Field Equations and Radial Solution in a Noncommutative Spherically Symmetric Geometry

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 dimensional reduction of noncommutative gauge theory and that the yielded diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity. Field equations are derived in the framework of teleparallel gravity through Weitzenbock geometry. We solve these field equations by considering a mass that is distributed spherically symmetrically 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 noncommutative Schwarzschild black hole. For the first time, we derive the Newtonian gravitational force equation in the commutative relativity framework, and this result could provide the possibility to investigate examples in various topics in quantum and ordinary theories of gravity.


Introduction
Field equations of gravity and radial solutions have been previously derived in noncommutative geometry [1,2]. The generalization of quantum field theory by noncommutativity based on coordinate coherent state formalism also cures the short distance behavior of point-like structures [3][4][5][6][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 function, 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-like gravitational source can not be a delta function distribution and will be given by a Gaussian distribution of minimal width √ θ as follows: Furthermore, noncommutative gauge theory appears in string theory [8][9][10][11][12][13]: the boundary the-

ory of an open string is noncommutative when it ends on D-bran with a constant B-field or an
Abelian gauge field (particularly see Ref. [8]). Therefore, closed string theories are expected to remain commutative as long as the background is geometric. Recent evidence has found a connection between non-geometry and closed string noncommutativity and even non-associativity [14][15][16]; approaches using dual membrane theories [17] and matrix models [18] arrive at the same conclusion.
The ordinary quantum field theory is unable to present an exact description of exotic effects of the inherent non-locality of interactions, so we need a model to provide an effective description of many of the non-local effects in string theory within a simpler setting [19]. This model is given appropriately by a noncommutative Lagrangian and introduced by authors in Ref. [2]. Admittedly, This Lagrangian and the relevant explanations will be the basis of our next general calculations. In this paper are going to use the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote indices related to spacetime, and the first half of the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) to denote indices related to the tangent space. A Minkowski spacetime whose Lorentz metric is assumed to have the form of η ab = diag(−1, +1, +1, +1).
The middle letters of the Latin alphabet (i, j, k, ... = 1, 2, 3) will be reserved for space indices.
The noncommutative Lagrangian is expressed aṡ In the usual way, having a Lagrangian, which describes gravitation based on noncommutative background, is like those of gauge theories written in terms of contractions of its field strength, here represented by torsion of Weitzenböck 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 the Planck scale of n-dimensional spacetime is given by In mass dimension 2 the weight constant χ 0 is In which the commutative limit, it reduces to gravitational constant. therefore, Θ AB is a noncommutative parameter, defined as By considering the calculation of superpotential and energy-momentum current with respect to noncommutative gauge potential, given by B µ a = | det(θ µ ′ a ′ )| 1/2nθνµ ω aν , the version of noncommutative gravitational field equations are produced. ω aν are gauge fields corresponding to the gauging of the translation group, i.e., replacing R n by the Lie algebra g of local gauge transformations with gauge functions and its relation with the verbien field is expressed as: h µ a = δ µ a − eθ νµ ω aν and δ µ a has the perturbative effect in the trivial holonomic tetrad fields of flat space.
It is important to note that by applying the "dimensional reduction of gauge theories", noncommutative electrodynamics gauge field; shown by the noncommutative Yang-Miles theory, reduces to the gauge theories of gravitation , which naturally yields Weitzenböck geometry on the spacetime. Also, the induced diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity macroscopically describing general relativity. In section 2 we show that our Lagrangian can be made equivalent with general relativity. In section 3 we 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 spherically symmetric space-time in section 4. We continue our discussion with investigations about the limiting cases of our line element and horizons of noncommutative Schwarzschild black hole in this method. Finally we show how the Newtonian gravitational force equation can be derived from our line element in the commutative limit in section 5.

