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
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 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)
1
dd  1
A, B= 0, 1, 2, 3, 5
-the
10
infinitesimal
MAB  MBA ,
2
generators of DS group;
Pa  Ma5
translations
a, b = 0, 1, 2, 3;
Mab  Mba
Lorentz rotations
the DS group is important for matter
couplings (see for ex. A.H. Chamseddine,
V. Mukhanov, J. High Energy Phys., 3,
033, 2010)
• the commutative 4-dimensional Minkowski space-time, endowed with spherical
symmetry as base manifold:
ds 2  dt 2  dr 2  r 2 (d2  sin 2 d2 )
[] G. Zet, V. Manta, S. Babeti (Pretorian), Int. J. Mod. Phys. C14, 41, 2003;
Commutative gauge theory of
gravitation
10 gauge fields (or potentials)
AB (x)  BA (x)
A, B= 0, 1, 2, 3, 5
a 5 (x)  ea (x)
the four tetrad fields
a, b = 0, 1, 2, 3;
ab (x )  ba (x )
the six antisymmetric spin connection
Commutative gauge theory of
gravitation
The field strength tensor associated with the gauge fields ωμAB(x) (Lie algebravalued tensor):
AB
AC DB
F
 [AB


]
[  ] CD,
ηAB = diag(-1, 1, 1, 1, 1)
(the brackets indicate
antisymmetrization of indices)
• the torsion:
a
a
c
F
 T
 [ea]  [ab
e
  ]bc
ηab = diag(-1, 1, 1, 1)
• and the curvature tensor:
ab
ab
ac db
2 a b
F
 R 
 [ab





4

e[e]
]
[ ] cd
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
g  abea eb
We define
e  det (eμa )
FF e e
ab
μν
μ
a
ν
b
The gauge invariant action associated to the gravitational gauge fields is
S
1
d4x e F

16 G
Although the action appears to depend on the non-diagonal 
AB
it is a function on
g  only
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
a
ab
F
F
The non-null components of the strength tensor   and  
a
ab
F

(x)



If components
vanish
the spin connection components
a
are determined by tetrads e
Commutative gauge theory of
gravitation
• gauge fields created by a point like source of mass m and constant electric charge:
e0
 (A(r ), 0, 0, 0),
e1
1
 (0,
, 0, 0),
A( r )
e2  (0, 0, rC(r), 0),
e3  (0, 0, 0, rC(r)sin θ),
10
  ( U(r ), 0, 0, 0),
12
  (0, 0, W(r), 0),
23  (V(r ), 0, 0, - cos),
A, C, U, V, W, Z
functions only of the
3D radius r.
13
  (0, 0, 0, Z(r ) sin ),
02  03  0, 0, 0, 0,
Depends on r and θ
Commutative gauge theory of
gravitation
F010  
AA  U
,
A
F032  rCV sin  ,
W
,
A
Z
) sin  ,
A
F133  (C  rC  
F023  rCV ,
01
F01
  U  42 ,
F122  C  rC  
23
F01
  V,
02
F02
  WU  42rCA ,
13
F02
 VW ,
03
F03
 ( ZU  42rCA ) sin ,
C
12
2
F12  W  4 r ,
A
C
13
2
F13  ( Z  4 r ) sin ,
A
12
F23
 ( Z  W ) cos ,
12
F03
 VZ sin ,
23
F23
 (1  ZW  42r 2C2 ) sin ,
Commutative gauge theory of
gravitation
WU AW 1  ZW ZU ZA


F  2  U 

 2 2 

 242 .
rCA rC
rCA rC
r C


F  Fμνab eaμ ebν
For gauge fields of the de-Sitter group DS created by a point like source of mass
m and constant electric charge Q, with Fa   0
constraints
U  AA
V0
W Z  A(C rC),
and we can impose the supplementary condition C = 1 .
Commutative gauge theory of
gravitation
a
e

The solution of field equations for gravitational gauge potentials ( x ) with energy-
momentum tensor for electromagnetic field is:
2
2
m

Q


A2  1 
   r2  2 ,
r  
r
2
 AA'

