Field Equations and Radial Solution in a Noncommutative Spherically Symmetric Geometry
aa r X i v : . [ phy s i c s . g e n - ph ] D ec Field Equations and Radial Solutions in aNon-commutative Spherically Symmetric Geometry
Aref Yazdani Department of Physics, Faculty of Basic Sciences,University of Mazandaran,P. O. Box 47416-95447, Babolsar, IRAN
Abstract
We study a noncommutative theory of gravity in the framework of torsional spacetime.This theory is based on a Lagrangian obtained by applying the technique of dimensionalreduction of non-commutative gauge theory, and that the yielded diffeomorphism invariantfield theory can be made equivalent to a teleparallel formulation of gravity. Field equa-tions are derived in the framework of teleparallel gravity through Weitzenb¨ok geometry.We solve these field equations by considering a mass that is distributed spherically sym-metric in a stationary static spacetime in order to obtain a noncommutative line element.This new line element interestingly reaffirms the coherent state theory for a noncommu-tative Schwarzschild black hole. For the first time, we derive the Newtonian gravitationalforce equation in the commutative relativity framework, and this result could provide thepossibility to investigate examples in various topics in quantum and ordinary theories ofgravity. [email protected] Introduction
Field equations of gravity and radial solutions have been previously derived in noncommutativegeometry [1,2]. The generalization of quantum field theory by noncommutativity based on co-ordinate coherent state formalism also cures the short distance behavior of point-like structures[3-7]. In this method, the particle mass M instead of being completely localized at a point,is dispensed throughout a region of linear size √ θ , substituting the position Dirac-delta func-tion, describing point-like structures, with a Gaussian function, describing smeared structures.In other words, we assume the energy density of a static, spherically symmetric, particle-likegravitational source can not be a delta function distribution and will be given by a Gaussiandistribution of minimal width √ θ as follows: ρ θ ( r ) = M (4 πθ ) / exp( − r / θ ) (1)Furthermore, noncommutative gauge theory appears in string theory [8-13]: the boundary the-ory of an open string is noncommutative when it ends on D-bran with a constant B-field or anAbelian gauge field (particularly see Ref. [8]). Therefore, closed string theories are expected toremain commutative as long as the background is geometric. Recent evidence has found a con-nection between non-geometry and closed string noncommutativity and even non-associativity[14-16]; approaches using dual membrane theories [17] and matrix models [18] arrive at thesame conclusion.The ordinary quantum field theory is unable to present an exact description of exotic effects ofthe inherent non-locality of interactions, so we need a model to provide an effective descriptionof many of the non-local effects in string theory within a simpler setting [19]. The model leadsto the gauge theories of gravitation through an ordinary class of dimensional reductions ofnoncommutative electrodynamics on flat space, which then can be made equivalent to a formu-lation of teleparallel gravity, macroscopically describing general relativity. Moreover, this modelis developed by the parallel theories of gravitation, giving a clear