Equivalence with General Relativity
In order to continue our discussion to achieve to noncommutative field equations, we should show how our model can be coupled with general relativity. With respect to the given relation for the vanishing curvature of the Weitzenböck connection, we havė where is the curvature of the Levi-Civita connection, The above equations show that, whereas in general relativity torsion vanishes, in teleparallel gravity it is curvature that vanishes. We rewrite the Eq.(7) based on their components in order to find the scaler ofṘ ρ θµν , therefore we havė that is the tensor written in terms of the Weitzenböck connection only. Like the Riemanian curvature tensor, it is a 2-form assuming values in the Lie algebra of the Lorentz group (see Ref. [21]). By taking appropriate contractions it is easy to show thaṫ By considering the Eq.(20) and the following term we achieve to the scalar version of Eq. (7), The Lagrangian of Eq.(2) can be written in a simple form oḟ with a combination of Eqs. (12) and (13),L takes the following forṁ By considering the Eqs.(3,4)L exchanges tȯ up to a divergence at the commutative limit; therefore, the Lagrangian of Eq.(2)L is equivalent to the Lagrangian of general relativity as followṡ is the Einstein-Hilbert Lagrangian of general relativity. However, this result could be extended with many further terms, but this is enough to derive a valid field equations.

Noncommutative Field Equations
In this section, we are going to present a reformulation of teleparallel gravity, (is made equivalent to general relativity). Due to the introduced noncommutative Lagrangian (2), we are able to derive the field equations similarly to the teleparallel method. Weitzenböck geometric definitions and some well-known concepts of general relativity [22][23][24] and teleparallel gravity are required, (more explanations about these equations can be found in Ref. [23], [25], [26]). In 4-dimension, the noncommutative action integral is given by Under an arbitrary variation δh µ a of the tetrad field, the action variation is written in the following form Where is the matter energy-momentum tensor. (More definitions about this tensor can be found in Ref. [27]). Now, consider first an infinitesimal Lorentz transformation as With ε b a = −ε b a . because of such transformation the tetrad should be changed as The requirement of invariance of the action under local Lorentz transformation therefore yields Since ε b a is antisymmetric, symmetric of energy-momentum tensor yields some specific results that can be seen in Ref. [22]. Consider spacetime coordinates that are transformed as follows Whereby, we retrieve the tetrad in the form of Substituting in Eq.(18), we have or equivalently Substituting the identity whereȦ is the spin connection in teleparallel gravity. The important property of teleparallel gravity is its spin connection is related only to the inertial properties of the frame, not to gravitation. In fact, it is possible to choose an appropriate frame in which it vanishes everywhere.
We know the above formula vanishes by the Eq.(42), (see also Ref. [28]), and making use of the symmetric of the energy-momentum tensor, the action variation assumes the form of Integrating the first term by parts and neglecting the surface term, the invariance of the action From arbitrariness of ζ c , under the covariant derivativeD µ , it follows thaẗ By identity of the above conservation law becomes In a purely spacetime form, it reads This is the conservation law of the source of energy-momentum tensor. Variation with respect to the noncommutative gauge potential B µ a yields the noncommutative teleparallel version of the gravitational field equations where which defines the superpotential, For the gauge current we have Note that the matter energy-momentum tensor which is defined in this relation appears as the source of torsion; similarly, the energy-momentum tensor appears as the source of curvature in general relativity. Our computation has led us to the following results: for noncommutative superpotential and gauge current. The lagrangianL appears again in our equations, but notice that this term cross its coefficient yields a term purely based on field strength. This simplified expression maintains equivalence to general relativity. We can observe that the gravitational field equations depend on the torsion only. Finally the field equations can be written as: Where k = χ 0 e 2 is a constant. These field equations are similar to teleparallel field equations. Although it would be distinguished with different field strengthṪ µσ a which is given by the covariant rotational of noncommutative gauge potential of B µ a . By considering the following equations from the teleparallel theory (see for instance, [20], [26], [28]) The field equations take the exact following form which unlike the left hand side of Eq.(39), is written purely based on noncommutative field strength, the above field equation is written in terms of Weitzenböck connection only. Regarding to the equivalency between corresponding Lagrangians and the above simplified field equations and applying the Eq.(34), we have therefore as equivalent with Einstein's field equations. Note that the equation (45) is not Einstein's field equations but the teleparallel field equations made equivalent to general relativity. And equivalent model of teleparallel field equations with general relativity expressed in references in detail, (see for instance [26], [28]). We continue our discussion to derive noncommutative line element by solving these field equations.

