On noncommutative corrections in a de Sitter gauge theory of gravity

1. On noncommutative corrections in a de Sitter gauge theory of gravity SIMONA BABEŢI (PRETORIAN) “Politehnica” University, Timişoara 300223, Romania, E-mail simona.pretorian@et.upt.ro

2. ON NONCOMMUTATIVE CORRECTIONS IN A DE SITTER GAUGE THEORY OF GRAVITY • Commutative de Sitter gauge theory of gravitation • gauge fields • the field strength tensor • Noncommutative corrections • noncommutative generalization for the gauge theory of gravitation • noncommutative gauge fields • noncommutative field strength tensor • corrections to the noncommutative analogue of the metric tensor • Analytical program in GRTensorII under Maple • procedure to implements particular gauge fields and field strength tensor calculation • procedure with noncommutative tensors

3. Commutative de Sitter gauge theory of gravitation • Gauge theory of gravitation with • the de-Sitter (DS) group SO(4,1) (10 dimensional) as local symmetry (gauge theory with tangent space respecting the Lorentz symmetry) A, B= 0, 1, 2, 3, 5 -the 10 infinitesimal generators of DS group; the DS group is important for matter couplings (see for ex. A.H. Chamseddine, V. Mukhanov, J. High Energy Phys., 3, 033, 2010) translations a, b = 0, 1, 2, 3; Lorentz rotations • the commutative 4-dimensional Minkowski space-time, endowed with spherical symmetry as base manifold: [] G. Zet, V. Manta, S. Babeti (Pretorian), Int. J. Mod. Phys. C14, 41, 2003; Commutative gauge theory of gravitation

4. 10 gauge fields (or potentials) A, B= 0, 1, 2, 3, 5 the four tetrad fields a, b = 0, 1, 2, 3; the six antisymmetric spin connection Commutative gauge theory of gravitation

5. The field strength tensor associated with the gauge fields ωμAB(x) (Lie algebra-valued tensor): ηAB = diag(-1, 1, 1, 1, 1) (the brackets indicate antisymmetrization of indices) • the torsion: ηab = diag(-1, 1, 1, 1) • and the curvature tensor: λ is a real parameter. For λ→0 we obtain the ISO(3,1), i.e., the commutative Poincaré gauge theory of gravitation. Commutative gauge theory of gravitation

6. We define The gauge invariant action associated to the gravitational gauge fields is Although the action appears to depend on the non-diagonal it is a function on only Commutative gauge theory of gravitation

7. We can adopt particular forms of spherically gauge fields of the DS group : • created by a point like source of mass m and constant electric charge Q • that ofRobertson-Walker metric • of a spinning source of mass m The non-null components of the strength tensor and If components vanish the spin connection components are determined by tetrads Commutative gauge theory of gravitation

8. gauge fields created by a point like source of mass m and constant electric charge: A, C, U, V, W, Z functions only of the 3D radius r. Depends on r and θ Commutative gauge theory of gravitation

9. Commutative gauge theory of gravitation

10. Forgauge fields of the de-Sitter group DS created by a point like source of mass m and constant electric charge Q, with constraints and we can impose the supplementary condition C = 1 . Commutative gauge theory of gravitation

11. The solution of field equations for gravitational gauge potentials with energy-momentum tensor for electromagnetic field is: For Λ→0 the solution becomes the Reissner-Nordström one. [] Math.Comput.Mod. 43 (2006) 458 Commutative gauge theory of gravitation

12. gauge fields of equivalentRobertson-Walker metric k is a constant; U, V, W, Y and Z are functions of time t and 3D radius r, Depends on t, r, θ [] G. Zet, C.D. Oprisan, S.Babeti,, Int. J. Mod. Phys. C15, 7, 2004; Commutative gauge theory of gravitation

13. Commutative gauge theory of gravitation

14. With N=1, For Λ→0 Commutative gauge theory of gravitation

15. gauge fields of a spinning source of mass m B, C, E, H, P, Q, R, S, W and Y are functions of 3D radius r and θ Depends on r and θ Commutative gauge theory of gravitation

16. If components vanish the spin connection components are determined by tetrads Commutative gauge theory of gravitation