understanding of Einstein’sprinciple of absolute parallelism. It is defined by a non-trivial vierbein field and formed by alinear connection. For carrying non-vanishing torsion, this connection is known as Wietzenb¨ockgeometry on spacetime.This model is given appropriately by a noncommutative Lagrangian and introduced by au-thors in Ref. [2]. Admittedly, This Lagrangian and the relevant explanations will be thebasis of our next general calculations. In this paper are going to use the Greek alphabet( µ, ν, ρ, ... = 0 , , ,
3) to denote indices related to spacetime, and the first half of the Latin2lphabet ( a, b, c, ... = 0 , , ,
3) to denote indices related to the tangent space. A Minkowskispacetime whose Lorentz metric is assumed to have the form of η ab = diag ( − , +1 , +1 , +1).The middle letters of the Latin alphabet ( i, j, k, ... = 1 , ,
3) will be reserved for space indices.The noncommutative Lagrangian is expressed as˙ L Gr = χ e det( h σ ′ σ ) η µµ ′ h η νν ′ η λλ ′ ˙ T λµν ˙ T λ ′ µ ′ ν ′ − ˙ T νµν ˙ T ν ′ µ ′ ν ′ + 12 ˙ T ν ′ µν ˙ T νµ ′ ν ′ i (2)In the usual way, having a Lagrangian, which describes gravitation based on noncommutativebackground, is like those of gauge theories written in terms of contractions of its field strength,here represented by torsion of Weitzenb¨ock connection. Its behavior under a local change of∆ µ is the main invariance property of the particular combination torsion tensor fields. Here e is Yang-Mills coupling constant, noncommutative scale determines the Planck length, and thePlanck scale of n-dimensional spacetime is given by k = q πG N = e | P f af f (Θ AB ) | / n (3)In mass dimension 2 the weight constant χ is χ = | P f af f (Θ AB ) | − /n (4)In which the commutative limit, it reduces to gravitational constant. therefore, Θ AB is anoncommutative parameter, defined asΘ AB = θ µν θ µb θ µb θ ab ! → θ µν = θ ab = 0 (5)By considering the calculation of superpotential and energy-momentum current with respectto noncommutative gauge potential, given by B µa = | det( θ µ ′ a ′ ) | / n ˆ θ νµ ω aν , the version of non-commutative gravitational field equations are produced. ω aν are gauge fields corresponding tothe gauging of the translation group, i.e., replacing R n by the Lie algebra g of local gaugetransformations with gauge functions and its relation with the verbien field is expressed as: h µa = δ µa − eθ νµ ω aν and δ µa has the perturbative effect in the trivial holonomic tetrad fields offlat space.It is important to note that by applying the ”dimensional reduction of gauge theories”, non-commutative electrodynamics gauge field; shown by the noncommutative Yang-Miles theory,reduces to the gauge theories of gravitation , which naturally yields Weitzenb¨ock geometry onthe spacetime. Also, the induced diffeomorphism invariant field theory can be made equivalent3o a teleparallel formulation of gravity macroscopically describing general relativity. In section2 we show that our Lagrangian can be made equivalent with general relativity. In section 3we are going to derive the field equations by utilizing various definitions of teleparallel gravity.By simplifying and solving the field equations, we obtain the line element in the sphericallysymmetric space-time in section 4. We continue our discussion with investigations about thelimiting cases of our line element and horizons of noncommutative Schwarzschild black hole