Dual graviton#Massive dual gravity

The dual formulations of linearized gravity are described by a mixed Young symmetry tensor $T\_\{\backslash lambda\_1\; \backslash lambda\_2\; \backslash cdots\; \backslash lambda\_\{D-3\}\; \backslash mu\}$, the so-called dual graviton, in any spacetime dimension D > 4 with the following characters:

{{cite journal

|last=Danehkar |first=A.

|date=2019

|title=Electric-magnetic duality in gravity and higher-spin fields

|journal=Frontiers in Physics

|volume=6 |pages=146

|bibcode=2019FrP.....6..146D

|doi=10.3389/fphy.2018.00146

|doi-access=free

}}

:$T\_\{\backslash lambda\_1\; \backslash lambda\_2\; \backslash cdots\; \backslash lambda\_\{D-3\}\; \backslash mu\}\; =\; T\_\{[\backslash lambda\_1\; \backslash lambda\_2\; \backslash cdots\; \backslash lambda\_\{D-3\}]\; \backslash mu\},$

:$T\_\{[\backslash lambda\_1\; \backslash lambda\_2\; \backslash cdots\; \backslash lambda\_\{D-3\}\; \backslash mu]\}\; =\; 0.$

where square brackets show antisymmetrization.

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

:$T\_\{\backslash alpha\backslash beta\backslash gamma\}\; =\; T\_\{[\backslash alpha\backslash beta]\backslash gamma\},$

:$T\_\{[\backslash alpha\backslash beta]\backslash gamma\}+T\_\{[\backslash beta\backslash gamma]\backslash alpha\}+T\_\{[\backslash gamma\backslash alpha\; ]\backslash beta\}\; =\; 0.$

The Lagrangian action for the spin-2 dual graviton $T\_\{\backslash lambda\_1\; \backslash lambda\_2\backslash mu\}$ in 5-D spacetime, the Curtright field, becomes

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

where $F\_\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}$ is defined as

:$F\_\{[\backslash alpha\backslash beta\backslash gamma]\backslash delta\}=\backslash partial\_\{\backslash alpha\}\; T\_\{[\backslash beta\backslash gamma\; ]\backslash delta\}+\backslash partial\_\{\backslash beta\}T\_\{[\backslash gamma\backslash alpha]\backslash delta\}\; +\backslash partial\_\{\backslash gamma\}T\_\{[\backslash alpha\backslash beta]\backslash delta\},$

and the gauge symmetry of the Curtright field is

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

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

:$E\_\{[\backslash alpha\backslash beta\backslash delta][\backslash varepsilon\backslash gamma]\}\backslash equiv\backslash frac\{1\}\{2\}$

(\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

:$E\_\{[\backslash alpha\backslash beta]\backslash gamma\}=\; g^\{\backslash varepsilon\backslash delta\}\; E\_\{[\backslash alpha\backslash beta\backslash delta][\backslash varepsilon\backslash gamma]\},$

:$E\_\{\backslash alpha\}=g^\{\backslash beta\backslash gamma\}E\_\{[\backslash alpha\backslash beta]\backslash gamma\}.$

They fulfill the following Bianchi identities

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

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

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

$\backslash mathcal\{L^\{\backslash rm\; spinless\}\_\{\backslash rm\; dual,\; massive\}\}\; =\; -\backslash frac\{1\}\{2\}u+\backslash frac\{1\}\{2\}(v-gu)^2+\backslash frac\{1\}\{3\}g(v-gu)^3\; \backslash sideset\{\_3\}\{\_2\}F(1,\backslash frac\{1\}\{2\},\backslash frac\{3\}\{2\};2,\backslash frac\{5\}\{2\};-4g^2(v-gu)^2),$

where $V^\{\backslash mu\}=\backslash frac\{1\}\{6\}\backslash epsilon^\{\backslash mu\backslash alpha\backslash beta\backslash gamma\}V\_\{\backslash alpha\backslash beta\backslash gamma\}~,\; v=V\_\{\backslash mu\}V^\{\backslash mu\}\; \backslash text\{and\}\; ~u=\backslash partial\_\{\backslash mu\}V^\{\backslash mu\}.${{Cite journal|last=Curtright|first=Thomas L.|date=2019-10-01|title=Massive dual spinless fields revisited|journal=Nuclear Physics B|volume=948|pages=114784|doi=10.1016/j.nuclphysb.2019.114784|issn=0550-3213|bibcode=2019NuPhB.94814784C|arxiv=1907.11530|s2cid=198953144}} The coupling constant $g/m$ appears in the equation of motion to couple the trace of the conformally improved energy momentum tensor $\backslash theta$ to the field as in the following equation

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

And for the spin-2 massive dual gravity in 4-D, the Lagrangian is formulated in terms of the Hessian matrix that also constitutes Horndeski theory (Galileons/massive gravity) through

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

where $K\_\backslash mu^\backslash nu=3\; \backslash partial\_\backslash alpha\; T\_\{[\backslash beta\backslash gamma]\backslash mu\}\backslash epsilon^\{\backslash alpha\backslash beta\backslash gamma\backslash nu\}$.

