We define a theory of noncommutative general relativity for canonical
noncommutative spaces. We find a subclass of general coordinate transformations
acting on canonical noncommutative spacetimes to be volume-preserving
transformations. Local Lorentz invariance is treated as a gauge theory with the
spin connection field taken in the so(3,1) enveloping algebra. The resulting
theory appears to be a noncommutative extension of the unimodular theory of
gravitation. We compute the leading order noncommutative correction to the
action and derive the noncommutative correction to the equations of motion of
the weak gravitation field.Comment: v2: 10 pages, Discussion on noncommutative coordinate transformations
  has been changed. Corresponding changes have been made throughout the pape

A. Einstein

Archil Kobakhidze

R. Jackiw

Xavier Calmet

English

arXiv

We define a theory of noncommutative general relativity for canonical noncommutative spaces. We find a subclass of general coordinate transformations acting on canonical noncommutative spacetimes to be volume-preserving transformations. Local Lorentz invariance is treated as a gauge theory with the spin connection field taken in the so(3,1) enveloping algebra. The resulting theory appears to be a noncommutative extension of the unimodular theory of gravitation. We compute the leading order noncommutative correction to the action and derive the noncommutative correction to the equations of motion of the weak gravitation field

Calmet, Xavier

Kobakhidze, Archil

Carolina Digital Repository

arXiv:hep-th/0506157v2  27 Jun 2005Noncommutative General RelativityXavier Calmet1∗ and Archil Kobakhidze2†1Université Libre de BruxellesBoulevard du Triomphe (Campus plaine)B-1050 Brussels, Belgium2Department of Physics and AstronomyUniversity of North Carolina at Chapel HillChapel Hill, NC 27599, USAJune, 2005AbstractWe define a theory of noncommutative general relativity for canonical noncom-mutative spaces. We find a subclass of general coordinate transformations acting oncanonical noncommutative spacetimes to be volume-preserving transformations. LocalLorentz invariance is treated as a gauge theory with the spin connection field taken inthe so(3,1) enveloping algebra. The resulting theory appears to be a noncommutativeextension of the unimodular theory of gravitation. We compute the leading order non-commutative correction to the action and derive the noncommutative correction to theequations of motion of the weak gravitation field.∗calmet@physics.unc.edu†kobakhid@physics.unc.edu11 IntroductionGeneral Relativity [1] is a very successful theory when it comes to describe macroscopic effectsof gravitation. However, it is widely believed that an unification of Quantum Mechanics andGeneral Relativity requires a short distance modification of spacetime. It can be shownthat classical General Relativity considered together with Quantum Mechanics implies theexistence of a fundamental length [2]. A class of models that incorporate the notion ofa fundamental length in gauge theories are gauge theories formulated on noncommutativespaces.It is a challenge to formulate General Relativity on noncommutative spaces and thereare thus different approaches in the literature. In [3] for example a deformation of Einstein’sgravity was studied using a construction based on gauging the noncommutative SO(4,1) deSitter group and the Seiberg-Witten map with subsequent contraction to ISO(3,1). Mostrecently another construction of a noncommutative gravitational theory [4] was proposedbased on a twisted Poincaré algebra [5, 6]. These approaches although mathematicallyconsistent, are not minimal formulations of Einstein’s General Relativity on noncommutativespaces. The main problem in formulating a theory of gravity on noncommutative manifoldsis that it is difficult to implement symmetries such as general coordinate covariance and localLorentz invariance and to define derivatives which are torsion-free and satisfy the metricitycondition.Similar obstacles appear in constructing models of particle physics on flat spacetimewith canonical noncommutativity defined by the algebra [x̂a, x̂b] = iθab (θab is constant andantisymmetric). Indeed, it turns out to be rather difficult to implement most symmetriesparticle physicists are