Dual graviton

Dual graviton
Composition	Elementary particle
Interactions	Gravitation
Status	Hypothetical
Antiparticle	Self
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 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 $T_{\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu }$ , the so-called dual graviton, in any spacetime dimension D > 4 with the following characters:^[2]

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^[2]

{\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:^[2]

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^[7]

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.^[8] 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 $T_{abc}$ :^[9]

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

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

and scalar curvature $R$ with dual scalar curvature $E$ :^[9]

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.^[10]

References

1 2 Hull, C. M. (2001). "Duality in Gravity and Higher Spin Gauge Fields". Journal of High Energy Physics. 2001 (9): 27. arXiv:hep-th/0107149. Bibcode:2001JHEP...09..027H. doi:10.1088/1126-6708/2001/09/027.
1 2 3 4 5 Bekaert, X.; Boulanger, N.; Henneaux, M. (2003). "Consistent deformations of dual formulations of linearized gravity: A no-go result". Physical Review D. 67 (4): 044010. arXiv:hep-th/0210278. Bibcode:2003PhRvD..67d4010B. doi:10.1103/PhysRevD.67.044010.
1 2 de Wit, B.; Nicolai, H. (2013). "Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions". Journal of High Energy Physics. 2013 (5): 77. arXiv:1302.6219. Bibcode:2013JHEP...05..077D. doi:10.1007/JHEP05(2013)077.
↑ Curtright, T. (1985). "Generalised Gauge Fields". Physics Letters B. 165 (4–6): 304. Bibcode:1985PhLB..165..304C. doi:10.1016/0370-2693(85)91235-3.
↑ West, P. (2012). "Generalised geometry, eleven dimensions and E₁₁". Journal of High Energy Physics. 2012 (2): 18. arXiv:1111.1642. Bibcode:2012JHEP...02..018W. doi:10.1007/JHEP02(2012)018.
↑ Godazgar, H.; Godazgar, M.; Nicolai, H. (2014). "Generalised geometry from the ground up". Journal of High Energy Physics. 2014 (2): 75. arXiv:1307.8295. Bibcode:2014JHEP...02..075G. doi:10.1007/JHEP02(2014)075.
1 2 Bizdadea, C.; Cioroianu, E. M.; Danehkar, A.; Iordache, M.; Saliu, S. O.; Sararu, S. C. (2009). "Consistent interactions of dual linearized gravity in D = 5: couplings with a topological BF model". European Physical Journal C. 63 (3): 491–519. arXiv:0908.2169. Bibcode:2009EPJC...63..491B. doi:10.1140/epjc/s10052-009-1105-0.
↑ Henneaux, M.; Teitelboim, C. (2005). "Duality in linearized gravity". Physics Letters B. 71 (2): 024018. arXiv:gr-qc/0408101. Bibcode:2005PhRvD..71b4018H. doi:10.1103/PhysRevD.71.024018.
1 2 Henneaux, M., "E₁₀ and gravitational duality" https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_workshops_conferences/jointerc_2014/henneaux.pdf
↑ Hull, C. M. (2000). "Symmetries and Compactifications of (4,0) Conformal Gravity". Journal of High Energy Physics. 2000 (0012): 007. arXiv:hep-th/0011215. Bibcode:2000JHEP...12..007H. doi:10.1088/1126-6708/2000/12/007.

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[Hull-1] 1 2 Hull, C. M. (2001). "Duality in Gravity and Higher Spin Gauge Fields". Journal of High Energy Physics. 2001 (9): 27. arXiv:hep-th/0107149. Bibcode:2001JHEP...09..027H. doi:10.1088/1126-6708/2001/09/027.

[Bekaert-2] 1 2 3 4 5 Bekaert, X.; Boulanger, N.; Henneaux, M. (2003). "Consistent deformations of dual formulations of linearized gravity: A no-go result". Physical Review D. 67 (4): 044010. arXiv:hep-th/0210278. Bibcode:2003PhRvD..67d4010B. doi:10.1103/PhysRevD.67.044010.

[deWit-3] 1 2 de Wit, B.; Nicolai, H. (2013). "Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions". Journal of High Energy Physics. 2013 (5): 77. arXiv:1302.6219. Bibcode:2013JHEP...05..077D. doi:10.1007/JHEP05(2013)077.

[Curtright-4] Curtright, T. (1985). "Generalised Gauge Fields". Physics Letters B. 165 (4–6): 304. Bibcode:1985PhLB..165..304C. doi:10.1016/0370-2693(85)91235-3.

[West-5] West, P. (2012). "Generalised geometry, eleven dimensions and E₁₁". Journal of High Energy Physics. 2012 (2): 18. arXiv:1111.1642. Bibcode:2012JHEP...02..018W. doi:10.1007/JHEP02(2012)018.

[Godazgar-6] Godazgar, H.; Godazgar, M.; Nicolai, H. (2014). "Generalised geometry from the ground up". Journal of High Energy Physics. 2014 (2): 75. arXiv:1307.8295. Bibcode:2014JHEP...02..075G. doi:10.1007/JHEP02(2014)075.

[Bizdadea-7] 1 2 Bizdadea, C.; Cioroianu, E. M.; Danehkar, A.; Iordache, M.; Saliu, S. O.; Sararu, S. C. (2009). "Consistent interactions of dual linearized gravity in D = 5: couplings with a topological BF model". European Physical Journal C. 63 (3): 491–519. arXiv:0908.2169. Bibcode:2009EPJC...63..491B. doi:10.1140/epjc/s10052-009-1105-0.

[Henneaux-8] Henneaux, M.; Teitelboim, C. (2005). "Duality in linearized gravity". Physics Letters B. 71 (2): 024018. arXiv:gr-qc/0408101. Bibcode:2005PhRvD..71b4018H. doi:10.1103/PhysRevD.71.024018.

[Henneaux2-9] 1 2 Henneaux, M., "E₁₀ and gravitational duality" https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_workshops_conferences/jointerc_2014/henneaux.pdf

[Hull2-10] Hull, C. M. (2000). "Symmetries and Compactifications of (4,0) Conformal Gravity". Journal of High Energy Physics. 2000 (0012): 007. arXiv:hep-th/0011215. Bibcode:2000JHEP...12..007H. doi:10.1088/1126-6708/2000/12/007.

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