Corrections to Schwarzschild Solution in Noncommutative Gauge Theory of Gravity

A deformed Schwarzschild solution in noncommutative gauge theory of gravitation is obtained. The gauge potentials (tetrad fields) are determined up to the second order in the noncommutativity parameters $\Theta^{\mu\nu}$. A deformed real metric is defined and its components are obtained. The noncommutativity correction to the red shift test of General Relativity is calculated and it is concluded that the correction is too small to have observable effects. Implications of such a deformed Schwarzschild metric are also mentioned.


Introduction
The noncommutativity of space-time is a compelling option for the description of quantized space-time and its study is significant for answering the ultimate question about the quantum nature of space-time at very high energy scales. If nature has chosen such a course, it is most sensible to search for manifestations of the noncommutativity of space-time at the "natural laboratories" of the highest energy, i.e. the gravitational singularities.
The noncommutativity of space-time is intrinsically connected with gravity [1,2]. Gauge theories of gravitation have been intensively studied up to now, both on commutative [3,4] (see also the reviews [5,6]) [7] and noncommutative [8,9] space-times. Many recent investigations are oriented towards a formulation of General Relativity on noncommutative spacetimes. In Ref. [8] for example, a deformation of Einstein's gravity was studied by gauging the noncommutative SO(4, 1) de Sitter group and using the Seiberg-Witten map [2,10,11] with subsequent contraction to the Poincaré (inhomogeneous Lorentz) group ISO (3,1). Another construction of noncommutative gravitational theory, based on the twisted Poincaré algebra [12] was proposed in Ref. [13] . The twisting procedure insures the invariance of the algebra [x µ , x ν ] = i Θ µν (canonical structure) defining the noncommutativity of the space-time; however, it turned out that the dynamics of the noncommutative gravity coming from string theory [14] is much richer than the one in this version of deformed gravity [13]. In Refs. [15] a noncommutative version of General Relativity was proposed for a restrictive class of coordinate transformations which preserve the canonical structure. By gauging the Lorentz algebra so(3, 1) within the enveloping algebra approach one obtains a theory of noncommutative General Relativity restricted to the volume-preserving transformations (unimodular theory of gravity). Another attempted approach was to twist the gauge Poincaré algebra [16]. It is worthwhile to emphasize that there remains one more important unsolved problem in all these theories: to establish a Leibniz rule for gauge transformations of fields [17,18], since the star product is not invariant under the diffeomorphism transformations. Steps towards this goal have been taken in a geometrical approach to noncommutative gravity [19].
In this paper, proceeding along the approach in Ref. [8], we present a deformed Schwarzschild solution in noncommutative gauge theory of gravitation. Although this version of noncommutative gravity is certainly not a final one, we believe that the complete theory will retain the main features of this approach. First, we recall the results of a previous study, in which a de Sitter gauge theory of gravitation over a spherically symmetric commutative Minkowski space-time was developed [7]. Then, a deformation of the gravitational field is constructed by gauging the noncommutative de Sitter SO(4, 1) group [8] and using Seiberg-Witten map [2]. The space-time of noncommutative theory will be also of Minkowski type but it will be endowed with spherical noncommutative coordinates. The deformed gauge fields are determined up to the second order in the noncommutativity parameters Θ µν .
Finally, the deformed gravitational gauge potentials (tetrad fields)ê a µ (x, Θ) are obtained by contraction of the noncommutative gauge group SO(4, 1) to the Poincaré (inhomogeneous Lorentz) group ISO(3, 1). As an application, we calculate these potentials for the case of a Schwarzschild solution and define the corresponding deformed metricĝ µν (x, Θ). It is for the first time when such a deformed metric is given for a 4-dimensional noncommutative space-time. The corrections appear only in the second order of the expansion in Θ, i.e. there are no terms of the first order in Θ. We will give also an evaluation of the noncommutativity corrections to the red shift test of General Relativity, which turns out to be extremely small for the case of the Sun.
The calculations are very tedious, so that we have used an analytical program conceived for the GRTensor II package of the Maple platform. Specific routines have been written and adapted for Maple. Section 2 is devoted to the commutative gauge theory of the de Sitter group SO(4, 1) formulated on a 4-dimensional Minkowski space-time endowed with a spherical metric. The Section 3 contains the results regarding the noncommutative theory. The deformed gauge potentials (tetrad fields) are obtained up to the second order of the expansion in Θ. Based on these results, we define a deformed real metric and calculate its components in the case of a Schwarzschild solution. Using the results we determine in Section 4 the deformed Schwarzschild metric. The corrections are obtained up to the second order of the noncommutativity parameters Θ µν . An evaluation of the value for the correction to the red shift test of General Relativity is also given. Some concluding remarks and further directions of investigation are given in Section 5.