inthis method. Finally we show how the Newtonian gravitational force equation can be derivedfrom our line element in the commutative limit in section 5. In order to continue our discussion to achieve to noncommutative field equations, we shouldshow how our model can be coupled with general relativity. With respect to the given relationof ˙Γ ρµν = Γ ρµν + ˙ K ρµν (6)for the vanishing curvature of the Weitzenb¨ock connection, we have˙ R ρθµν = R ρθµν + ˙ Q ρθµν ≡ R ρθµν = ∂ µ Γ ρθν − ∂ ν Γ ρθν + Γ ρσµ Γ σθν − Γ ρσν Γ σθµ (8)is the curvature of the Levi-Civita connection, The above equations show that, whereas ingeneral relativity torsion vanishes, in teleparallel gravity it is curvature that vanishes. Werewrite the Eq.(7) based on their components in order to find the scaler of ˙ R ρθµν , therefore wehave˙ Q ρθµν = ( ∂ µ ˙ K ρθν − ∂ ν ˙ K ρθµ + ˙ T ρσµ ˙ K σθν − ˙Γ ρσν ˙ K σθµ − ˙Γ σθµ ˙ K ρσν + ˙Γ σθν ˙ K ρσµ ) + ˙ K ρσν ˙ K σθµ − ˙ K ρσµ ˙ K σθν (9)that is the tensor written in terms of the Weitzenb¨ock connection only. Like the Riemaniancurvature tensor, it is a 2-form assuming values in the Lie algebra of the Lorentz group (seeRef. [21]). By taking appropriate contractions it is easy to show that˙ Q ρθµν = ( ˙ D µ ˙ k ρθν − ˙ D ν ˙ k ρθµ ) + ˙ K ρσν ˙ K σθµ − ˙ K ρσµ ˙ K σθν (10)4y considering the Eq.(20) and the following term − R = ˙ Q ≡
12 Λ θρ ˙ Q ρθµν dx µ ∧ dx ν (11)we achieve to the scalar version of Eq.(7), R ≡ ( ˙ K µνρ ˙ K ρνµ − ˙ K νµρ ˙ K µρν ) + 2 h ∂ µ ( h ˙ T νµν ) . (12)The Lagrangian of Eq.(2) can be written in a simple form of˙ L = χ e det( h σ ′ σ ) (cid:18) ˙ K µνρ ˙ K ρνµ − ˙ K νµρ ˙ K µρν (cid:19) (13)with a combination of Eqs.(12) and (13), ˙ L takes the following form˙ L = χ e det( h σ ′ σ ) (cid:18) R − h ( ∂ µ ( h ˙ T νµν )) (cid:19) . (14)By considering the Eqs.(3,4) ˙ L exchanges to˙ L = L − ∂ µ ( h πG ˙ T νµν ) (15)up to a divergence at the commutative limit; therefore, the Lagrangian of Eq.(2) ˙ L is equivalentto the Lagrangian of general relativity as follows˙ L = − πG √− gR (16)is the Einstein-Hilbert Lagrangian of general relativity. However, this result could be extendedwith many further terms, but this is enough to derive a valid field equations. In this section, we are going to present a reformulation of teleparallel gravity, (is made equiv-alent to general relativity). Due to the introduced noncommutative Lagrangian (2), we areable to derive the field equations similarly to the teleparallel method. Weitzenb¨ock geometricdefinitions and some well-known concepts of general relativity [22-24] and teleparallel gravityare required, (more explanations about these equations can be found in Ref. [23],[25],[26]). In4-dimension, the noncommutative action integral is given by S = Z ˙ L Gr d x (17)5nder an arbitrary variation δh µa of the tetrad field, the action variation is written in thefollowing form δS = Z Ξ aµ δh µa hd x (18)Where h Ξ aµ = δ ˙ L Gr δB µa ≡ δ ˙ L Gr δh µa = ∂ ˙ L Gr ∂h µa − ∂ λ ∂ ˙ L Gr ∂ λ ∂h µa (19)is the matter energy-momentum tensor. (More definitions about this tensor can be found inRef. [27]). Now, consider first an infinitesimal Lorentz transformation asΛ ba = δ ba + ε ba (20)With ε ba = − ε ba . because of such transformation the tetrad should be changed as δh µa = ε ba h µb (21)The requirement of invariance of the action under local Lorentz transformation therefore yields Z Ξ ba ε ba hd x = 0 (22)Since ε ba is antisymmetric, symmetric of energy-momentum tensor yields some specific resultsthat can be seen in Ref. [22]. Consider spacetime coordinates that are transformed as follows x ′ ρ = x ρ + ζ ρ (23)Whereby, we retrieve the tetrad in the form of δh µa ≡ h ′ µa ( x ) − h µa ( x ) = h ρa ∂ ρ ζ µ − ζ ρ ∂ ρ h µa (24)Substituting in Eq.