on noncommutative corrections in a de sitter gauge theory of gravity simona babeŢi (pretorian)...
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On noncommutative corrections in a de Sitter gauge theory of gravity
SIMONA BABEŢI (PRETORIAN)
“Politehnica” University, Timişoara 300223, Romania, E-mail [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 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)
,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)
1dd2
1 A, B= 0, 1, 2, 3, 5
a, b = 0, 1, 2, 3;
Commutative gauge theory of gravitation
)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
Commutative gauge theory of gravitation
• 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)
Commutative gauge theory of gravitation
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
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
aFabF
aFIf components vanish the spin connection components are determined by tetrads
ae
)x(ab
Commutative gauge theory of gravitation
•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
Commutative gauge theory of gravitation
, ,sin , 212
203
001 A
WCrCFrCVF
A
UAAF
,sin)( , 313
302
A
ZCrCFrCVF
,VF ,4UF 2301
20101
,VWF ,rCA4WUF 1302
20202
,sin)rCA4ZU(F 20303 ,sinVZF12
03
,sin)Cr4ZW1(F 2222323 ,cos)WZ(F 12
23
,A
Cr4WF 212
12
,sin)A
Cr4Z(F 213
13
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
), CrA(CZ W 0 V AAU
and we can impose the supplementary condition C = 1 .
0Fa
νb
μa
abμν eeFF
.24rC
AZ
rCA
ZU
Cr
ZW1
rC
WA
rCA
WUU2F 2
22
Commutative gauge theory of gravitation
,r
Qr
r
m21A
2
222
2R
44
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
r
1A''AA'A
r
'AA42F
[] Math.Comput.Mod. 43 (2006) 458
Commutative gauge theory of gravitation
• gauge fields of equivalent Robertson-Walker metric
),0 ,0 ,kr1
)t(a ,0(e ),0 ,0 ,0 ),t(N(e
2
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
Commutative gauge theory of gravitation
,sin)WNar(F ,VNarF ,UNkr1
aF 3
032022
101
,kr1
Z1sinaF ,
kr1
VaF
23132
212
,4 , ,1
4 20202
12022
201
01 Nrat
VF
t
YF
kr
Na
t
UF
,UYr
VF ,sin
t
ZF ,sinNra4
t
WF 02
121303
20303
,sinUZr
WF ,
kr1
ra4UV
r
YF 03
132
221212
,cos)VW(F ,sinra4UWr
ZF 03
232213
13
.sin)WVar4ZY1(F ,sint
ZF 22223
231323
Commutative gauge theory of gravitation
.a
a8akaa6F
2
222
For Λ→0
2
2
a
akaa6F
)t(N
)t(ar)r,t(W)r,t(V ,
kr1)t(N
)t(a)r,t(U
2
.kr1)r,t(Z)r,t(Y 2
0Fa
.Na
Na8NakNNaaNaa6F
32
32223
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
,sin
ar,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
m
Ja,mr2ar)r(
cosar),r(
22
222
Commutative gauge theory of gravitation
aFIf components vanish the spin connection components are determined by tetrads
ae
)x(ab
2
22
2
222
2
2sinar2'C
2
arr2'sinaB
0E
YrH
0Q
cossinaP
arcosaS
2sinaar
cosR
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
,dr
d' r
cosaV
sinarU
,cossina
W2
Commutative gauge theory of gravitation
The non-null components of the strength tensor abF
UF0101 Z
2
sinaF02
01
X2
sinaF13
01
ZF2301
ZsinaF0102 X
2F02
02
Z2
F1302
TsinaF23
02
X2
sinF03
03
Z2
sinF12
03
3
22 'r2arr4cosaZ
3
2222 'rr22sinr2a2X
3
2222
2
'r4''sinar42U
3
2222 cosa4arT
Z2
F0312
X
2F1212
UsinaF 20113
Z2
sinarF
220213
X2
sinarF
221313
ZsinaF 223
13
ZarsinF 220123 X
2
sinaF
20223
TarsinF 2223
23 Z2
sinaF
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 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.