17. The non-null components of the strength tensor Commutative gauge theory of gravitation

18. Noncommutative (NC) de Sitter gauge theory of gravitation • noncommutative scalars fields coupled to gravity • the source is not of a δ function form but a Gaussian distribution • noncommutative analogue of the Einstein equations subject to the appropriate boundary conditions • one maps (via Seinberg-Witten map – a gauge equivalence relation) the known solutions of commutative theory to the noncommutative theory Noncommutative gauge theory of gravitation

19. Noncommutative (NC) de Sitter gauge theory of gravitation corrected solutions in the NC theory solutions of commutative theory via Seinberg-Witten map (a gauge equivalence relation) Defining a NC analogue of the metric tensor one can interpret the results. The Seinberg-Witten map in a NC theory construction allows to have the same gauge group and degrees of freedom as in the commutative case. ordinary gauge variations of ordinary gauge fields inside NC gauge fields produce the NC gauge variation of NC gauge fields -infiniresimal variations under the NC gauge transformations -infiniresimal variations under the commutative gauge transformations A.H. Chamseddine, Phys. Lett. B504 33, 2001; Noncommutative gauge theory of gravitation

20. The noncommutative structure of the Minkowski space-time is determined by: real constant parameters NC field theory on such a space-time is constructed by a * product (associative and noncommutative) -- the Groenewold-Moyal product: Noncommutative gauge theory of gravitation

21. The gauge fields corresponding to de Sitter gauge symmetry for the NC case: The NC field strength tensor associated with the gauge fields separated in the two parts: ^ ‘hat’ for NC Noncommutative gauge theory of gravitation

22. The gauge fields can be expanded in powers of Θμν (with the (n) subscript indicates the n-th order in Θμν ), power series defined using the Seinberg-Witten map from the commutative gauge theory: the zeroth order agrees with the commutative theory [] K. Ulker, B. Yapiskan, Seiberg-Witten Maps to All Orders, Phys.Rev. D 77, 065006, 2008; Noncommutative gauge theory of gravitation

23. The first order expressions for the gauge fields are: Noncommutative gauge theory of gravitation

24. The noncommutative field strength tensor: (Moyal) * product having the first order corrections (of the curvature and torsion): undeformed undeformed Noncommutative gauge theory of gravitation

25. Using a relatively simple recursion relation, the second order terms for the gauge fields are first order first order first order [] S. Fabi, B. Harms, A. Stern, Phys.Rev.D78:065037, 2008. Noncommutative gauge theory of gravitation

26. The noncommutative field strength tensor: (Moyal) * product Noncommutative gauge theory of gravitation

27. The noncommutative analogue of the metric tensor is: (Moyal) * product hermitian conjugate Noncommutative gauge theory of gravitation

28. The NC scalar is where is the *-inverse of Noncommutative gauge theory of gravitation

29. Taking (r-θ noncommutativity) the noncommutative analogue of the metric tensor for the particular form of spherically gauge fields of the de-Sitter group DS created by a point like source of mass m and constant electric charge Q is: 11 For arbitrary Θμν , the deformed metric is not diagonal even if the commutative one has this property the noncommutativity modifies the structure of the gravitational field. [] M.Chaichian, A. Tureanu, G. Zet, Phys.Lett. 660, 2008; Noncommutative gauge theory of gravitation

30. The first order corrections to the NC scalar vanish when we take space-space noncommutative parameter Having The NC scalar curvature for Reissner-Nordström de Sitter solution is non-zero for deformed Schwarzschild (Λ=0, Q=0) and Reissner-Nordström (Λ=0) solution and corrected for de Sitter solution. Noncommutative gauge theory of gravitation

31. Noncommutative corrections are too small to be detectable in present day experiments, but important to study the influence of quantum space-time on gravitational effects. • red shift of the light propagating in a gravitational field (see for example Phys.Lett. 660, 2008) • For example, if we consider the red shift of the light [] propagating in a deformed Schwarzschild gravitational field (Q=0, Λ=0), then we obtain for the case of the Sun: • Δλ/λ= 2 · 10−6 − 2.19 · 10−2 4Θ2 + O(Θ4). • thermodynamical quantities of black holes (see for example [] J.HIGH ENERGY PHYS., 4, 064, 2008; • Corrected horizons radius corrected distance between horizons • Corrected Hawking-Bekenstein temperature and horizons area, corrected thermodynamic entropy of black-hole. [] S. Weinberg, Gravitation and Cosmology, John Wiley and Sons, Inc, N.Y. 1972 Noncommutative gauge theory of gravitation

