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tal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 dual linearized gravity 2 massive dual gravity 3 dual graviton coupling with bf theory 4 dual gravitoelectromagnetism 5 dual graviton in conformal gravity 6 see also 7 references toggle the table of contents dual graviton 1 language 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia hypothetical particle found in supergravity dual graviton composition elementary particle family gauge boson interactions gravitation status hypothetical antiparticle self theorized 2000s 1 2 electric charge 0 e spin 2 ħ string theory fundamental objects string cosmic string brane d brane perturbative theory bosonic superstring type i type ii heterotic non perturbative results s duality t duality u duality m theory f theory ads cft correspondence phenomenology phenomenology cosmology brane cosmology landscape mathematics geometric langlands correspondence mirror symmetry monstrous moonshine vertex algebra k theory related concepts theory of everything conformal field theory quantum gravity supersymmetry supergravity twistor string theory n 4 supersymmetric yang mills theory kaluza klein theory multiverse holographic principle theorists aganagić arkani hamed atiyah banks berenstein bousso costello curtright dijkgraaf distler douglas duff dvali ferrara fischler friedan gates gliozzi gopakumar green greene gross gubser gukov guth hanson harvey hořava horowitz gibbons kachru kaku kallosh kaluza kapustin klebanov knizhnik kontsevich klein linde maldacena mandelstam marolf martinec minwalla moore motl mukhi myers nanopoulos năstase nekrasov neveu nielsen van nieuwenhuizen novikov olive ooguri ovrut polchinski polyakov rajaraman ramond randall randjbar daemi roček rohm sagnotti scherk schwarz seiberg sen shenker siegel silverstein sơn staudacher steinhardt strominger sundrum susskind t hooft townsend trivedi turok vafa veneziano verlinde verlinde wess witten yau yoneya zamolodchikov zamolodchikov zaslow zumino zwiebach history glossary v t e beyond the standard model simulated large hadron collider cms particle detector data depicting a higgs boson produced by colliding protons decaying into hadron jets and electrons standard model evidence hierarchy problem dark matter dark energy quintessence phantom energy dark radiation dark photon cosmological constant problem strong cp problem neutrino oscillation theories brans dicke theory cosmic censorship hypothesis fifth force f theory theory of everything unified field theory grand unified theory technicolor kaluza klein theory 6d 2 0 superconformal field theory noncommutative quantum field theory quantum cosmology brane cosmology string theory superstring theory m theory mathematical universe hypothesis mirror matter randall sundrum model n 4 supersymmetric yang mills theory twistor string theory dark fluid doubly special relativity de sitter invariant special relativity causal fermion systems black hole thermodynamics unparticle physics graviphoton graviscalar graviton gravitino massive gravity gauge gravitation theory gauge theory gravity cpt symmetry supersymmetry mssm nmssm superstring theory m theory supergravity supersymmetry breaking extra dimensions large extra dimensions quantum gravity false vacuum string theory spin foam quantum foam quantum geometry loop quantum gravity quantum cosmology loop quantum cosmology causal dynamical triangulation causal fermion systems causal sets canonical quantum gravity semiclassical gravity superfluid vacuum theory experiments annie gran sasso ino lhc sno super k tevatron nova v t e 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 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 e 11 generalized geometry in eleven dimensions 5 and the e 7 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 ds and anti de sitter ads spaces 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 11 dual linearized gravity edit the dual formulations of linearized gravity are described by a mixed young symmetry tensor t λ 1 λ 2 λ d 3 μ 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 t λ 1 λ 2 λ d 3 μ t λ 1 λ 2 λ d 3 μ displaystyle t_ lambda _ 1 lambda _ 2 cdots lambda _ d 3 mu t_ lambda _ 1 lambda _ 2 cdots lambda _ d 3 mu t λ 1 λ 2 λ d 3 μ 0 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 t α β γ displaystyle t_ alpha beta gamma the symmetry properties imply that t α β γ t α β γ displaystyle t_ alpha beta gamma t_ alpha beta gamma t α β γ t β γ α t γ α β 0 displaystyle t_ alpha beta gamma t_ beta gamma alpha t_ gamma alpha beta 0 the lagrangian action for the spin 2 dual graviton t λ 1 λ 2 μ displaystyle t_ lambda _ 1 lambda _ 2 mu in 5 d spacetime the curtright