)(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
Noncommutative gauge theory of gravitation
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(f2
i
!2
1
)x(g)x(f2
i)x(g)x(f
)x(g2
iexp)x(f)x(g)x(f
2
Noncommutative gauge theory of gravitation
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^
Noncommutative gauge theory of gravitation
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
2
a
1
aa
...)x()x()x(),x(ˆ ab
2
ab
1
abab
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
bc
cababccaba
1eFFe
4
ie
abab
1F,
4
i
The first order expressions for the gauge fields are:
BC
ACBC
ACAB,
Noncommutative gauge theory of gravitation
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(f2
i
!n
1)x(g)x(f
n1n1
nn11
n
)n(
BC)n(
ACBC)n(
ACAB)n(,
bcb
]1ac[
b]
1
ac[
b]
ac[
1
a]
1[
a
1eeeeF
ab
1
ab
1,
ab
1
ab]
1[
ab
1,,F
Noncommutative gauge theory of gravitation
bc
c1
ababc
1
ababcab
1
ab
1
cc1
abc
1
c
1
abccab
1
a
2
eFeFeF
FeFeFe8
ie
ab
1
ab
11
ab
1
ab
2F,F,F,
8
i
[] S. Fabi, B. Harms, A. Stern, Phys.Rev.D78:065037, 2008.
)x(g)x(f2
i
!n
1)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
Noncommutative gauge theory of gravitation
...)x(F)x(F)x(F),x(F ab
2
ab
1
abab
ab
11
ab
2
ab
2,
ab
11
ab
11,
ab2
ab]
2[
ab
2
,,
,,F
cddb
]ac[
ab][
abF
ab
1,
ab
1
ab1
ab]
1[
ab
1,,F
CDDB
]AC[
AB][
AB ˆˆˆF
(Moyal) * product
The noncommutative field strength tensor:
Noncommutative gauge theory of gravitation
abba
ab eeee2
1g
The noncommutative analogue of the metric tensor is:
(Moyal) * product
hermitian conjugate
baab eeg
Noncommutative gauge theory of gravitation
bab
a eFeF
The NC scalar is
ba
ba ee
aewhere is the *-inverse ofae
a2
a1
aa eeee
b1a
b
1aba
1eeeeee
b
1a1
b
11a
b2a
b
2a
b
1a
1ba2
eeeeeeeeeeee
F
Noncommutative gauge theory of gravitation
Taking
[] M.Chaichian, A. Tureanu, G. Zet, Phys.Lett. 660, 2008;
3,2,1,0,,
0000
000
000
0000
42332223200 '''ArA"AA"A'ArA5'AA2'rAA2
4
1A,xg
42211 A
''A
4
1
A
1,xg
422222222 ''AAr12'Ar16'rAA12A
16
1r,xg
42222222233 cossin'Ar2''AAr
A
'Ar'rAA24
16
1sinr,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
Noncommutative gauge theory of gravitation
F The first order corrections to the NC scalar vanish when we take space-space noncommutative parameter
2sin,'''A,''A,'A,AfFF
Having,r
r
Q
r
m21A 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 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 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
Noncommutative gauge theory of gravitation
Taking for RW
3,2,1,0,,
0000
0000
000
000
42
2
2
00kr116
aa5a61,xg
4232
222222242
2
2
11
kr116
1krkaa4ak4kr3aa16aaa12aaa13akr1
kr1
a,xg
422
22222
22kr1
aaa4ak4aa9aa8aa5aa4
16
1ar,xg
422
222
2222
33kr1
aaa4ak4aa9aa8aa5aa4
16
sinsinar,xg
4222
01kr12
aakr,xg
the noncommutative analogue of the metric tensor has only one off diagonal component.
(t-r noncommutativity)
Noncommutative gauge theory of gravitation
FThe first order corrections to the NC scalar vanish
2sin,r,a,a,a,afFF
22
2sin,r,a,a,a,af
a
akaa6F
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.
Commutative gauge theory of gravitation
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(`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 foraF
abF
νb
μa
abμν eeFF
Famn{^a miu niu} Fabmn{^a ^b miu niu}
and scalar F
Noncommutative gauge theory of gravitation
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
2
a
1
aa
...)x()x()x(),x(ˆ ab
2
ab
1
abab
>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
bc
cababccaba
1eFFe
4
ie
abab
1F,
4
i
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
ab
1F
a
1F
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(_);
abba
ab eeee2
1g
aeThe hermitian conjugate