(18), we have δS = Z Ξ aµ [ h aρ ∂ µ ζ ρ − ζ ρ ∂ ρ h µa ] hd x (25)or equivalently δS = Z [Ξ ρc ∂ ρ ζ c + ζ c Ξ ρµ ∂ ρ h µc − ζ ρ ∂ ρ h µa ] hd x (26)Substituting the identity ∂ ρ h µa = ˙ A baρ h µb − ˙Γ µλρ h λa (27)where ˙ A is the spin connection in teleparallel gravity. The important property of teleparallelgravity is its spin connection is related only to the inertial properties of the frame, not to grav-itation. In fact, it is possible to choose an appropriate frame in which it vanishes everywhere.6e know the above formula vanishes by the Eq.(42), (see also Ref. [28]), and making use ofthe symmetric of the energy-momentum tensor, the action variation assumes the form of δS = Z Ξ ρc [ ∂ ρ ζ c + ( ˙ A cbρ − ˙ K cbρ ) ζ b ] hd x (28)Integrating the first term by parts and neglecting the surface term, the invariance of the actionyields Z [ ∂ µ ( h Ξ µa ) − ( ˙ A baµ − ˙ K baµ )( h Ξ µb )] ζ a hd x = 0 (29)From arbitrariness of ζ c , under the covariant derivative ¨ D µ , it follows that¨ D µ h Ξ µa ≡ ∂ µ ( h Ξ µa ) − ( ˙ A baµ − ˙ K baµ )( h Ξ µb ) = 0 (30)By identity of ∂ ρ h = h ˙Γ ννρ ≡ h ( ˙Γ νρν − ˙ K νρν ) (31)the above conservation law becomes ∂ µ Ξ µa + ( ˙Γ µρµ − ˙ K µρµ )Ξ ρa − ( ˙ A baµ − ˙ K baµ )Ξ µb = 0 (32)In a purely spacetime form, it reads¨ D µ Ξ µλ ≡ ∂ µ Ξ µλ + ( ˙Γ µµρ − ˙ K µρµ )Ξ ρλ − ( ˙Γ ρλµ − ˙ K ρλµ )Ξ µρ = 0 (33)This is the conservation law of the source of energy-momentum tensor. Variation with respectto the noncommutative gauge potential B µa yields the noncommutative teleparallel version ofthe gravitational field equations ∂ σ ( h ˙ S µσa ) − kh ˙ J µa = kh Ξ µa (34)where h ˙ S µσa = hh λa ˙ S µσλ ≡ − k ∂ ˙ L∂ ( ∂ σ h aµ ) (35)which defines the superpotential, For the gauge current we have h ˙ J µa = − ∂ ˙ L∂B aµ ≡ − ∂ ˙ L∂h aµ (36)Note that the matter energy-momentum tensor which is defined in this relation appears as thesource of torsion; similarly, the energy-momentum tensor appears as the source of curvature ingeneral relativity. Our computation has led us to the following results:˙ S µσa = 2 ˙ T µσa − ˙ T σµa − h σa ˙ T ηµη + h µa ˙ T ηση (37)7nd ˙ J µa = 1 k h λa ˙ S νµc ˙ T cνλ − h µa h ˙ L + 1 k ˙ A caσ ˙ S µσc (38)for noncommutative superpotential and gauge current. The lagrangian ˙ L appears again inour equations, but notice that this term cross its coefficient yields a term purely based on fieldstrength. This simplified expression maintains equivalence to general relativity. We can observethat the gravitational field equations depend on the torsion only. Finally the field equationscan be written as: ∂ σ h (2 ˙ T µσa − ˙ T σµa − h σa ˙ T ηµη + h µa ˙ T ηση ) ! − kh k h λa ˙ S νµc ˙ T cνλ − h µa h ˙ L + 1 k ˙ A caσ ˙ S µσc ! = kh Ξ µa (39)Where k = χ e is a constant. These field equations are similar to teleparallel field equations.Although it would be distinguished with different field strength ˙ T µσa which is given by thecovariant rotational of noncommutative gauge potential of B µa . By considering the followingequations from the teleparallel theory (see for instance,[20],[26],[28])˙ T aµν = ∂ ν h aµ − ∂ µ h aν + ˙ A aeν h eµ − ˙ A aeµ h eν (40)˙Γ ρνµ = h ρa ∂ µ h aν + h ρa ˙ A abµ h bν (41) ∂ µ h aν − ˙Γ ρνµ h aρ + ˙ A abµ h bν = 0 (42)and ˙ T ρνµ = ˙Γ ρµν − ˙Γ ρνµ (43)The field equations take the exact following form ∂∂x σ ( ˙Γ σaµ − ˙Γ σµa ) − ∂∂x µ ˙Γ λaλ + ∂∂x λ ˙Γ λaµ − ˙Γ ηaλ ˙Γ λµη + ˙Γ ηaµ ˙Γ λλη = χ e ρ ( r ) ∂∂x a ∂∂x µ , (44)which unlike the left hand side of Eq.