A

1
2

F  2  4
 A'  AA' '
 r
r 2 

F  4
For Λ→0 the solution becomes the Reissner-Nordström one.
[] Math.Comput.Mod. 43 (2006) 458
Commutative gauge theory of
gravitation
4  
 4 
 2 
 R 
• gauge fields of equivalent Robertson-Walker metric
e0  ( N(t ), 0, 0, 0),
e2  (0, 0, ra(t), 0),
e1  (0,
a (t )
, 0, 0),
1  kr2
e3  (0, 0, 0, ra(t)sin θ),
ωμ01  (0, U(t,r),0, 0),
ωμ02  (0, 0, V(t,r),0),
ωμ03  (0, 0, 0, W(t,r)sin θ),
ω12
μ  (0,0,Y(t,r),0),
23
ω13

(
0
,
0
,
0
,
Z(t,r)
sin
θ),
ω
μ
μ  (0, 0, 0,  cosθ),
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
1
F01

a
1  kr2
2
F12
a
2
F02
 ra  VN ,
 UN,
V
1  kr2
,
3
F03
 (ra  WN) sin ,

Z
3
F13
 a sin 1 

1  kr2


,


U
42 Na
Y
V
12
02
F 

,
F02 
,
F02 
 42 Nra,
t
t
t
1  kr 2

03  W
2
13 Z
02 V
F03  
 4 Nra  sin ,
F03 
sin , F12 
 UY,
t
r
 t

2 2 


Y
4

ra 

12
03  W
F12
 
 UV 
,
F13

 UZ sin ,
 r
2
 r

1

kr



13  Z
03
F13  
 UW  42ra2  sin ,
F23
 ( W  V) cos,
 r

13 Z
23
F23

sin ,
F23
 (1  ZY  42r 2a 2  WV) sin .
t
01
01
Commutative gauge theory of
gravitation
Fa   0
F  6
U( t , r )  
a ( t )
2
,
N( t ) 1  kr
Y( t , r )  Z( t , r )  1  kr 2 .
  kN3  a 2 N  82a 2 N3
aaN  aaN
2 3
.
a N
With
N=1,
aa  k  a 2  82a 2
F6
.
2
a
For Λ→0
F6
ra ( t )
V( t , r )  W ( t , r )  
N( t )
aa  k  a 2
a2
Commutative gauge theory of
gravitation
• gauge fields of a spinning source of mass m
 


a sin  

1



e 
, 0 ,0 ,
, e    0,
, 0 ,0 ,


 
 





r2  a2
2
3
2

e   0,0,  ,0 , e     a
sin , 0, 0,
sin  ,




0





(r, )  r 2  a 2 cos2 
(r )  r 2  a 2  2mr ,
01  C(r, ), 0, 0, B(r, ) ,
02  Q(r, ), 0, 0, P(r, ) ,
03  0, U(r, ), V(r, )0,
12
  0, W (r, ), Y(r, ),0 ,
13
  E(r, ), 0, 0, H(r, ) sin ,
23  S(r, ),0,0, R (r, ) 
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
a
J
m
a
ab
F

 vanish
If components
the spin connection components  (x)
a
are determined by tetrads e
r
R   cos

H  r
Y

E0
2
B  a sin 
C


'2r r 2  a 2  
2 2
2
2
'2r a sin   2

S  a cos

P
2
2

a sin  cos

ar sin 
 
V
a cos 

02  0, 0, 0, P ,
12
  0, W, Y,0 ,
13
  0, 0, 0, H sin ,
  a 2 sin 2  
Q0
2 2
U
03
  0, U, V, 0 ,
a
22
r2  a2  
a 2 sin  cos
W
,
 
01  C, 0, 0, B,
2
23  S,0,0, R ) Commutative gauge theory of
gravitation
' 
d

,  r 
,
dr
r

  

ab
F

The non-null components of the strength tensor
02
F01

01
F01
U
01
F02
 a sin  Z
03
F03

03
F12

2 
01
F23
Z
X
U
2 

2
  sin  r  a

Z

X
2
13
F01

13
F02

a sin 
2 

Z
2

2
23
F13
2
 a sin  Z

a  sin 2 
02
Z F23 
X
2
 

23
F02
 a sin  T
sin  
Z
2

12
F12

X
2 
01
F13
 a sin 2  U


r 2  a 2 sin 
02
F13 
Z
2 


2
13 a  sin 
23
F23 
Z F23
 sin  r 2  a 2 T
2

2a 2   2r 2 sin 2   2   2r 2  r' 
3
23
F01
Z
X
12
F03

2
2
13 r  a sin 
F 
X
13
2 
02
F02

sin  
X
2

a sin 

2   4r 2   a 2 sin 2   ' ' 2  4r'
23
Commutative gauge theory of
gravitation
Z  a cos