so familiar with. In particular, Lorentz invariance is explicitly violatedby the noncommutative algebra. However, it has been shown in [7] that there is anotherexact symmetry, the noncommutative Lorentz invariance, based on the usual Lorentz algebraso(3,1) which is undeformed. Another example is the implementation of noncommutativelocal gauge theories. A formulation of noncommutative gauge theories within the envelopingalgebra approach has been proposed in [8]. The fields taken in the enveloping algebra areexpanded in term of a series in θ. Each of the terms of this series is however a function ofthe classical variables [9, 10]. The number of degrees of freedom is thus finite and the sameas in the corresponding commutative gauge theory.In this work, based partially on the above achievements in implementing symmetries onflat noncommutative spacetimes, we would like to propose a theory of General Relativityon curved spacetimes with canonical noncommutativity. We shall use the tetrad approachto General Relativity. This formalism applied to noncommutative General Relativity allows2to follow closely the usual construction of noncommutative gauge theories. This requiresto implement two gauge symmetries: local Lorentz transformations which can be seen asa local gauge theory based on the algebra so(3,1) for the spin connection field and generalcoordinate transformations which are inhomogenous translations with the tetrad as a gaugefield.The gauging of noncommutative so(3,1) algebra is only possible if the correspondinggauge field, the spin connection, is assumed to be in the enveloping algebra. Hence, to im-plement local Lorentz invariance we follow the approach developed in [8]. The invarianceunder the general coordinate transformations, however, is explicitly violated by the canonicalnoncommutative algebra. Nevertheless, we find a restricted class of coordinate transforma-tions which preserve the canonical structure. It turns out that this transformations corre-spond to volume-preserving diffeomorphisms. Thus, the basic new ideas for constructing atheory of noncommutative General Relativity are to formulate local Lorentz invariance bygauging so(3,1) within the enveloping algebra approach and to introduce volume-preservingcoordinate transformations in place of general coordinate transformations, which is indeedan exact symmetry of the canonical noncommutative spacetime.2 Noncommutative General RelativityLet us start from a noncommutative spacetime and assume that the coordinates fulfill canon-ical commutation relations:[x̂µ, x̂ν ] = iθµν . (1)Obviously, the commutator (1) explicitly violates general coordinate covariance since θµν isconstant in all reference frames. However, we can identify a subclass of general coordinatetransformations,x̂µ′ = x̂µ + ξ̂µ(x̂), (2)which are compatible with the algebra given by (1). The hat on the function ξ̂(x̂) indicatesthat it is in the enveloping algebra. Under the change of coordinates (2) the commutator(1) transforms as:[x̂µ′, x̂ν′] = x̂µ′x̂ν′ − x̂ν′x̂µ′ = iθµν + [x̂µ, ξ̂ν] + [ξ̂µ, x̂ν ] + O(ξ̂2) (3)Requiring that θ remains constant yields the following partial differential equations:θµα∂̂αξ̂ν(x̂) = θνβ∂̂β ξ̂µ(x̂). (4)3A nontrivial solution to this condition can be easily found:ξ̂µ(x̂) = θµν ∂̂ν f̂(x̂), (5)where f̂(x̂) is an arbitrary field. This noncommutative general coordinate transformationcorresponds to the following classical transformation: ξ̂µ(x) = θµν∂ν f̂(x). The Jacobian ofthis restricted coordinate transformations is equal to 1, meaning that the volume elementis invariant: d4x′ = d4x. The version of General Relativity based on volume-preservingdiffeomorphism is known as the unimodular theory of gravitation [11]. Thus we came to theconclusion that symmetries of canonical noncommutative spacetime naturally lead to thenoncommutative version of unimodular gravity.Now we need to implement two gauge symmetries mentioned above. A noncommutativegauge transformation Λ̂(x̂) valued in the iso(3,1) Lie algebra can be decomposed using thegenerators of the inhomogeneous translations pµ = −i∂µ, which are anti-Hermitian, and thegenerators of the Local Lorentz algebra so(3,1) Σab, which are Hermitian. One findsΛ̂(x̂) = ξ̂(x̂) + Λ̂(x̂) = ξ̂µ(x̂)pµ +12λ̂ab(x̂)Σab, (6)where ξ̂µ is subject to the constraint (5). Note that pµ acts on the coordinates and functions,including λ̂ab, and a, b, ... run over the tangent space indices. As in [7], the algebra of gener-ators is undeformed. It is easy to verify that the commutator of two noncommutative gaugetransformations [Λ̂1(x̂), Λ̂2(x̂)] is in general not a noncommutative gauge transformation ifthe transformations are Lie algebra valued. As in the Yang-Mills case, the solution is toassume that the noncommutative gauge transformations are in the enveloping algebra. Letus introduce a noncommutative vector potential which corresponds to the noncommutativegauge transformation (6)Âa(x̂) = (D̂a) = iÊµa (x̂)pµ +i2ω̂(x̂) bca Σbc (7)where Êµa (x̂) are the components of the noncommutative tetrad Êa(x̂) , i.e. the gauge fieldscorresponding to general coordinate transformations and ω̂(x̂) bca are the spin connectionsfields associated with local Lorentz invariance. Note that Âa(x̂) plays a dual role. It canbe viewed as a covariant derivative as well. It is also worth noticing that Êa = Êµa ∂̂µ = ∂̂a,which implies that the noncommutative tetrad is mapped trivially on the commutative one:Êa = ea to all orders in θ.Let us now assume that the gauge transformations and the spin connection field are inthe enveloping algebra:Λ̂ = Λ(x) + Λ(1)(x, ωa) + O(θ2), (8)4andω̂a = ωa(x) + ω(1)a (x, ωa) + O(θ2), (9)respectively, with Λ(x) = ξµ(x)pµ +12λab(x)Σab and ωa(x) =12ω bca Σbc. We require thatthe commutator of two noncommutative gauge transformations with Λ̂1 and Λ̂2 be a gaugetransformation Λ̂ ̂Λ1×Λ2 :(δ̂Λ̂1 δ̂Λ̂2 − δ̂Λ̂2 δ̂Λ̂1)⋆ φ̂(x) =(iδ̂Λ̂1Λ̂2[ωa] − iδ̂Λ̂2Λ̂1[ωa] + [Λ̂1[ωa]⋆, Λ̂2[ωa])⋆ φ̂(x) (10)= Λ̂ ̂Λ1×Λ2 ⋆ φ̂(x).One finds as usual[Λ1, Λ2] = iΛΛ1×Λ2 (11)in the zeroth order in θ andiδΛ1Λ(1)2 − iδΛ2Λ(1)1 + iθab{∂aΛ1, ∂bΛ2} + [Λ1, Λ(1)2 ] − [Λ2, Λ(1)1 ] = Λ(1)Λ1×Λ2 (12)in the leading order in θ. A solution to this consistency equation isΛ(1)1 =14θab{∂aΛ1, ωb} (13)and analogously for Λ(1)2 and Λ(1)Λ1×Λ2 and where we have used: θab = θµνeaµebν and ∂a =eµa∂µ. Note that the consistency condition is derived in the leading order in θ and ξ(x).However since ξ(x) is itself proportional to θ, the relevant part of the volume-preservingdiffeomorphism transformation is trivial. In other words, the terms proportional to ξ(x) canbe dropped in equation (13) and it actually determines λ(1) which is the first non-trivialterm in the Seiberg-Witten map for the so(3,1) gauge transformation.The consistency condition for the spin connection is given byδΛω(1)a = ∂aΛ(1) −12θbc (∂bλ∂cωa − ∂bωa∂cΛ) + [Λ, ω(1)a ] + i[Λ(1), ωa]. (14)A solution isω(1)a = −14θbc{ωb, ∂cωa + Fca} (15)One thus has Êµa = eµa to all orders in θ and ω̂a = ωa + ω(1)a + O(θ2).The Seiberg-Witten map for the field strength is given by F̂ab = Fab +F(1)ab +O(θ2), whereF(1)ab =12θcd{Fac, Fbd} −14θcd{ωc, (∂d + Dd)Fab}, (16)5where Da = Aa = ieµapµ +i2ω bca Σbc. is the commutative covariant derivative.The commutative field strength Fab contains the Riemann tensor Rcdab as well as a torsionT cab :Fab = i[Da, Db] =12R cdab Σcd + Tcab Dc (17)with Rab =12R cdab Σcd and Tcab = (Daeνb − Dbeνa)ecν . The commutative covariant deriva-tive Da is torsion free (Tcab = 0) and compatible with a metric: eaµDaebν = 0. We nowhave all the required tools to consider actions that are invariant under general coordinatetransformations.3 Action for noncommutative General RelativityThe Seiberg-Witten map for the Riemann tensor Rab which is the field strength tensorcorresponding to a local noncommutative Lorentz transformation can be read from equations(16) and (17) where the classical torsion is being set to zero:R̂ab = Rab + R(1)ab + O(θ2), (18)withR(1)ab =12θcd{Rac, Rbd} −14θcd{ωc, (∂d + Dd)Rab}. (19)The noncommutative Riemann tensor is then given byR̂ab(x̂) =12R̂ cdab (x̂)Σcd, (20)from which we can determine the corresponding noncommutative Ricci tensor, R̂ bdab , and aRicci scalar R̂ = R̂ abab in terms of the classical fields using the above Seiberg-Witten map.The noncommutative action is then given byS =∫d4x12κ2R̂(x̂) =∫d4x12κ2(R(x) + R(1)(x))+ O(θ2). (21)In the second line we have made use of the Weyl quantization procedure which allows toreplace the noncommutative variables by commuting ones by expanding the noncommutativefields using the Seiberg-Witten maps. The only correction to