(39), is written purely based on noncommutative fieldstrength, the above field equation is written in terms of Weitzenb¨ock connection only. Regardingto the equivalency between corresponding Lagrangians and the above simplified field equationsand applying the Eq.(34), we have therefore R aµ − h aµ R = k Ξ aµ (45)as equivalent with Einstein’s field equations. Note that the equation (45) is not Einstein’sfield equations but the teleparallel field equations made equivalent to general relativity. Andequivalent model of teleparallel field equations with general relativity expressed in referencesin detail, (see for instance [26],[28]). We continue our discussion to derive noncommutative lineelement by solving these field equations. 8 Noncommutative Line Element
Teleparallel versions of the stationary, static, spherically, axis-symmetric, and symmetric of theSchwarzschild solution have been previously obtained [31],[32]. Within a framework inspiredby noncommutative geometry, We solve the field equations for a distribution of sphericallysymmetrically mass in a stationary static spacetime, like the exterior solution of Schwarzschild(see also Ref. [23]). Then it is natural to assume that the line element is as follows ds = − f (˜ r ) dt + g (˜ r ) dr + h (˜ r )˜ r ( dθ + sin θdφ ) . (46)With a new radial coordinate defined as r = ˜ r q h (˜ r ), the line element becomes ds = − A ( r ) dt + B ( r ) dr + r ( dθ + sin θdφ ) . (47)Usually one replaces the functions A ( r ) and B ( r ) with exponential functions to obtain some-what simpler expressions for the noncommutative tensor components. Hence, we introduce thefunctions α ( r ) and β ( r ) by e α ( r ) = A ( r ) and e β ( r ) = B ( r ) to get ds = − e α dt + e β dr + r ( dθ + sin θdφ ) . (48)Tetrad components of the above metric takes the following form: h aµ = − e α e β sin θ cos φ r cos θ cos φ − r cos θ sin φ e β sin θ sin φ r cos θ sin φ r sin θ cos φ e β cos θ − r sin θ (49)Weitzenb¨ock connection ˙Γ ρµν has following expression:˙Γ ρµν = h ρa ∂ ν h aµ (50)Now, we can calculate the non-vanishing components of Weitzenb¨ock connection as follows:Γ = − α ′ , Γ = 2 β ′ , Γ = − re − β , Γ = − re − β sin θ, Γ = e α r = Γ , Γ = 1 r = Γ , Γ = − sin θ cos θ, Γ = Γ = cot θ. (51)9y replacing these components in Eq. (44), the noncommutative tensors of Eqs.(52-54) for theleft-hand side of the field equations will produce the following expression N ˆ t ˆ t = 1 r ( − e − β + 1 − ψ θ ) − r β ′ e − β = χ e ρ ( r ) δ ˆ t ˆ t (52) N ˆ r ˆ r = 1 r (2 e − β + 1 − ψ θ ) + 2 r α ′ e − β = χ e ρ ( r ) δ ˆ r ˆ r (53) N ˆ θ ˆ θ = N ˆ φ ˆ φ = 1 r e − β ( rα ′′ + rα ′ − rα ′ β ′ + α ′ − β ′ −