4r r 2  a 2    2r  ' 
3

r 2  a 2     4a 2 cos2 
T
3
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
corrected solutions
solutions of
commutative theory
via Seinberg-Witten map
(a gauge equivalence relation)
in the NC theory
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 AB ()  ˆ 
ˆ AB ()  
ˆ AB (   )

ordinary gauge variations of ordinary
gauge fields inside NC gauge fields
produce the NC gauge variation of NC
gauge fields
ˆ
-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
The noncommutative structure of the Minkowski space-time is determined by:
x


, x   i
     real constant parameters
NC field theory on such a space-time is constructed by a * product (associative and
noncommutative) -- the Groenewold-Moyal product:
 i    
f ( x )  g ( x )  f ( x ) exp      g ( x ) 
2

i
 f ( x )g ( x )     f ( x )  g ( x )
2
2
1i
        f ( x )    g ( x )  
2!  2 
Noncommutative gauge theory of
gravitation
The gauge fields corresponding to de Sitter gauge symmetry for the NC case:
eˆ a ( x, ),
ˆ ab (x, )

The NC field strength tensor associated with the gauge fields separated in the two
parts:
Fˆa  ,
^
Fˆ ab
‘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:
eˆ a ( x , )  e a ( x )  e a ( x )  e a ( x )  ...
1
2 
the zeroth order agrees with the commutative theory
ˆ ab ( x, )  ab ( x )   ab ( x )   ab ( x )  ...

1
2 
[] K. Ulker, B. Yapiskan, Seiberg-Witten Maps to All Orders, Phys.Rev. D 77, 065006, 2008;
Noncommutative gauge theory of
gravitation
The first order expressions for the gauge fields are:
 



 
i
c
ab c
e a    ab  ec  F
  ab  F
e bc
4
1
i
 ab    ,    F ab
4
1
, AB  ACCB  ACCB
Noncommutative gauge theory of
gravitation
The noncommutative field strength tensor:
AB
AC ˆ DB
ˆ AB
ˆ
Fˆ
 [ 


]
[  ] CD
(Moyal) * product
having the first order corrections (of the curvature and torsion):
undeformed
ab
ab
ac dbAC
AB
B
B
AC
F
 R 
 [ab







,









(
n
)
C
(
n
)
C
[( n)  ] cd
]

ab
ab 
F  [  ]    ,  
1
1
1

ab


 ,   
1 

ab

1
  ,  ab
a
a  ac b
ac b
ac
b 
F  [ e ]   [e]  [ e ] [ 1 e] bc
1
1  1
1

f ( x ) (n )
a i n a
a
ab c
1
F


e


e (x]
0
 
 T
g( x ) 
  [  ] [ f 
)bc g( x )
 
n!  2 
1
1
n
n
1
n
Noncommutative gauge theory of
gravitation
1
n
undeformed
Using a relatively simple recursion relation, the second order terms for the gauge fields are
first order
ab  ec  Fc












i
c
c
e a      ab  ec  F
 ab    e c  F c   ab 1  ec  F

8
2 
 1 1 
 1
first order
ab c
 ab  F
e






 c
ab
ab
c
ab
ab
c 

     ab  F ab
e




F
e




F

e
  
 


 
 1  bc
1 
1
 1

 ,    F ab

 
first order

i
ab
ab
ab
 ab     ,    F   ,      F    ,    F 1 
1
2 
1
 1

8
AC
, AB
(n ) CB  AC (n )  C B
(n )  
n
1i
f ( x ) (n ) g( x )    11  n  n  1   n f ( x ) 1   n g( x )
n!  2 
[] S. Fabi, B. Harms, A. Stern, Phys.Rev.D78:065037, 2008.
Noncommutative gauge theory of
gravitation
The noncommutative field strength tensor:
Fˆab( x , )  Fab ( x )  F ab( x )  F ab( x )  ...
1
2 
AB
ˆ AB
ˆ [AC
ˆ DB
Fˆ
 [ 
] 
  ] CD
ab
ac db
F
 [ab
 ]  [  ] cd

(Moyal) * product



ab
ab
ab
F  [  ]   , 
   ,  



1
1
1
1


ab


 ,   
1 

ab

ab
ab




ab
ab
F 
 [  ab


,






,

]
  2   ,  
  
2
2


1

1 1
1


 ,   
2 

ab


   ,  
2 

Noncommutative gauge theory of
gravitation
ab


   ,   
1 1 
ab
The noncommutative analogue of the metric tensor is:
(Moyal) * product
gˆ  