field becomes 2 15 l d u a l 1 12 f α β γ δ f α β γ δ 3 f α β ξ ξ f α β λ λ 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 f α β γ δ displaystyle f_ alpha beta gamma delta is defined as f α β γ δ α t β γ δ β t γ α δ γ t α β δ 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 δ σ α t α β γ 2 α σ β γ α α β γ γ α α β 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 e α β δ ε γ 1 2 ε f α β δ γ γ f α β δ ε 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 e α β γ g ε δ e α β δ ε γ displaystyle e_ alpha beta gamma g varepsilon delta e_ alpha beta delta varepsilon gamma e α g β γ e α β γ displaystyle e_ alpha g beta gamma e_ alpha beta gamma they fulfill the following bianchi identities α e α β γ g γ α e β 0 displaystyle partial _ alpha e alpha beta gamma g gamma alpha e beta 0 where g α β displaystyle g alpha beta is the 5 d spacetime metric massive dual gravity edit in 4 d the lagrangian of the spinless massive version of the dual gravity is l d u a l m a s s i v e s p i n l e s s 1 2 u 1 2 v g u 2 1 3 g v g u 3 f 3 f 2 1 1 2 3 2 2 5 2 4 g 2 v g u 2 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 v μ 1 6 ϵ μ α β γ v α β γ v v μ v μ and u μ v μ 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 g m displaystyle g m appears in the equation of motion to couple the trace of the conformally improved energy momentum tensor θ displaystyle theta to the field as in the following equation m 2 v μ g m μ θ displaystyle left box m 2 right v_ mu frac g m partial _ mu theta 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 δ ν μ g m k ν μ 1 1 2 g m 2 k α β k β α 1 3 g m 3 k α β k β γ k γ α 1 8 g m 4 k α β k β α 2 2 k α β k β γ k γ δ k δ α 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 k μ ν 3 α t β γ μ ϵ α β γ ν 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 k α β θ β α displaystyle k_ alpha beta theta _ beta alpha so the equation of motion becomes m 2 t α β γ g m p α β γ λ μ ν λ θ μ ν 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 p α β γ λ μ ν 2 ϵ α β λ μ η γ ν ϵ α γ λ μ η β ν ϵ β γ λ μ η α ν 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 t λ 1 λ 2 λ n 3 μ displaystyle t_ lambda _ 1 lambda _ 2 lambda _ n 3 mu the symmetrizer becomes that of so n 2 9 dual graviton coupling with bf theory edit dual gravitons have interaction with topological bf model in d 5 through the following lagrangian action 7 s l d 5 x l d u a l l b f displaystyle s_ rm l int d 5 x cal l _ rm dual cal l _ rm bf where l b f t r b f displaystyle cal l _ rm bf tr mathbf b wedge mathbf f here f d a r a b displaystyle mathbf f equiv d mathbf a sim r_ ab is the curvature form and b e a e b 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 s b f d 5 x l b f s e h 1 2 d 5 x r g 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 g det g μ ν displaystyle g det g_ mu nu is the determinant of the metric tensor matrix and r displaystyle r is the ricci scalar dual gravitoelectromagnetism edit in similar manner while we define gravitoelectromagnetism for the graviton we can define electric and magnetic fields for the dual graviton 17 there are the following relation between the gravitoelectric field e a b h a b displaystyle e_ ab h_ ab and gravitomagnetic field b a b h a b displaystyle b_ ab h_ ab of the graviton h a b displaystyle h_ ab and the gravitoelectric field e a b t a b c displaystyle e_ ab t_ abc and gravitomagnetic field b a b t a b c displaystyle b_ ab t_ abc of the dual graviton t a b c displaystyle t_ abc 18 15 b a b t a b c e a b h a b displaystyle b_ ab t_ abc e_ ab h_ ab e a b t a b c b a b h a b displaystyle e_ ab t_ abc b_ ab h_ ab and scalar curvature r displaystyle r with dual scalar curvature e displaystyle e 18 e r displaystyle e star r r e displaystyle r star e where displaystyle star denotes the hodge star operator dual graviton in conformal gravity edit the free 4 0 conformal gravity in d 6 is defined as s d 6 x g c a b c d c a b c d displaystyle mathcal s int mathrm d 6 x sqrt g c_ abcd c abcd where c a b c d 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 see also edit 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 references edit 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 s2cid 14739195 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 s2cid 119201330 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 e 11 journal of high energy physics 2012 2 18 arxiv 1111 1642 bibcode 2012jhep 02 018w doi 10 1007 jhep02 2012 018 s2cid 119240022 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 s2cid 15873396 ogievetsky v i polubarinov i v 1965 11 01 interacting field of spin 2 and the einstein equations annals of physics 35 2 167 208 bibcode 1965anphy 35 167o doi 10 1016 0003 4916 ...