1) + α ′ e − β = χ e ρ ( r ) δ ˆ θ ˆ θ = χ e ρ ( r ) δ ˆ φ ˆ φ (54)Adding equations (52) and (53) we get simply1 r ( ψ θ − e − β + e − β ( α ′ − β ′ ) + 1) = χ e ρ ( r ) (55)where α ( r ) = β ( r ). It should also be noted that by recalling Eq.(48), we can consider thelimiting case for our solution assuming ( α ′ − β ′ ) = k , which k is a constant, and by consideringthe time-coordinate, we can shift this constant to an arbitrary value. It is possible, therefore,without loss of generality to choose k = 0. It does not contradict with Eq.(48) to set α ′ = β ′ .According to this analysis, the equation N ˆ t ˆ t = χ e ρ ( r ) δ ˆ t ˆ t can be written as − r ddr [ r ( e − β − ψ θ − χ e ρ ( r ) (56)For a perfect fluid in thermodynamic equilibrium, the stress-energy tensor takes on a particu-larly simple form Ξ µν = ( ρ + P ) u µ u ν + pg µν (57)where the pressure P can be neglected due to the distribution of mass and the gravitationaleffects; consequently, only one term will remain in the above formula as followsΞ aµ = ρ ( r ) dx a dt dx µ dt (58)or Ξ aµ = ρ ( r ) δ aµ (59)Therefore, for spherically symmetric distribution of mass that depends on r-coordinate, we canwrite m ( r ) = Z r πr ρ ( r ) dr (60)Note that the ρ ( r ) is defined by Eq.(1). Indeed, we introduce the same energy density indicatedin the noncommutative perturbation theory [30] m ( r ) = M θ ( r ) = 2 M √ π γ ( 32 , r θ ) . (61)10q.(56) can be integrated to find e − β = 1 − χ πe m ( r ) r + ψ θ (62)Where ψ θ is a function that carries the tetrad field factor and will be defined later by Eqs.(67,68). Now by considering e − β = − h = h (63)the noncommutative line element for a spherically symmetric matter distribution is therefore ds = − (1 − χ πe m ( r ) r + ψ θ ) dt + (1 − χ πe m ( r ) r + ψ θ ) − dr + r ( dθ + sin θdφ ) . (64)The constant field of χ e in terms of Eqs.(3),(4),(5) can be retrieved as χ e = | θ µb | πG N (65)where G N is the Newtonian constant, and | θ µb | is determined by θ µb = θ = − θ ≡ θ . Where θ is a real, antisymmetric and constant tensor, therefore, the above equation can be simplifiedto yield: χ e = θ πG N (66)New line element (64) in particular depends on ψ θ , and naturally ψ θ has its origin on the quan-tum fluctuations of the noncommutative background geometry and originally comes from thefield equations. The presented solution for our field equations produces naturally some addi-tional terms in comparison with the solution of noncommutative version of general relativity,( naturally , because it has some additional terms in its components). These terms appear in thenew line element because ψ θ relates to the noncommutative torsional spacetime and algebraicproperties in spherically symmetric solution of the tetrad fields. We have therefore, ψ θ in asimplified following equation ψ θ = ε ˆ r ˆ θ ˆ φ ε ˆ r ˆ θ ˆ φ h ˆ r ˆ r e − β (67)Definition ε ˆ r ˆ θ ˆ φ ε ˆ r ˆ θ ˆ φ = − h is applied here. (see also Ref. [29]). According to this definition andEqs.(49) and (51), Through simplification, we find the following form of ψ θ ψ θ ∼ = X k =2 n X n =1 ( χ πe m ( r ) r ) k − X k =2 n +1 X n =1 ( χ πe m ( r ) r ) k . (68)Note that ψ θ is considered with the lower bound of P . If we want to consider at least the secondorder of θ (which is proposed by Ref. [7]) for ψ θ , then it is natural to assume n = 1. Therefore,11wo states for our line element will be produced: the imperfect state and the perfect state. Letus now consider the perfect state. There is a proof for this state in terms of some theorems inmathematics that allows us to introduce our line-element as an appropriate description for anoncommutative spacetime. Combination of these theorems with regard to our results is givenby: (Following the Ref.[33]) Theorem.