Noncommutative Line Element
Teleparallel versions of the stationary, static, spherically, axis-symmetric, and symmetric of the Schwarzschild solution have been previously obtained [31], [32]. Within a framework inspired by noncommutative geometry, We solve the field equations for a distribution of spherically symmetrically 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 With a new radial coordinate defined as r =r h(r), the line element becomes Usually one replaces the functions A(r) and B(r) with exponential functions to obtain somewhat simpler expressions for the noncommutative tensor components. Hence, we introduce the functions α(r) and β(r) by e 2α(r) = A(r) and e β(r) = B(r) to get ds 2 = −e 2α dt 2 + e 2β dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ).
Tetrad components of the above metric takes the following form: Weitzenböck connectionΓ ρ µν has following expression: Now, we can calculate the non-vanishing components of Weitzenböck connection as follows: By replacing these components in Eq. (44), the noncommutative tensors of Eqs.(52-54) for the left-hand side of the field equations will produce the following expression χ 0 e 2 ρ(r)δφφ (54) Adding equations (52) and (53) we get simply where α(r) = β(r). It should also be noted that by recalling Eq.(48), we can consider the limiting case for our solution assuming (α ′ − β ′ ) = k, which k is a constant, and by considering the 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 Ntt = χ 0 e 2 ρ(r)δtt can be written as For a perfect fluid in thermodynamic equilibrium, the stress-energy tensor takes on a particularly simple form where the pressure P can be neglected due to the distribution of mass and the gravitational effects; consequently, only one term will remain in the above formula as follows Therefore, for spherically symmetric distribution of mass that depends on r-coordinate, we can write m(r) = r 0 4πr 2 ρ(r)dr (60) Note that the ρ(r) is defined by Eq. (1). Indeed, we introduce the same energy density indicated in the noncommutative perturbation theory [30] Eq.(56) can be integrated to find Where ψ θ is a function that carries the tetrad field factor and will be defined later by Eqs.
(67,68). Now by considering the noncommutative line element for a spherically symmetric matter distribution is therefore The constant field of χ 0 e 2 in terms of Eqs. (3), (4), (5) can be retrieved as where G N is the Newtonian constant, and |θ µb | is determined by θ µb = θ 21 = −θ 12 ≡ θ. Where θ is a real, antisymmetric and constant tensor, therefore, the above equation can be simplified to yield: New line element (64) in particular depends on ψ θ , and naturally ψ θ has its origin on the quantum fluctuations of the noncommutative background geometry and originally comes from the field equations. The presented solution for our field equations produces naturally some additional 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 the new line element because ψ θ relates to the noncommutative torsional spacetime and algebraic properties in spherically symmetric solution of the tetrad fields. We have therefore, ψ θ in a simplified following equation Definition εrθφεrθφ = −6 h 2 is applied here. (see also Ref. [29]). According to this definition and Eqs.(49) and (51), Through simplification, we find the following form of ψ θ Note that ψ θ is considered with the lower bound of . If we want to consider at least the second order of θ (which is proposed by Ref. [7]) for ψ θ , then it is natural to assume n = 1. Therefore, two states for our line element will be produced: the imperfect state and the perfect state. Let us now consider the perfect state. There is a proof for this state in terms of some theorems in mathematics that allows us to introduce our line-element as an appropriate description for a noncommutative spacetime. Combination of these theorems with regard to our results is given by: (Following the Ref. [33]) It is easy to see that a polynomial is additive if and only if it is of the form 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 not commute with the x j ǫf (x). Note that a can be a constant field and it has given as ≈ χ 0 4πe 2 here. (see [33] and references cited therein). It is clear that components of f (x) can be exactly replaced with components of h 00 .
Regarding to other investigations toward descriptions of noncommutative spacetime, we should expand our discussion into a comparison method with the other line elements presented for noncommutative spacetime. Ref. [7] suggests the following line element for noncommutative Schwarzschild spacetime suggested where dΩ 2 = dθ 2 + sin 2 θdφ 2 and γ(3/2, r 2 /4θ) is the lower incomplete gamma function We note that non-vanishing radial pressure is a consequence of the quantum fluctuation of the spacetime manifold leading to an inward gravitational pull and preventing the matter collapsing into 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 is described 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 element of Eq.(70), and it is expected to happen when ψ θ vanishes. Assuredly it is due to vanishing of the tetrad components h a µ in Eq.(49) or even Wietzenböck connections in Eq.(51). It means that in absence of torsional spacetime, the coordinate coherent state will be produced in the noncommutative field theory. It is completely reasonable since coherent state theory is derived in the noncommutative framework of general relativity, and the torsion is not defined in general relativity. This equivalency is shown with the following relation According to this proof, the solution of the presented noncommutative field equations in the imperfect state of itself results in the exact solution of noncommutative general relativity field equations through coordinate coherent state of our line element.

