Note on the noncommutative correction to gravity

A note on the noncommutative correction to gravity Pradip Mukherjee∗ a,b Anirban Saha a,c a Department of Physics, Presidency College 86/1 College Street, Kolkata - 700 073, India b [email protected] and c ani [email protected] An apparent contradiction in the leading order correction to noncommutative (NC) gravity reported in the literature has been pointed out. We show by direct computation that actually there is no such controvarsy and all perturbative NC corrections start from the second order in the NC parameter. The role of symmetries in the vanishing of the first order correction is manifest in our calculation. arXiv:hep-th/0605287v2 13 Jul 2006 PACS numbers: 11.10.Nx, 04.20.-q I. INTRODUCTION The idea of fuzzy space time where the coordinates xµ satisfy the noncommutative (NC) algebra [xµ , xν ] = iθµν (1) with constant anti-symmetric θµν , was mooted long ago1. This idea has been revived in the recent past and field theories defined over this NC space has been studied extensively2 . As always, a colossal challenge in this context comes from general theory of relativity (GTR). There are various attempts to fit GTR in the context of NC space time. Broadly there are two different approaches to the analysis of NC theories. One approach is to treat the fields as operators in some Hilbert space. The other approach is to define the fields over phase space with ordinary multiplication replaced by the Gronewald– Moyal product. In the later approach the original theory can be mapped to an equivalent commutative theory in the framework of perturbative expansion in the NC parameter, using the Seiberg–Witten-type maps3,4 for the fields. This commutative equivalent approach has been used to analyse many gauge theories in the recent past5 . Since gravity can be viewed effectively as a gauge theory the commutative equivalent approach seems to be a promising one . Indeed, a minimal theory of NC gravity6 has been constructed recently based on this approach where the NC correction appears as a series expansion in the NC parameter. The leading order correction is reported to be linear in θ in this work. Construction of a theory of NC gravity remains a topic of considerable current interest in the literature and various authors have approached the problem from different angles. In7 for example a deformation of Einstein’s gravity was studied using a construction based on gauging ∗ Also Visiting Associate, S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City, Calcutta -700 098, India and IUCAA, Post Bag 4, Pune University Campus, Ganeshkhind, Pune 411 007,India the noncommutative SO(4,1) de Sitter group and the SW map3 with subsequent contraction to ISO(3,1). Another construction of a noncommutative gravitational theory was proposed in8 . Very recently noncommutative gravity has been connected with stringy perspective9 . In all these works the leading order noncommutative effects appear in the second order in the NC parameter θ. It seems thus that the result of6 is in contradiction. In this brief report we extend their work to show that actually there is no such controversy. The organisation of this report is as follows. In the next section we present a brief review of the results of6 . This will be useful as the starting point of our calculation as well as to fix the notations. In section 3 details of computation of the first order correction term has been given. Section 4 contains the concluding remarks. II. REVIEW OF MINIMAL FORMULATION OF NC GRAVITY The main problem of implementing GTR on NC platform is that the algebra (1) is not invariant under general coordinate transformation. However, we can identify a subclass of general coordinate transformations, x̂µ′ = x̂µ + ξ̂ µ (x̂), (2) which are compatible with the algebra given by (1). This imposes a restriction on ξ µ θµα ∂ˆα ξˆν (x̂) = θνβ ∂ˆβ ξ̂ µ (x̂). (3) and the theory corresponds to the version of General Relativity based on volume-preserving diffeomorphism known as the unimodular theory of gravitation10 . Thus the symmetries of canonical noncommutative space time naturally lead to the noncommutative version of unimodular gravity6. With the symmetries preserved in this manner the extension of GTR to noncommutative perspective is done using the tetrad formalism and invoking the enveloping algebra method11 . The theory is then cast in the commutative equivalent form by the use of appropriate Seiberg–Witten (SW) maps. The final form of NC 2 action is S= Z d4 x 1 R̂(x̂) 2κ2 (4) R̂ is the noncommutative version of the Ricci scalar R̂ = R̂ab ab (5) where R̂ab cd are the components of the NC Riemann tensor appearing in R̂ab (x̂) = 1 R̂ cd (x̂)Σcd , 2 ab (6) The latin indices refer to the vierbein and Σcd are the generators of the local Lorentz algebra SO (3, 1). R̂ab (x̂) can be expanded as6 (1) R̂ab = Rab + Rab + O(θ2 ) (7) with (1) Rab 1 1 = θcd {Rac , Rbd } − θcd {ωc , (∂d + Dd )Rab }. (8) 2 4 In the above expression ωabc are the spin connection fields ωa (x) = 1 bc ω Σbc 2 a (9) which are antisymmetric under the exchange of b and c. Again Da is the covariant derivative i Da = ∂a + ωabc Σbc 2 (10) Note that all quantities appearing on the rhs of (8) are ordinary