1
ab eˆ a  eˆ b  eˆ b  eˆ a
2

hermitian conjugate
g  abea eb
Noncommutative gauge theory of
gravitation
The NC scalar
Fˆ
is
ab ˆ 
Fˆ  eˆ a  Fˆ 
 eb

a
where eˆ a is the *-inverse of eˆ 
eˆ a  eˆ b   ab
eˆ a  e a  e a  e a  
1
2 


e a   e b  e a e b  e a 1 e b 
1
 1


  b


b

b

b

b
e a   e b  e a e   e a e   e a 2 e   e a 1 e   e a 1 e  
2
2
1 1
 1 1

Noncommutative gauge theory of
gravitation
Taking


0 0
0 0


 0 0  0

,
0   0 0


0 0
0 0 

(r-θ noncommutativity)
,   0,1, 2, 3
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:

gˆ 00 x,    A 2 

 
1
2rAA '3 2A 2 A'2 5rA 2 A' A" A3A" rA 3A' ' '  2    4
4
1
1 A' ' 2
gˆ 11x,  

   4
A2 4 A
 
11
gˆ 22 x,    r 2 


 
1 2
A  12 rAA '16 r 2 A'2 12 r 2 AA ' '  2    4
16

1 
A'

gˆ 33 x,   r 2 sin 2    4 2rAA' r  r 2 AA' '2r 2 A' 2  sin 2   cos2   2    4
16  
A


μν
For arbitrary Θ , 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
The first order corrections to the NC scalar
space-space noncommutative parameter
Fˆ
vanish when we take
Fˆ  F  f A, A' , A' ' , A' ' ' , sin  2  
Having
2
2
m
Q

A2  1 

 r2,
r
r2 
The NC scalar curvature for Reissner-Nordström de Sitter solution is
Fˆ  4  f r, sin , m, Q2  
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 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
 0  0 0




0
0
0


  
,
0
0 0 0


 0

0
0
0


Taking for RW
(t-r noncommutativity)
,   0,1, 2, 3
the noncommutative analogue of the metric tensor has only one off diagonal component.

6a2  5a a 2
gˆ 00 x ,    1 
   4 
16 1  kr2 
gˆ 11x ,   
a2
1  kr

2
1  kr2 a 4  13aa 2a  12a 2a a  16a 2a 2  3kr2  4ka 2  4aakkr2  12  4 
3
161  kr2 
1 
a 8aa2  9a 2a  4ka  4aa a 2
2
2
2
gˆ 22 x,   r a 
4aa  5a 
  4 
16 


a 8aa 2  9a 2a  4ka  4aa a  2
sin 2  
2
2
2
2
gˆ 33 x,   r a sin  
4aa  5a 
   4

16 
1  kr 2

gˆ 01x,    

kra a
2 1  kr

2 2
1  kr2

 
2   4
Noncommutative gauge theory of
gravitation

 
The first order corrections to the NC scalar
Fˆ
vanish
Fˆ  F  f a, a , a,a, r, sin 2  
Fˆ  6
aa  k  a 2
a2
 f a, a , a,a, r, sin 2  
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
The 10 (non-deformed) gauge fields (or potentials) must be defined:
ea
the four tetrad fields
>grdef(`ev {^a miu}`); grcalc(ev(up,dn));
grdisplay(_);
ab (x )  ba (x )
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
We implemented the GRTensor II commands for
Fab
Famn{^a miu niu}
Fa 
and scalar
ab μ ν
F  Fμν
ea e b
Fabmn{^a ^b miu niu}
F
>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
The gauge fields expanded in powers of Θμν , with the (n) subscript indicates the n-th
order in Θμν
eˆ a ( x , )  e a ( x )  e a ( x )  e a ( x )  ...
1
2 
ˆ ab ( x, )  ab ( x )   ab ( x )   ab ( x )  ...

1
2 
>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:
 

 
i
c
ab c
e a    ab  ec  F
  ab  F
e bc
4
1
i 
ab
     ,    F 
1
4
ab

>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
The GRTensor II commands for
F ab
1
and
F a 
1
>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
>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
The hermitian conjugate
eˆ a 
gˆ   

1
ab eˆ a  eˆ b  eˆ b  eˆ a
2

>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