leading order in θ comes fromthe Seiberg-Witten map for the so(3,1) gauge field. It is easy to verify that this actionis Hermitian and invariant under unimodular coordinate transformations and local Lorentz6transformations. This noncommutative general relativity theory is, by construction, torsionfree. In the leading order in the expansion in θ we can use the classical relations:ωabµ (x) =12ecµ(x)(Ωabc(x) − Ωb ac (x) − Ωabc (x))(22)withΩabc(x) = eaµ(x)ebν(x) (∂µeνc (x) − ∂νeµc (x)) . (23)The equation (21) represents an action for the noncommutative version of the unimodulartheory of gravitation. The unimodular theory is known [11] to be classically equivalent toEinstein’s General Relativity with a cosmological constant. Indeed, we can rewrite the action(21) in the form of an Einstein-Hilbert action by introducing a Lagrange multiplier Λ whichappears to be an arbitrary integration constant:S =∫d4x(12κ2det(eaµ(x))(R(x) + R(1)(x))+ Λ(det(eaµ(x)) − 1)+ O(θ2)). (24)In deriving (24) we have used:det⋆(eaµ(x))def=14!ǫµνρσǫabcdeaµ(x) ⋆ ebν(x) ⋆ ecρ(x) ⋆ edσ(x) = det(eaµ(x)) + O(θ2), (25)where ⋆ is the star product.Let us now consider the weak field approximation of the noncommutative action (24).Although the theory defined by the action (24) does not admit flat spacetime as a backgroundsolution, we can still locally (for regions with volume V << 1/Λ) expand the tetrad aroundflat spacetime:eµa(x) = ηµa −12hµa(x). (26)Here hµa(x) is a weak gravitational field subject to the traceless condition (det(eaµ(x)) = 1follows from (24)). The noncommutative correction to Einstein’s action in the weak fieldlimit reads18θlnR̃ abkl R̃cdmn dkmabcd −116θlnR̃ abln R̃cdkm dkmabcd (27)where dkmabcd are the structure constants defined by dkmabcdΣkm = 2{Σab, Σcd}. Notice thatR̃ abkm = −12∂k(∂ahbm +12∂bham) − ∂m(∂ahbk − ∂bhak) (28)is the leading order of the weak field approximation of R cdab . This modification of thelinearized noncommutative action implies a noncommutative correction of the equations ofmotion for the weak gravitational field:Rab −12ηabR =18θan(∂r∂kR̃cdmn + ∂r∂nR̃cdkm)dkmrbcd −18θln∂r∂lR̃cdmn damrbcd (29)7where Rab is the usual Ricci tensor and R is the corresponding Ricci scalar in the linearizedapproximation (we have omitted the contribution coming from the cosmological constant).This modification might have some interesting physical implications that will be studiedelsewhere.Finally we briefly discuss the relation between the tetrad formalism considered here and asecond-order formalism which involves the metric tensor. The noncommutative metric tensordefined naively as ĝµν(x̂) = Eaµ(x̂)Ebν(x̂)ηab is neither real nor symmetric. This raises a moregeneric question of the geometrical interpretation of the noncommutative deformation ofEinstein’s General Relativity considered above. The simple prescription to define the metricin our case is to solve the deformed equations of motion for the classical tetrads at eachgiven order of θ expansion and then to determine the metric in the standard way.4 ConclusionsWe have constructed a theory of noncommutative General Relativity on a canonical non-commutative spacetime. The general coordinate transformations is shown to be restricted tothe volume-preserving transformations. Thus the General Relativity on canonical noncom-mutative spacetimes is the noncommutative version of the unimodular theory of gravitation.The local Lorentz invariance is described as a noncommutative gauge theory by taking thespin-connection field in the enveloping algebra.The action for noncommutative General Relativity was constructed and the expansion infirst order of the noncommutative parameter θ has been calculated. We derived the noncom-mutative correction to the equations of motion of the weak gravitational field. An interestingquestion is whether the effects of spacetime noncommutativity coming from the noncommu-tative modifications of gravity are stronger than the ones coming from the modifications ofthe interactions of the standard model [12].It will also be interesting to consider classical solutions of the noncommutative action de-fined in this work. The minimal length introduced in our version of General Relativity couldhave interesting consequences for the horizon and the singularity of black holes. It will alsobe worth studying cosmological implications and the quantization of the noncommutativeaction.8AcknowledgmentsThe work of X. C. was supported in part by a scholarship of the Université libre de Bruxelles.The work of A. K. was partially supported by the GRDF grant 3305.References[1] A. Einstein, “The Foundation Of The General Theory Of Relativity,” Annalen Phys.49, 769 (1916).