Let L be a perfect field. Recall that a polynomial f ( x ) ∈ L [ x ] is called additive if f ( x + y ) = f ( x ) + f ( y ) identically. It is easy to see that a polynomial is additive if and only ifit is of the form f ( x ) = 1 − a x + a x − ... ± a nn x n = X n =0 a nn x n − X n =0 a n +1 n x n +1 (69)The set of additive polynomials forms a noncommutative field in which ( f ◦ g )( x ) = f ( g ( x )).This field is generated by scalar multiplications x ax for a ∈ L and x i ∈ f ( x ) does notcommute with the x j ǫf ( x ). Note that a can be a constant field and it has given as ≈ χ πe here. (see [33] and references cited therein). It is clear that components of f ( x ) can be exactlyreplaced with components of h .Regarding to other investigations toward descriptions of noncommutative spacetime, we shouldexpand our discussion into a comparison method with the other line elements presented fornoncommutative spacetime. Ref. [7] suggests the following line element for noncommutativeSchwarzschild spacetime suggested ds = − (cid:18) − Mr √ π γ (3 / , r / θ ) (cid:19) dt + (cid:18) − Mr √ π γ (3 / , r / θ ) (cid:19) − dr + r d Ω (70)where d Ω = dθ + sin θdφ and γ (3 / , r / θ ) is the lower incomplete gamma function γ (3 / , r / θ ) ≡ Z r / θ dt √ te − t (71)We note that non-vanishing radial pressure is a consequence of the quantum fluctuation of thespacetime manifold leading to an inward gravitational pull and preventing the matter collapsinginto a point. According to the line element (64), in a neighborhood of the origin at r ≤ θ ,the energy density distribution of a static symmetric and noncommutative fuzzy spacetime isdescribed by Eq.(1), which replaces the Dirac δ distribution by a smeared Gaussian profile.Meanwhile, in the imperfect state, our line element can be made equivalent to the line elementof Eq.(70), and it is expected to happen when ψ θ vanishes. Assuredly it is due to vanishing ofthe tetrad components h aµ in Eq.(49) or even Wietzenb¨ock connections in Eq.(51). It means hat in absence of torsional spacetime , the coordinate coherent state will be produced in thenoncommutative field theory. It is completely reasonable since coherent state theory is derivedin the noncommutative framework of general relativity, and the torsion is not defined in generalrelativity. This equivalency is shown with the following relation1 − M r √ π γ ( 32 , r θ ) ∼ = g coherent state = 1 − Mr √ π γ ( 32 , r θ ) . (72)According to this proof, the solution of the presented noncommutative field equations in theimperfect state of itself results in the exact solution of noncommutative general relativity fieldequations through coordinate coherent state of our line element. In this paper we have not extended our discussion into black holes, but our introduced equationscan be the basis of a subject on noncommutative back holes. Indeed the calculation of eventhorizons of a noncommutative Schwarzschild black hole would be done by the horizon equation − h r H = h ( r H ) = 0. Answers to this equation are illustrated by Figs.(1),(2). Fig.(1) showsthe behavior of h versus the horizon radii when ψ θ vanishes. It is clear that ψ θ vanishingapproximately results in g of Eq.(70), Fig.(2) shows the behavior of h at the same conditionswhen we have ψ θ . As we can see from these figures, there is a different behavior in the perfect13igure 1: The imperfect state in a noncommutative spherically symmetric geometry. Thefunction of h vs r √ θ , for various values of M √ θ , The upper curve corresponds to M = 1 . √ θ (without horizon), the middle curve corresponds to M = M ≈ . √ θ (with one horizon at r H = r ≈ . √ θ ) and finally the lowest curve corresponds to M = 3 . √ θ (two horizons at r H = r − ≈ . √ θ and r H = r + ≈ . √ θ ).Figure 2: The perfect state in a noncommutative spherically symmetric geometry. The functionof h vs r √ θ , for various values of M √ θ . The upper curve corresponds to M = 1 . √ θ (without horizon), the middle one corresponds to M = M ≈ . √ θ (with one horizon at r H = r ≈ . √ θ ) and finally the lowest curve corresponds to M = 3 . √ θ (two horizons at r H = r − ≈ . √ θ and r H = r + ≈ . √ θ ). 14tate in comparison with the imperfect state near the horizon radii which is due to the natureof torsional spacetime. Although, the same behaviors have been indicated in the origin and thehigher bound of r . In teleparallel gravity, the Newtonian force equation is obtained by assuming the class of framesin which the teleparallel spin connection ˙ A vanishes, and the gravitational field is stationaryand weak [29], [34]. In our model, the Newtonian gravitational force equation directly derivesfrom torsion components by ψ θ in its commutative limit. When we write the expansion of newline element in the noncommutative limit, we have: h = 1 − χ e m ( r ) r + ( χ e m ( r ) r ) − ( χ e m ( r ) r ) (73)Or equivalent with h = 1 − A m ( r ) r + B m ( r ) r − C m ( r ) r (74)Due to noncommutative effects, r in the denominator vanishes, but in the limit case, when itgoes to the commutative limit, it is modified to the commutative g of Schwarzschild solutionin addition to a force equation much similar to Newtonian gravitational force equation. Notethat the induced gravitational constant of Eq.