commutative functions. Using the above expansion we can write from (4) Z  1  S = d4 x 2 R(x) + R(1) (x) + O(θ2 ). (11) 2κ R(x) is the usual Ricci scalar and R(1) (x) is its first order correction. In the following section we will explicitly compute this correction term . III. EXPLICIT COMPUTATION OF THE FIRST ORDER CORRECTION TERM For the computation of the first order term we need an explicit form of Σcd , the generators of the local Lorentz algebra SO (3, 1). This is given by12 [Σcd ]a b = δ a c ηdb − δ a d ηcb (12) where ηab = diag (−, +, +, +). As stated, the latin indices refer to the vierbein. Our plan of calculation is as follows. Using (12) we compute the first order correction to the Ricci tensor (8). Since by finding the abcomponent of R̂ab we get R̂ on contraction, the corresponding first order correction to the NC action can now be calculated. We now proceed to compute the correction term (1) R(1) (x). First note that R(1) (x) = Rab ab . From (6) h iab (1) and (7) this is equal to Rab . We thus have to calculate the corresponding matrix element of the rhs of (8) and contract. For convenience we write the result as h iab (1) Rab = T1 + T2 (13) where T1 and T2 denote respectively the contributions coming from the first and the second terms. Now after some computation we get   T1 = 2θcd Racg a Rbd bg + Rac b g Rbd ga (14) The computation of the second term is somewhat involved. We first compute the part containing the covariant derivative as [(∂d + Dd ) Rab ]e f = 2∂d Rab e f + iωd eg Rabgf (15) Using this expression for the derivative term we compute the second term to be 
cd 1 ωc aj ∂d Rabj b − ωc aj ∂d Rba b j T2 = −θ 2 
i b aj bg aj g ωc ωdj Rabg − ωd ωc Rabjg (16) + 4 One can easily see that the first two terms cancel remembering the fact that the indices of the Riemann tensor refer to the tetrad and hence raised or lowered by ηab . Collecting all the nonvanishing terms from (14, 16) we get the correction term R(1) (x) as
 R(1) (x) = θcd 2 Racg a Rbd bg + Rac b g Rbd ga 
i − ωc aj ωdj g Rabg b − ωd aj ωc bg Rabjg (17) 4 This concludes our computation. Now one can show that all the terms of the above equation (17) individually vanishes exploiting the antisymmetry of θab and the various symmetry properties of the Riemann tensor and the spin connection fields. IV. CONCLUSION Formulation of gravity in the perspective of noncommutative (NC) space time remains a topic of considerable current interest. There is an apparent contradiction in the results reported in the literature in the sense that 3 where as in most of the works the noncommutative correction starts from the second order7,8,9 , a minimal formulation of NC gravity6 reports a first order correction. We have explicitly computed the first order correction term of the later work and demonstrated that it vanishes. This has been shown to be due to the symmetries of the various factors involved in the correction term. It appears that in the perturbative framework the order θ correction must vanish because the zero order theory carries full local Lorentz symmetry. V. support. The authors also acknowledge the hospitality of IUCAA where part of the work has been done. PS After the submission of our comment to the Physical review D there appeared a paper13 by the authors of6 where they have given results of calculation to the second order. Naturally they have also found that the first order correction vanishes. ACKNOWLEDGEMENT AS would like to thank the Council for Scientific and Industrial Research (CSIR), Govt. of India, for financial 1 2 3 4 5 6 Heisenberg first suggested this idea which was later developed by Snyder; H. S. Snyder, Phys. Rev. 71 (1947) 38; ibid 72 (1947) 874. See R. J. Szabo, Phys. Rep. 378 (2003) 207 and the references therein. N. Seiberg, E. Witten, JHEP 9909, 032 (1999) [hep-th/9908142]. A. A. Bichl, J. M. Grimstrup, L. Popp, M. Schweda, R. Wulkenhaar, [hep-th/0102103]. O. F. Dayi, B. Yapiskann, JHEP 10 (2002) 022, [hep-th/0208043]; S. Ghosh, Nucl.Phys. B 670 (2003) 359, [hep-th/0306045]; B. Chakraborty, S. Gangopadhyay, A. Saha, Phys. Rev. D 70 (2004) 107707, [hep-th/0312292]; S. Ghosh, Phys.Rev.D70 (2004) 085007, [hep-th/0402029]; P. Mukherjee, A. Saha, Mod.Phys.Lett.A21 (2006) 821, [hep-th/0409248]; P. Mukherjee, A. Saha, [hep-th/0605123]; A. Saha, A. Rahaman, P. Mukherjee, [hep-th/0603050]. X. Calmet, A. Kobakhidze, Phys. Rev. D 72 045010,2005, [hep-th/0506157] 7 8 9 10 11 12 13 A. H. Chamseddine, Phys. Lett. B 504, 33 (2001) [hep-th/0009153]. P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, J. Wess Class.Quant.Grav. 22 3511 (2005), [hep-th/0504183] L. Alvarez-Gaume, F. Meyer, M. A. Vazquez-Mozo, [hep-th/0605113 ] J. J. van der Bij, H. van Dam and Y. J. Ng, Physica 116A, 307 (1982), F. Wilczek, Phys. Rept. 104, 143 (1984); W. Buchmuller and N. Dragon, Phys. Lett. B 207, 292 (1988); M. Henneaux and C. Teitelboim,Phys. Lett. B 222, 195 (1989); W. G. Unruh, Phys. Rev. D 40, 1048 (1989). B. Jurco, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C 17, 521 (2000), [hep-th/0006246]. S. Weinberg, Gravitation and Cosmology: Principles and Applications opf the General Theory of Relativity, John Wiley & Sons. , New York, 1972. X. Calmet, A. Kobakhidze, [hep-th/0605275]