[2] X. Calmet, M. Graesser and S. D. H. Hsu, “Minimum length from quantummechanics and classical general relativity,” Phys. Rev. Lett. 93, 211101 (2004)[arXiv:hep-th/0405033].[3] A. H. Chamseddine, “Deforming Einstein’s gravity,” Phys. Lett. B 504, 33 (2001)[arXiv:hep-th/0009153].[4] P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp and J. Wess, “A gravitytheory on noncommutative spaces,” arXiv:hep-th/0504183.[5] J. Wess, “Deformed coordinate spaces: Derivatives,” in Lectures given at BW2003Workshop on Mathematical, Theoretical and Phenomenological Challenges Beyond theStandard Model: Perspectives of Balkans Collaboration, Vrnjacka Banja, Serbia, 29Aug - 2 Sep 2003, arXiv:hep-th/0408080.[6] M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu, “On a Lorentz-invariantinterpretation of noncommutative spacetime and its implications on noncommutativeQFT,” Phys. Lett. B 604, 98 (2004) [arXiv:hep-th/0408069].[7] X. Calmet, “Spacetime symmetries of noncommutative spaces,” Phys. Rev. D 71,085012 (2005) [arXiv:hep-th/0411147].[8] B. Jurco, S. Schraml, P. Schupp and J. Wess, “Enveloping algebra valued gauge trans-formations for non-Abelian gauge groups on non-commutative spaces,” Eur. Phys. J. C17, 521 (2000) [arXiv:hep-th/0006246].[9] J. Madore, S. Schraml, P. Schupp and J. Wess, “Gauge theory on noncommutativespaces,” Eur. Phys. J. C 16, 161 (2000) [arXiv:hep-th/0001203].[10] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP 9909,032 (1999) [arXiv:hep-th/9908142].[11] The equations of motion corresponding to this theory have first been written down byA. Einstein in: A. Einstein, Siz. Preuss. Acad. Scis., (1919); “Do Gravitational FieldsPlay an essential Role in the Structure of Elementary Particle of Matter,” by A. Ein-stein et al (Dover, New York, 1952)[Eng. translation].The theory has been rediscovered in J. J. van der Bij, H. van Dam and Y. J. Ng, “TheoryOf Gravity And The Cosmological Term: The Little Group Viewpoint,” Physica 116A,307 (1982), and further developed by a number of authors, see e.g., F. Wilczek, “Founda-tions And Working Pictures In Microphysical Cosmology,” Phys. Rept. 104, 143 (1984);9W. Buchmuller and N. Dragon, “Einstein Gravity From Restricted Coordinate Invari-ance,” Phys. Lett. B 207, 292 (1988); M. Henneaux and C. Teitelboim,“The Cosmolog-ical Constant And General Covariance,” Phys. Lett. B 222, 195 (1989); W. G. Unruh,“A Unimodular Theory Of Canonical Quantum Gravity,” Phys. Rev. D 40, 1048 (1989).[12] X. Calmet, “What are the bounds on space-time noncommutativity?,” Eur. Phys. J.C 41, 269 (2005) [arXiv:hep-ph/0401097].10

Noncommutative general relativity

Xavier Calmet (4462561)

Archil Kobakhidze (16114781)

Sussex Research Online

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We define a theory of noncommutative general relativity for canonical noncommutative spaces. We find a subclass of general coordinate transformations acting on canonical noncommutative spacetimes to be volume-preserving transformations. Local Lorentz invariance is treated as a gauge theory with the spin connection field taken in the so(3,1) enveloping algebra. The resulting theory appears to be a noncommutative extension of the unimodular theory of gravitation. We compute the leading order noncommutative correction to the action and derive the noncommutative correction to the equations of motion of the weak gravitation field. © 2005 The American Physical Society.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

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in Lectures given at the BW2003 Workshop on Mathematical, Theoretical and Phenomenologicalwith volume V 1=) expand the tetrad around flat spacetime: ea x

The equations of motion corresponding to this theory had first been written down by Albert Einstein

http://dx.doi.org/10.1103/PhysRevD.72.045010

Noncommutative General Relativity

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