(3) vanishes in the commutative limit and agreeswith that found in [35] using the supergravity dual of noncommutative Yang-Mills theory infour dimensions. Newton was the first to consider in his Principia an extended expression ofhis law of gravity including an inverse-cube term of the form F = G m m r + B m m r , B is a constant. (75)He attempts to explain the Moon’s apsidal motion by above relation. In the commutative limitour metric can be defined in the form of: h commutative ∼ = 1 − Mr + 4 M r − M r (76)Where m ( r ) is given by Eq. (61), and in the commutative limit it has the form oflim θ −→ m ( r ) = 2 M (77)By considering the following terms in the Equations of (75), • relativistic limits G = 1, 15 set the m = m = 2 M, B = − M ,therefore, for our line element we can set h commutative = ( g commutative Schwarszchild solution + F ( r ) Newton ) (78)As we can see from the Eq.(64) and (68), (expansion of new line element) the h has two parts:torsional and non-torsional parts, the above relation states that in the limit of commutativity,torsional parts reduce to force equation of F ( r ) and non-torsional part yields the g of com-mutative Schwarzschild solution.Einstein’s theory of general relativity attributes gravitation to curved spacetime instead of be-ing due to a force propagated between bodies. Energy and momentum distort spacetime intheir vicinity, and other particles move in trajectories determined by the geometry of space-time. Therefore, descriptions of the motions of light and mass are consistent with all availableobservations. Meanwhile, according to general relativity’s definition the gravitational force is afictitious force due to the curvature of spacetime because the gravitational acceleration of a bodyin free fall is due to its world line being a geodesic of spacetime [36]. Whereas, through a weakequivalence principle assumed initially in teleparallel gravity [37], our results are reasonableand we can conclude that: The presented solution in its commutative limit attributesthe gravitation to a force propagated between bodies, and the curved spacetime,or sum of torsion and curvature. This result is Similar to Einstein-Cartan theoryof gravity [38].
Moreover, the different behaviour of Schwarzschild black hole horizon, which is absent in theprevious method, is due to force of heavy pulling from the black hole in terms of this introducedforce. As can be seen in fig.(2), the intensity of this force has a direct relation with mass M ,so the heavier the black hole is, the stronger force it has near its horizon. In this letter, we have utilized a noncommutative Lagrangian which gives us possibilities to useteleparallel gravity to derive field equations. Solution of these field equations in the sphericallysymmetric geometry yields a new noncommutative line element. In the limit cases when thetorsion vanishes, we have obtained an interesting result: in absence of torsional spacetimethe version of coordinate coherent state in noncommutative field theory will be roduced . Incidentally, Figs.(1),(2) show other limit cases in our solution at the large dis-tances and different range of masses.As we expressed before, there are conceptual differences, in general relativity, curvature is usedto geometrize the gravitational interaction, geometry replaces the concept of force, and thetrajectories are determined, not by force equations, but by geodesics. Teleparallel Gravity, onthe other hand, attributes gravitation to torsion. Torsion, however, accounts for gravitationnot by geometrizing the interaction, but by acting as a force [29]. This is a definition used inteleparallel gravity, whereas our model do not exactly coincide with teleparalel gravity (in thelimit case only); therefore, it is natural to have more complex results especially the definitionof existing force in torsion for gravitational interactions is approved clearly in thelimit of commutativity in our model. Absolutely, attributing the gravitation to the forceequation in the relativity framework which is shown directly through the commutative limit ofour line element, can be utilized in the various branches of physics.
Acknowledgment
I would like to give very special thanks to Professor Kourosh Nozari for valuable comments on thiswork. I also would like to thank Mr. Arya Bandari for effective proofreading of this paper.
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