32. Taking for RW (t-r noncommutativity) the noncommutative analogue of the metric tensor has only one off diagonal component. Noncommutative gauge theory of gravitation

33. The first order corrections to the NC scalar vanish • No second order corrections if the scale factor a(t)=constant; • In the case of linear expansion (a(t)=vt) we have • diagonal noncommutative analogue of the metric tensor, • small “t” can be defined using second order analysis of singular points of ordinary space time scalar curvature; • More realistic scale parameter can be analyzed in the NC model. [] S. Fabi, B. Harms, A. Stern, Phys.Rev.D78:065037, 2008. Noncommutative gauge theory of gravitation

34. The 10 (non-deformed) gauge fields (or potentials) must be defined: >grdef(`ev {^a miu}`); grcalc(ev(up,dn)); grdisplay(_); the four tetrad fields the six antisymmetric spin connection >grdef(`omega{^a ^b miu}`); grcalc(omega(up,up,dn)); >grload(mink2,`c:/grtii(6)/metrics/mink2.mpl`); >grdef(`ev {^a miu}`); grcalc(ev(up,dn)); grdisplay(_); >grdef(`omega{^a ^b miu}`); grcalc(omega(up,up,dn)); >grdef(`eta1{(a b)}`); grcalc(eta1(dn,dn)); Commutative gauge theory of gravitation

35. We implemented the GRTensor II commands for Fabmn{^a ^b miu niu} Famn{^a miu niu} F and scalar >grdef(`Famn{^a miu niu} := ev{^a niu,miu} - ev{^a miu,niu} + omega{^a ^b miu}*ev{^c niu}*eta1{b c} - omega{^a ^b niu}*ev{^c miu}*eta1{b c}`); grcalc(Famn(up,dn,dn)); grdisplay(_); >grdef(`Fabmn{^a ^b miu niu} := omega{^a ^b niu,miu}- omega{^a ^b miu,niu} + (omega{^a ^c miu} *omega{^d ^b niu} - omega{^a ^c niu}*omega{^d ^b miu})*eta1{c d} +4*lambda^2*(kdelta{^b c}*kdelta{^a d} - kdelta{^a c}*kdelta{^b d})*ev{^c miu}*ev{^d niu}`)`); grcalc(Fabmn(up,up,dn,dn)); grdisplay(_); >grdef(`evi{^miu a}`); grcalc(evi(up,dn)); >grdef(`F:=Fabmn{^a^b miu niu}*evi{^miu a}*evi{^niu b}`); grcalc(F); grdisplay(_); Commutative gauge theory of gravitation

36. The gauge fields expanded in powers of Θμν , with the (n) subscript indicates the n-th order in Θμν >grdef(`hatev{^a miu}:=ev{^a miu}+ev1{^a miu}+ev2{^a miu}`); grcalc(hatev(up,dn)); grdisplay(_); >grdef(`hatomega{^a^b miu}:=omega{^a^b miu}+omega1{^a^b miu} + omega2{^a^b miu}`); grcalc(hatomega(up,up,dn)); grdisplay(_); Noncommutative gauge theory of gravitation

37. The first order expressions for the gauge fields: >grdef(`ev1{^a miu}:=(-I/4)*Tnc{^rho^sigma}* ((omega{^a^c rho}*ev{^d miu,sigma}+ (omega{^a^c miu,sigma}+Fabmn{^a^c sigma miu} )*ev{^d rho}) *eta1{c d})`); grcalc(ev1(up,dn)); grdisplay(_); >grdef(`omega1{^a^b miu}:= (-I/4)*Tnc{^rho^sigma}* ((omega{^a^c rho}*(omega{^d^b miu,sigma}+Fabmn{^d^b sigma miu}) +(omega{^a ^c miu,sigma}+ Fabmn{^a^c sigma miu})*omega{^d^b rho} )*eta1{c d})`); grcalc(omega1(up,up,dn)); grdisplay(_); Noncommutative gauge theory of gravitation