Schwarzschild Black hole, Horizons
In this paper we have not extended our discussion into black holes, but our introduced equations

Force Equation in Commutative Limit
In teleparallel gravity, the Newtonian force equation is obtained by assuming the class of frames in which the teleparallel spin connectionȦ vanishes, and the gravitational field is stationary and weak [29], [34]. In our model, the Newtonian gravitational force equation directly derives from torsion components by ψ θ in its commutative limit. When we write the expansion of new line element in the noncommutative limit, we have: Or equivalent with Due to noncommutative effects, r in the denominator vanishes, but in the limit case, when it goes to the commutative limit, it is modified to the commutative g 00 of Schwarzschild solution in addition to a force equation much similar to Newtonian gravitational force equation. Note that the induced gravitational constant of Eq.(3) vanishes in the commutative limit and agrees with that found in [35] using the supergravity dual of noncommutative Yang-Mills theory in four dimensions. Newton was the first to consider in his Principia an extended expression of his law of gravity including an inverse-cube term of the form He attempts to explain the Moon's apsidal motion by above relation. In the commutative limit our metric can be defined in the form of: Where m(r) is given by Eq. (61), and in the commutative limit it has the form of By considering the following terms in the Equations of (75), • relativistic limits G = 1, As we can see from the Eq. (64) and (68), (expansion of new line element) the h 00 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 00 of commutative Schwarzschild solution.
Einstein's theory of general relativity attributes gravitation to curved spacetime instead of being due to a force propagated between bodies. Energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. Therefore, descriptions of the motions of light and mass are consistent with all available observations. Meanwhile, according to general relativity's definition the gravitational force is a fictitious force due to the curvature of spacetime because the gravitational acceleration of a body in free fall is due to its world line being a geodesic of spacetime [36]. Whereas, through a weak equivalence principle assumed initially in teleparallel gravity [37], our results are reasonable and we can conclude that: The presented solution in its commutative limit attributes the gravitation to a force propagated between bodies, and the curved spacetime, or sum of torsion and curvature. This result is Similar to Einstein-Cartan theory of gravity [38].
Moreover, the different behaviour of Schwarzschild black hole horizon, which is absent in the previous method, is due to force of heavy pulling from the black hole in terms of this introduced force. 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.

Conclusion
In this letter, we have utilized a noncommutative Lagrangian which gives us possibilities to use teleparallel gravity to derive field equations. Solution of these field equations in the spherically symmetric geometry yields a new noncommutative line element. In the limit cases when the torsion vanishes, we have obtained an interesting result: in absence of torsional spacetime the version of coordinate coherent state in noncommutative field theory will be produced. Incidentally, Figs.(1),(2) show other limit cases in our solution at the large distances and different range of masses. As we expressed before, there are conceptual differences, in general relativity, curvature is used to geometrize the gravitational interaction, geometry replaces the concept of force, and the trajectories are determined, not by force equations, but by geodesics. Teleparallel Gravity, on the other hand, attributes gravitation to torsion. Torsion, however, accounts for gravitation not by geometrizing the interaction, but by acting as a force [29]. This is a definition used in teleparallel gravity, whereas our model do not exactly coincide with teleparalel gravity (in the limit case only); therefore, it is natural to have more complex results especially the definition of existing force in torsion for gravitational interactions is approved clearly in the limit of commutativity in our model. Absolutely, attributing the gravitation to the force equation in the relativity framework which is shown directly through the commutative limit of our line element, can be utilized in the various branches of physics.