On noncommutative corrections in a de Sitter gauge theory of gravity

On noncommutative corrections in a de Sitter gauge theory of gravity

SIMONA BABEŢI (PRETORIAN)

“Politehnica” University, Timişoara 300223, Romania, E-mail [email protected]

mailto:[email protected]

Commutative gauge theory of gravitation

ON NONCOMMUTATIVE CORRECTIONS IN A DE SITTER GAUGE THEORY OF GRAVITY

Commutative de Sitter gauge theory of gravitationgauge fields

the field strength tensor

Noncommutative corrections noncommutative generalization for the gauge theory of gravitation

noncommutative gauge fieldsnoncommutative field strength tensor

corrections to the noncommutative analogue of the metric tensor

Analytical program in GRTensorII under Mapleprocedure to implements particular gauge fields and field strength

tensor calculationprocedure with noncommutative tensors


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)

,MM BAAB -the 10 infinitesimal generators of DS group;

)dsind(rdrdtds 2222222

baab MM

5aa MP 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)

1dd21

A, B= 0, 1, 2, 3, 5

a, b = 0, 1, 2, 3;


)x(e)x( a5a the four tetrad fields

)x()x( baab the six antisymmetric spin connection

10 gauge fields (or potentials) )x()x( BAAB

a, b = 0, 1, 2, 3;

A, B= 0, 1, 2, 3, 5


• the torsion:

• and the curvature tensor:

,CDDB

]AC[

AB][

ABF

The field strength tensor associated with the gauge fields ωμAB(x) (Lie algebra-

valued tensor):

ηAB = diag(-1, 1, 1, 1, 1)

ηab = 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.

bcc

]ab[

a][

aa eeTF

b]

a[

2cd

db]

ac[

ab][

abab ee4RF

(the brackets indicate antisymmetrization of indices)


νb

μa

abμν eeFF

baab eeg

)(edete aμ

FexdG16

1S 4

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

AB

g

We define


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

aFabF

aFIf components vanish the spin connection components are determined by tetrads ae

)x(ab


•gauge fields created by a point like source of mass m and constant electric charge:

),0 ,0 ,)r(A

1 ,0(e ),0 ,0 ,0 ),r(A(e 10

),θsinrC(r) ,0 ,0 ,0(e ),0 ,rC(r) ,0 ,0(e 32

),sin)r(Z ,0 ,0 ,0( ),0 ,(r)W ,0 ,0( ),0 ,0 ,0 ),r(U( 131210

,0,0,0,0 ),osc- ,0 ,0 ),r(V( 030223

A, C, U, V, W, Z functions only of the 3D radius r.

Depends on r and θ


, ,sin , 212

203

001 A

WCrCFrCVFA

UAAF

,sin)( , 313

302

AZCrCFrCVF

,VF ,4UF 2301

20101

,VWF ,rCA4WUF 1302

20202

,sin)rCA4ZU(F 20303 ,sinVZF12

03

,sin)Cr4ZW1(F 2222323 ,cos)WZ(F 12

23

,ACr4WF 212

12

,sin)ACr4Z(F 213

13


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

), CrA(CZ W 0 V AAU

and we can impose the supplementary condition C = 1 .

0Fa

νb

μa

abμν eeFF

.24rC

AZrCAZU

CrZW1

rCWA

rCAWUU2F 2

22


,rQr

rm21A 2

222

2R44

The solution of field equations for gravitational gauge potentials with energy-momentum tensor for electromagnetic field is:

)x(ea

For Λ→0 the solution becomes the Reissner-Nordström one.