38. The GRTensor II commands for and >grdef(`F1abmn{^a^b miu niu}:= omega1{^a^b niu,miu}- omega1{^a^b miu,niu} +(omega1{^a^c miu}*omega{^d^b niu}-omega{^a^c niu} *omega1{^d^b miu} +omega{^a^c miu}*omega1{^d^b niu}-omega1{^a^c niu} *omega{^d^b miu} +(I/2)*Tnc{^rho ^sigma}*(omega{^a^c miu,rho} *omega{^d^b niu,sigma} -omega{^a^c niu,rho}*omega{^d^b miu,sigma}) )*eta1{c d}`); grcalc(F1abmn(up,up,dn,dn)); grdisplay(_); >grdef(`F1amn{^a miu niu}:= ev1{^a niu,miu}-ev1{^a miu,niu} +(omega1{^a^c miu}*ev{^d niu} -omega1{^a^c niu}*ev{^d miu} +omega{^a^c miu}*ev1{^d niu} -omega{^a^c niu}*ev1{^d miu} +(I/2)*Tnc{^rho ^sigma}*(omega{^a^c miu,rho}*ev{^d niu,sigma} -omega{^a^c niu,rho}*ev{^d miu,sigma}))*eta1{c d} `); grcalc(F1amn(up,dn,dn)); grdisplay(_); Noncommutative gauge theory of gravitation

39. >grdef(`ev2{^a miu}:=(-I/8)*Tnc{^rho^sigma}*(omega1{^a^c rho}*ev{^d miu,sigma} +omega{^a^c rho}*(ev1{^d miu,sigma}+F1amn{^d sigma miu}) +(I/2)*Tnc{^lambda^tau}*omega{^a^c rho,lambda}*ev{^d miu,sigma,tau} +(omega1{^a^c miu,sigma}+F1abmn{^a^c sigma miu})*ev{^d rho} +(omega{^a^c miu,sigma}+Fabmn{^a^c sigma miu})*ev1{^d rho} +(I/2)*Tnc{^lambda^tau}*((omega{^a^c miu,sigma,lambda} +Fabmn{^a^c sigma miu,lambda} )*ev{^d rho,tau}))*eta1{c d}`); grcalc(ev2(up,dn)); grdisplay(_); >grdef(`omega2{^a^b miu}:=(-I/8)* Tnc{^rho^sigma}* (omega1{^a^c rho}*(omega{^b^d miu,sigma}+Fabmn{^d^b sigma miu}) +(omega{^a^c miu,sigma}+Fabmn{^a^c sigma miu})*omega1{^d^b rho} +omega{^a^c rho}*(omega1{^d^b miu,sigma}+F1abmn{^d^b sigma miu}) +(omega1{^a^c miu,sigma}+F1abmn{^a^c sigma miu})*omega{^d^b rho} +(I/2)*Tnc{^lambda^tau}*(omega{^a^c rho,lambda}*(omega{^d^b miu,sigma,tau} +Fabmn{^d^b sigma miu,tau}) +(omega{^a^c miu,sigma,lambda}+Fabmn{^a^c sigma miu,lambda})*omega{^d^b rho,tau}))*eta1{c d}`); grcalc(omega2(up,up,dn)); grdisplay(_); Noncommutative gauge theory of gravitation

40. The hermitian conjugate >grdef(`hatevc{^a miu}:=ev{^a miu}-ev1{^a miu}+ev2{^a miu}`); grcalc(hatevc(up,dn)); grdisplay(_); >grdef(`hatg{miu niu}:=(1/2)*eta1{a b}* (hatev{^a miu}*hatevc{^b niu} +hatev{^b niu}*hatevc{^a miu}+(I/2)*Tnc{^rho^sigma}* (hatev{^a miu,rho}*hatevc{^b niu ,sigma} +hatev{^b niu,rho}*hatevc{^a miu ,sigma}) +(-1/8)*Tnc{^rho^sigma}*Tnc{^lambda^tau}* (hatev{^a miu,lambda,rho}*hatevc{^b niu,tau,sigma} +hatev{^b niu,lambda,rho}*hatevc{^a miu,tau,sigma} ))`); grcalc(hatg(dn,dn)); grdisplay(_); Commutative gauge theory of gravitation