On noncommutative corrections in a de Sitter gauge theory of gravity SIMONA BABEŢI (PRETORIAN) “Politehnica” University, Timişoara , Romania,

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

Commutative gauge theory of gravitation 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) -the 10 infinitesimal generators of DS group; translations 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; the DS group is important for matter couplings ( see for ex. A.H. Chamseddine, V. Mukhanov, J. High Energy Phys., 3, 033, 2010) A, B= 0, 1, 2, 3, 5 a, b = 0, 1, 2, 3;

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

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

Commutative gauge theory of gravitation 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 We define

Commutative gauge theory of gravitation 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 of Robertson-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 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

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

Commutative gauge theory of gravitation

For Λ→0 With N=1,

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

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

Noncommutative gauge theory of gravitation 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 Noncommutative (NC) de Sitter gauge theory of gravitation 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. -infiniresimal variations under the NC gauge transformations -infiniresimal variations under the commutative gauge transformations A.H. Chamseddine, Phys. Lett. B504 33, 2001; solutions of commutative theory ordinary gauge variations of ordinary gauge fields inside NC gauge fields produce the NC gauge variation of NC gauge fields via Seinberg-Witten map (a gauge equivalence relation) corrected solutions in the NC theory

Noncommutative gauge theory of gravitation real constant parameters The noncommutative structure of the Minkowski space-time is determined by: 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 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 The gauge fields can be expanded in powers of Θ μν (with the (n) subscript indicates the n-th order in Θ μν ), 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, , 2008;

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

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

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

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

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

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

Noncommutative gauge theory of gravitation Taking [] M.Chaichian, A. Tureanu, G. Zet, Phys.Lett. 660, 2008; 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: (r-θ noncommutativity) For arbitrary Θ μν, the deformed metric is not diagonal even if the commutative one has this property the noncommutativity modifies the structure of the gravitational field. 11

Noncommutative gauge theory of gravitation 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 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 4 Θ 2 + O(Θ 4 ). 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 Taking for RW the noncommutative analogue of the metric tensor has only one off diagonal component. (t-r noncommutativity)

Noncommutative gauge theory of gravitation 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.

using recursive relations to obtain leading order corrections in a NC gauge theory of gravitation for the NC field strength tensor the NC analogue of the metric tensor the NC scalar curvature NC corrections up to the second order for the gauge fields using the expansion in power of Θ (NC parameter) from commutative solution of the metric (tetrad fields of the Sitter gauge theory of gravitation over Minkovski spacetime) To study influence of quantum space-time on gravitational effects- procedures using GRtensorII of Maple START END to noncommutative solution of the metric (tetrad fields of the Sitter gauge theory of gravitation over NC Minkovski spacetime)

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

Commutative gauge theory of gravitation >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(_); We implemented the GRTensor II commands for Famn{^a miu niu} Fabmn{^a ^b miu niu} and scalar F

Noncommutative gauge theory of gravitation The gauge fields expanded in powers of Θ μν, with the (n) subscript indicates the n-th order in Θ μν >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 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 >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(_); The GRTensor II commands for and

Noncommutative gauge theory of gravitation >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(_);

Commutative gauge theory of gravitation >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(_); The hermitian conjugate