4F

2

22

r1A''AA'A

r'AA42F

[] Math.Comput.Mod. 43 (2006) 458


• gauge fields of equivalent Robertson-Walker metric

),0 ,0 ,kr1

)t(a ,0(e ),0 ,0 ,0 ),t(N(e2

10

),θsinra(t) ,0 ,0 ,0(e ),0 ,ra(t) ,0 ,0(e 32

), 0, V(t,r), 0, 0(ω), 0, 0, U(t,r), 0(ω 02μ

01μ

),0,Y(t,r),0,0( ωθ), sin, W(t,r)0, 0, 0(ω 12μ

03μ

θ), cos, 0, 0, 0( ωθ), sin, Z(t,r)0, 0, 0(ω 23μ

13μ

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


,sin)WNar(F ,VNarF ,UNkr1

aF 303

2022

101

,kr1

Z1sinaF ,kr1

VaF2

3132

212

,4 , ,1

4 20202

12022

201

01 NratVF

tYF

kr

Nat

UF

,UYrVF ,sin

tZF ,sinNra4

tWF 02

121303

20303

,sinUZr

WF ,kr1

ra4UVrYF 03

132

221212

,cos)VW(F ,sinra4UWrZF 03

232213

13

.sin)WVar4ZY1(F ,sintZF 22223

231323


.a

a8akaa6F 2

222

For Λ→0

2

2

aakaa6F

)t(N)t(ar)r,t(W)r,t(V ,

kr1)t(N

)t(a)r,t(U2

.kr1)r,t(Z)r,t(Y 2

0Fa

.Na

Na8NakNNaaNaa6F 32

32223

With N=1,


• 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

,sinar,0,0,sinae,0,,0,0e

,0,0,,0e,sina,0,0,e

22232

10

),r(R,0,0),,r(S,sin),r(H,0,0),,r(E

,0),,r(Y),,r(W,0,0),r(V),,r(U,0

,),r(P,0,0),,r(Q,),r(B,0,0),,r(C

2313

1203

0201

mJa,mr2ar)r(

cosar),r(

22

222


aFIf components vanish the spin connection components are determined by tetrads ae

)x(ab

2

22

2

222

22sinar2'C

2arr2'sinaB

0E

YrH

0Q

cossinaP

arcosaS

2sinaarcosR

2

22

22222

)R,0,0,S,sinH,0,0,0

,0,Y,W,0,0,V,U,0

,P,0,0,0,B,0,0,C

2313

1203

0201

,r

,drd' r

cosaVsinarU

,cossinaW2


The non-null components of the strength tensor abF

UF0101 Z

2sinaF02

01

X2sinaF13

01

ZF2301

ZsinaF0102 X

2F02

02

Z2

F1302

TsinaF23

02

X2

sinF0303

Z

2sinF12

03

3

22 'r2arr4cosaZ

3

2222 'rr22sinr2a2X

3

2222

2'r4''sinar42U

3

2222 cosa4arT

Z2

F0312

X

2F1212

UsinaF 20113 Z

2sinarF

220213

X

2sinarF

221313

ZsinaF 223

13

ZarsinF 220123 X

2sinaF

20223

TarsinF 2223

23 Z2sinaF

21323

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 (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.

)(ˆ)(ˆ)(ˆ ABABˆ

AB

ˆ

-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


real constant parameters ix,x

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:

�

)x(g)x(f2i

!21

)x(g)x(f2i)x(g)x(f

)x(g2iexp)x(f)x(g)x(f

2


The gauge fields corresponding to de Sitter gauge symmetry for the NC case:

),,x(ea ),x(ˆ ab

The NC field strength tensor associated with the gauge fields separated in the two parts:

,Fa

abF

‘hat’ for NC^


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:

...)x(e)x(e)x(e),x(e a

2a

1aa

...)x()x()x(),x(ˆ ab

2ab

1abab

the zeroth order agrees with the commutative theory

[] K. Ulker, B. Yapiskan, Seiberg-Witten Maps to All Orders, Phys.Rev. D 77, 065006, 2008;


bc

cababccaba1

eFFe4ie

abab

1F,

4i

The first order expressions for the gauge fields are:

BC

ACBC

ACAB,


0eeTF bcc

]ab[

a][

aa

cddb

]ac[

ab][

abab RF

having the first order corrections (of the curvature and torsion):

The noncommutative field strength tensor:

undeformed

undeformed

CDDB

]AC[

AB][

AB ˆˆˆF

(Moyal) * product

)x(g)x(f2i

!n1)x(g)x(f

n1n1

nn11

n

)n(

BC)n(

ACBC)n(

ACAB)n(,

bcb

]1ac[

b]

1ac[

b]

ac[

1a

]1

[a

1eeeeF

ab

1

ab

1,

ab

1ab

]1

[ab

1,,F


bc

c1

ababc1

ababcab1

ab1

cc1

abc1

c1

abccab1

a2

eFeFeF

FeFeFe8ie

ab

1ab

11

ab

1

ab

2F,F,F,

8i

[] S. Fabi, B. Harms, A. Stern, Phys.Rev.D78:065037, 2008.

)x(g)x(f2i

!n1)x(g)x(f

n1n1

nn11

n

)n(

BC)n(

ACBC)n(

ACAB)n(,

Using a relatively simple recursion relation, the second order terms for the gauge fields are

ccab Fe

cabab eF

abF,

first order

first order

first order


...)x(F)x(F)x(F),x(F ab

2ab

1abab

ab

11

ab

2

ab

2,

ab

11

ab

11,

ab2

ab]

2[

ab2

,,

,,F

cddb

]ac[

ab][

abF

ab

1,

ab

1ab

1ab

]1

[ab

1,,F

CDDB

]AC[

AB][

AB ˆˆˆF

(Moyal) * product

The noncommutative field strength tensor:


abba

ab eeee21g

The noncommutative analogue of the metric tensor is:

(Moyal) * product

hermitian conjugate

baab eeg


bab

a eFeF

The NC scalar is

ba

ba ee

aewhere is the *-inverse ofae

a2

a1

aa eeee

b1a

b1

aba1

eeeeee

b

1a1

b1

1ab

2ab

2a

b1

a1ba

2eeeeeeeeeeee

F


Taking

[] M.Chaichian, A. Tureanu, G. Zet, Phys.Lett. 660, 2008;

3,2,1,0,,

00000000000000

42332223200 '''ArA"AA"A'ArA5'AA2'rAA2

41A,xg

42211 A

''A41

A

1,xg

422222222 ''AAr12'Ar16'rAA12A

161r,xg

42222222233 cossin'Ar2''AAr

A'Ar'rAA24

161sinr,xg

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 Θμν , the deformed metric is not diagonal even if the commutative one has this property the noncommutativity modifies the structure of the gravitational field.

11


F The first order corrections to the NC scalar vanish when we take space-space noncommutative parameter

2sin,'''A,''A,'A,AfFF

Having,r

rQ

rm21A 2

2

22

The NC scalar curvature for Reissner-Nordström de Sitter solution is

2Q,m,sin,rf4F

non-zero for deformed Schwarzschild (Λ=0, Q=0) and Reissner-Nordström (Λ=0) solution and corrected for de Sitter solution.


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 Θ2 + O(Θ4).

thermodynamical quantities of black holes (see for example [] J.HIGH ENERGY PHYS., 4, 064, 2008;

Corrected horizons radius corrected distance between horizonsCorrected 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


Taking for RW3,2,1,0,,

00000000000000

42

2

200

kr116aa5a61,xg

4232

222222242

2

211

kr116

1krkaa4ak4kr3aa16aaa12aaa13akr1kr1

a,xg

422

22222

22kr1

aaa4ak4aa9aa8aa5aa4161ar,xg

422

222

2222

33kr1

aaa4ak4aa9aa8aa5aa416

sinsinar,xg

4222

01kr12

aakr,xg

the noncommutative analogue of the metric tensor has only one off diagonal component.

(t-r noncommutativity)


FThe first order corrections to the NC scalar vanish

2sin,r,a,a,a,afFF

22

2sin,r,a,a,a,af

aakaa6F

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.


the four tetrad fields

)x()x( baab the six antisymmetric spin connection

ae

The 10 (non-deformed) gauge fields (or potentials) must be defined:

>grload(mink2,`c:/grtii(6)/metrics/mink2.mpl`);

>grdef(èv {â miu}`); grcalc(ev(up,dn)); grdisplay(_);

>grdef(òmega{â ^b miu}`); grcalc(omega(up,up,dn));

>grdef(èta1{(a b)}`); grcalc(eta1(dn,dn));

>grdef(èv {â miu}`); grcalc(ev(up,dn));

grdisplay(_);

>grdef(òmega{â ^b miu}`); grcalc(omega(up,up,dn));


>grdef(`Famn{â miu niu} := ev{â niu,miu} - ev{â miu,niu}

+ omega{â ^b miu}*ev{^c niu}*eta1{b c}

- omega{â ^b niu}*ev{^c miu}*eta1{b c}`);

grcalc(Famn(up,dn,dn)); grdisplay(_);

>grdef(`Fabmn{â ^b miu niu} := omega{â ^b niu,miu}- omega{â ^b miu,niu} +

(omega{â ^c miu} *omega{^d ^b niu}

- omega{â ^c niu}*omega{^d ^b miu})*eta1{c d} +4*lambda^2*(kdelta{^b c}*kdelta{â d}

- kdelta{â c}*kdelta{^b d})*ev{^c miu}*ev{^d niu}`)`);

grcalc(Fabmn(up,up,dn,dn)); grdisplay(_);

>grdef(èvi{^miu a}`); grcalc(evi(up,dn));

>grdef(`F:=Fabmn{â^b miu niu}*evi{^miu a}*evi{^niu b}`);

grcalc(F); grdisplay(_);

We implemented the GRTensor II commands foraF

abFνb

μa

abμν eeFF

Famn{â miu niu} Fabmn{â ^b miu niu}

and scalar F


The gauge fields expanded in powers of Θμν , with the (n) subscript indicates the n-th order in Θμν

...)x(e)x(e)x(e),x(e a

2a

1aa

...)x()x()x(),x(ˆ ab

2ab

1abab

>grdef(`hatev{â miu}:=ev{â miu}+ev1{â miu}+ev2{â miu}`);grcalc(hatev(up,dn)); grdisplay(_);

>grdef(`hatomega{â^b miu}:=omega{â^b miu}+omega1{â^b miu} + omega2{â^b miu}`);

grcalc(hatomega(up,up,dn)); grdisplay(_);


bc

cababccaba1

eFFe4ie

abab

1F,

4i

The first order expressions for the gauge fields:

>grdef(èv1{â miu}:=(-I/4)*Tnc{^rho^sigma}*((omega{â^c rho}*ev{^d miu,sigma}+(omega{â^c miu,sigma}+Fabmn{â^c sigma miu} )*ev{^d rho})

*eta1{c d})`); grcalc(ev1(up,dn)); grdisplay(_); >grdef(òmega1{â^b miu}:= (-I/4)*Tnc{^rho^sigma}*((omega{â^c rho}*(omega{^d^b miu,sigma}+Fabmn{^d^b sigma miu}) +(omega{â ^c miu,sigma}+ Fabmn{â^c sigma miu})*omega{^d^b rho} )*eta1{c d})`); grcalc(omega1(up,up,dn)); grdisplay(_);


>grdef(`F1abmn{â^b miu niu}:= omega1{â^b niu,miu}-omega1{â^b miu,niu}

+(omega1{â^c miu}*omega{^d^b niu}-omega{â^c niu}*omega1{^d^b miu}

+omega{â^c miu}*omega1{^d^b niu}-omega1{â^c niu}*omega{^d^b miu}

+(I/2)*Tnc{^rho ^sigma}*(omega{â^c miu,rho}*omega{^d^b niu,sigma}

-omega{â^c niu,rho}*omega{^d^b miu,sigma}) )*eta1{c d}`); grcalc(F1abmn(up,up,dn,dn)); grdisplay(_);

>grdef(`F1amn{â miu niu}:= ev1{â niu,miu}-ev1{â miu,niu}+(omega1{â^c miu}*ev{^d niu} -omega1{â^c niu}*ev{^d miu}+omega{â^c miu}*ev1{^d niu} -omega{â^c niu}*ev1{^d miu}+(I/2)*Tnc{^rho ^sigma}*(omega{â^c miu,rho}*ev{^d niu,sigma}-omega{â^c niu,rho}*ev{^d miu,sigma}))*eta1{c d} `); grcalc(F1amn(up,dn,dn)); grdisplay(_);

The GRTensor II commands for and

ab

1F

a1F


>grdef(èv2{â miu}:=(-I/8)*Tnc{^rho^sigma}*(omega1{â^c rho}*ev{^d miu,sigma}+omega{â^c rho}*(ev1{^d miu,sigma}+F1amn{^d sigma miu})

+(I/2)*Tnc{^lambda^tau}*omega{â^c rho,lambda}*ev{^d miu,sigma,tau}+(omega1{â^c miu,sigma}+F1abmn{â^c sigma miu})*ev{^d rho}+(omega{â^c miu,sigma}+Fabmn{â^c sigma miu})*ev1{^d rho} +(I/2)*Tnc{^lambda^tau}*((omega{â^c miu,sigma,lambda}

+Fabmn{â^c sigma miu,lambda} )*ev{^d rho,tau}))*eta1{c d}`); grcalc(ev2(up,dn)); grdisplay(_);

>grdef(òmega2{â^b miu}:=(-I/8)* Tnc{^rho^sigma}*(omega1{â^c rho}*(omega{^b^d miu,sigma}+Fabmn{^d^b sigma miu})+(omega{â^c miu,sigma}+Fabmn{â^c sigma miu})*omega1{^d^b rho}+omega{â^c rho}*(omega1{^d^b miu,sigma}+F1abmn{^d^b sigma miu})+(omega1{â^c miu,sigma}+F1abmn{â^c sigma miu})*omega{^d^b rho}+(I/2)*Tnc{^lambda^tau}*(omega{â^c rho,lambda}*(omega{^d^b miu,sigma,tau}

+Fabmn{^d^b sigma miu,tau})+(omega{â^c miu,sigma,lambda}+Fabmn{â^c sigma miu,lambda})*omega{^d^b rho,tau}))*eta1{c d}`); grcalc(omega2(up,up,dn)); grdisplay(_);


>grdef(`hatevc{â miu}:=ev{â miu}-ev1{â miu}+ev2{â miu}`); grcalc(hatevc(up,dn)); grdisplay(_);

>grdef(`hatg{miu niu}:=(1/2)*eta1{a b}* (hatev{â miu}*hatevc{^b niu}

+hatev{^b niu}*hatevc{â miu}+(I/2)*Tnc{^rho^sigma}*(hatev{â miu,rho}*hatevc{^b niu ,sigma}+hatev{^b niu,rho}*hatevc{â miu ,sigma})

+(-1/8)*Tnc{^rho^sigma}*Tnc{^lambda^tau}*(hatev{â miu,lambda,rho}*hatevc{^b niu,tau,sigma}+hatev{^b niu,lambda,rho}*hatevc{â miu,tau,sigma} ))`); grcalc(hatg(dn,dn)); grdisplay(_);

abba

ab eeee21g

aeThe hermitian conjugate

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On noncommutative corrections in a de Sitter gauge theory of gravity