\sigma,\alphaT[\alpha\beta]\gamma=2(\partial[\alpha\sigma\beta]\gamma+\partial[\alpha\alpha\beta]\gamma-\partial\gamma\alpha\alpha\beta).
E[\alpha\beta\delta][\varepsilon\gamma]\equiv
| 1 | |
| 2 |
(\partial\varepsilonF[\alpha\beta\delta]\gamma-\partial\gammaF[\alpha\beta\delta]\varepsilon),
E[\alpha\beta]\gamma=g\varepsilon\deltaE[\alpha\beta\delta][\varepsilon\gamma],
E\alpha=g\beta\gammaE[\alpha\beta]\gamma.
\partial\alpha(E[\alpha\beta]\gamma+g\gamma[\alphaE\beta])=0,
where
g\alpha\beta
In 4-D, the Lagrangian of the spinless massive version of the dual gravity is
| \rmspinless | |
| l{L | |
| \rmdual,massive |
where
V\mu=
| 1 | |
| 6 |
\epsilon\mu\alpha\beta\gammaV\alpha\beta\gamma~,v=V\muV\muand~u=\partial\muV\mu.
g/m
\theta
| 2\right)V | ||
| \left(\Box+m | = | |
| \mu |
| 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
det
| \mu | ||||
| (\delta | ||||
|
| \mu | ||||
| K | ||||
|
| \beta | |
| (g/m) | |
| \alpha |
| ||||
| K | ||||
| \beta |
| \beta | |
| (g/m) | |
| \alpha |
| \gamma | |
| K | |
| \beta |
| ||||
| K | ||||
| \gamma |
| \beta | |
| (g/m) | |
| \alpha |
| \alpha) | |
| K | |
| \beta |
| \beta | |
| \alpha |
| \gamma | |
| K | |
| \beta |
| \delta | |
| K | |
| \gamma |
| \alpha\right], | |
| K | |
| \delta |
where
| \nu=3 | |
| K | |
| \mu |
\partial\alphaT[\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
| \beta | |
| K | |
| \alpha |
| \alpha | |
| \theta | |
| \beta |
| 2\right)T | ||
| \left(\Box+m | = | |
| [\alpha\beta]\gamma |
| g | |
| m |
P\alpha\beta\gamma,λ\mu\nu\partialλ\theta\mu\nu,
where the
P\alpha\beta\gamma,λ\mu\nu=2\epsilon\alpha\betaλ\muη\gamma\nu+\epsilon\alpha\gammaλ\muη\beta\nu-\epsilon\beta\gammaλ\muη\alpha\nu
For solutions of the massive theory in arbitrary N-D, i.e., Curtright field
| T | |
| [λ1λ2...λN-3]\mu |
Dual gravitons have interaction with topological BF model in D = 5 through the following Lagrangian action[7]
S\rm=\intd5x({\calL}\rm+{\calL}\rm).
{\calL}\rm=Tr[B\wedgeF]
F\equivdA\simRab
B\equivea\wedgeeb
In principle, it should similarly be coupled to a BF model of gravity as the linearized Einstein–Hilbert action in D > 4:
S\rm=\intd5x{\calL}\rm\simS\rm={1\over2}\intd5xR\sqrt{-g}.
g=\det(g\mu\nu)
R
In similar manner while we define gravitomagnetic and gravitoelectric for the graviton, we can define electric and magnetic fields for the dual graviton.[17] There are the following relation between the gravitoelectric field
Eab[hab]
Bab[hab]
hab
Eab[Tabc]
Bab[Tabc]
Tabc
Bab[Tabc]=Eab[hab]
Eab[Tabc]=-Bab[hab]
R
E
E=\starR
R=-\starE
\star
The free (4,0) conformal gravity in D = 6 is defined as
l{S}=\intd6x\sqrt{-g}CABCDCABCD,
where
CABCD
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.