So the zeroth interaction part, i.e., the third term in the Lagrangian, can be read as $K\_\backslash alpha^\backslash beta\; \backslash theta\_\backslash beta^\backslash alpha$ so the equation of motion becomes

$\backslash left(\backslash Box+m^2\backslash right)T\_\{[\backslash alpha\backslash beta]\backslash gamma\}=\backslash frac\{g\}\{m\}P\_\{\backslash alpha\backslash beta\backslash gamma,\backslash lambda\backslash mu\backslash nu\}\backslash partial^\{\backslash lambda\}\backslash theta^\{\backslash mu\backslash nu\},$

where the $P\_\{\backslash alpha\backslash beta\backslash gamma,\backslash lambda\backslash mu\backslash nu\}=2\backslash epsilon\_\{\backslash alpha\backslash beta\backslash lambda\backslash mu\}\backslash eta\_\{\backslash gamma\backslash nu\}+\backslash epsilon\_\{\backslash alpha\backslash gamma\backslash lambda\backslash mu\}\backslash eta\_\{\backslash beta\backslash nu\}-\backslash epsilon\_\{\backslash beta\backslash gamma\backslash lambda\backslash mu\}\backslash eta\_\{\backslash alpha\backslash nu\}$ is Young symmetrizer of such SO(2) theory.

For solutions of the massive theory in arbitrary N-D, i.e., Curtright field $T\_\{[\backslash lambda\_1\backslash lambda\_2...\backslash lambda\_\{N-3\}]\backslash mu\}$, the symmetrizer becomes that of SO(N-2).

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

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

where

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

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

:$S\_\{\backslash rm\; BF\}\; =\; \backslash int\; d^\{5\}x\; \{\backslash cal\; L\}\_\{\backslash rm\; BF\}\; \backslash sim\; S\_\{\backslash rm\; EH\}\; =\; \{1\; \backslash over\; 2\}\; \backslash int\; \backslash mathrm\{d\}^5x\; R\; \backslash sqrt\{-g\}.$

where $g=\backslash det(g\_\{\backslash mu\backslash nu\})$ is the determinant of the metric tensor matrix, and $R$ is the Ricci scalar.

In similar manner while we define gravitoelectromagnetism for the graviton, we can define electric and magnetic fields for the dual graviton.

{{cite journal

|last1=Henneaux |first1=M.

|last2=Teitelboim |first2=C.

|date=2005

|title=Duality in linearized gravity

|journal=Physical Review D

|volume=71 |issue=2 |page=024018

|bibcode=2005PhRvD..71b4018H

|doi=10.1103/PhysRevD.71.024018

|arxiv=gr-qc/0408101|s2cid=119022015

}} There are the following relation between the gravitoelectric field $E\_\{ab\}[h\_\{ab\}]$ and gravitomagnetic field $B\_\{ab\}[h\_\{ab\}]$ of the graviton $h\_\{ab\}$ and the gravitoelectric field $E\_\{ab\}[T\_\{abc\}]$ and gravitomagnetic field $B\_\{ab\}[T\_\{abc\}]$ of the dual graviton $T\_\{abc\}$:Henneaux, M., "E₁₀ and gravitational duality"

https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_workshops_conferences/jointerc_2014/henneaux.pdf

: $B\_\{ab\}[T\_\{abc\}]=E\_\{ab\}[h\_\{ab\}]$

: $E\_\{ab\}[T\_\{abc\}]=-B\_\{ab\}[h\_\{ab\}]$

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

: $E=\backslash star\; R$

: $R=-\backslash star\; E$

where $\backslash star$ denotes the Hodge dual.

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

: $\backslash mathcal\{S\}=\backslash int\; \backslash mathrm\{d\}^6x\; \backslash sqrt\{-g\}\; C\_\{ABCD\}C^\{ABCD\},$

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.

{{cite journal

|last1=Hull |first1=C. M.

|date=2000

|title=Symmetries and Compactifications of (4,0) Conformal Gravity

|journal=Journal of High Energy Physics

|volume=2000 |issue=12 |page=007

|arxiv=hep-th/0011215

|bibcode=2000JHEP...12..007H

|doi=10.1088/1126-6708/2000/12/007

|s2cid=18326976

}}

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,{{cite journal |last1=Bampi |first1=Franco |last2=Caviglia |first2=Giacomo |date=April 1983 |title=Third-order tensor potentials for the Riemann and Weyl tensors |journal=General Relativity and Gravitation |language=en |volume=15 |issue=4 |pages=375–386 |doi=10.1007/BF00759166 |issn=0001-7701 |bibcode=1983GReGr..15..375B |s2cid=122782358 }} meanwhile Curtright tensor is a field tensor in arbitrary dimensions.

• Graviton
• Gravitino
• Gravity
• Gravitoelectromagnetism
• Curtright field
• Taub–NUT space
• Nielsen–Olesen vortex
• 't Hooft loop
• Dual photon
• Massive gravity
• Horndeski's theory
• List of hypothetical particles

{{reflist|30em}}

{{Particles}}

{{quantum gravity}}

{{String theory topics |state=collapsed}}

{{Authority control}}

Category:Bosons

Category:Gauge bosons

Category:Quantum gravity

Category:String theory

Category:Hypothetical elementary particles

Category:Force carriers