4)
where η ab = diag (1, 1, 1, −1). The Poincaré gauge theory that we are using has the geometric structure of the Riemann-Cartan space U(4) with curvature and torsion [6]. The quantity T a µν is interpreted as the torsion tensor and R ab µν as the curvature tensor of the Riemann-Cartan space-time defined by the gravitational fields e a µ (x) and ω ab µ (x). By imposing the condition of null torsion T a µν = 0, one can solve for ω ab µ (x) in terms of e a µ (x), i.e. the spin connection components are determined by tetrads (they are not independent fields). Now, we consider a particular form of spherically gauge fields of the SO(4, 1) group given by the following Ansatz [7]: where A, U, V, W and Z are functions only of the three-dimensional radius. The non-zero components of T a µν and R ab µν were obtained in [7] using an analytical program designed for GRTensor II package of Maple: and respectively where A ′ , U ′ , V ′ , W ′ and Z ′ denote the derivatives of first order with respect to the rcoordinate. If we use (2.7), then the condition of null-torsion T a µν = 0 gives the following constraints: where α is an arbitrary constant of integration. For α = 2GM c 2 we obtain the commutative Schwarzschild solution (G is the Newton constant and M is the mass of the point-like source of the gravitational field). The corresponding metric has the following non-zero components We emphasize that this solution is obtained from the commutative SO(4, 1) gauge theory with a contraction k → 0 to the Poincaré group ISO(3, 1). We will follow now the Ref. [8] in order to obtain a deformation of gravitation by gauging the noncommutative de Sitter SO(4, 1) group. Then, by contraction to the Poincaré (inhomogeneous Lorentz) group ISO(3, 1) we will obtain the deformed gauge fieldsê a µ (x, Θ). In the next two Sections we will calculate these fields for the case of the Schwarzschild solution and define the corresponding deformed metricĝ µν (x, Θ) up to the second order of the expansion in Θ.

Deformed gauge fields
We assume that the noncommutative structure of the space-time is determined by the con- where Θ µν = − Θ ν µ are constant (canonical) parameters. To develop the noncommutative gauge theory, we introduce the star product "*" between the functions f and g defined over this space-time: The gauge fields for the noncommutative case are denoted byω AB µ (x, Θ), and they are subject to the reality conditions [8,10,11]: with "+" denoting the complex conjugate.
By expandingω ab µ (x, Θ) in powers of the noncommutative parameter Θ, Using the Seiberg-Witten map [2], one obtains the following noncommutative corrections up to the second order [8]: and As in the commutative case, we writeω a5 µ (x, Θ) = kê a µ (x, Θ) andω 55 µ (x, Θ) = k φ µ (x, Θ). Then we impose the condition of null torsion T a µν = 0 and notT a µν = 0, since by contraction k → 0 the quantity φ µ (x, Θ) will drop out [8]. Using (3.6) and (3.7) we obtain, in the limit k → 0, the deformed tetrad fieldsê a µ (x, Θ) up to the second order: We define also the complex conjugateê a+ µ (x, Θ) of the deformed tetrad fields given in Then we can introduce a deformed metric by the formula: (3.14) We can see that this metric is, by definition, a real one, even if the deformed tetrad fieldŝ e a µ (x, Θ) are complex quantities.

Second order corrections to Schwarzschild solution
Using the Ansatz (2.5)-(2.6), we can determine the deformed Schwarzschild metric. To this end, we have to obtain first the corresponding components of the tetrad fieldsê a µ (x, Θ) and their complex conjugatedê a µ + (x, Θ) given by the Eqs. (3.10) and (3.13). With the definition (3.14) it is possible then to obtain the components of the deformed metricĝ µν (x, Θ).
To simplify the calculations, we choose the coordinate system so that the parameters Θ µν are given as: The constant quantity Θ, which determines the noncommutativity of the space-time coordinates, has the dimension L 2 (square of length). The non-zero components of the tetrad fieldsê a µ (x, Θ) are: 3) where A ′ , A ′′ , A ′′′ are first, second and third derivatives of A(r), respectively. The complex conjugated components can be easily obtained from these expressions. Then, using the definition (3.14), we obtain the following non-zero components of the deformed metricĝ µν (x, Θ) up to the second order: g 22 (x, Θ) = r 2 + 1 16 For Θ → 0 we obtain the commutative Schwarzschild solution with A 2 = 1 − α r (see Eq. (2.11)).
It is interesting to remark that, if we choose the parameters Θ µν as in (4.1), then the deformed metricĝ µν (x, Θ) is diagonal as it is in the commutative case. But, in general, for arbitrary Θ µν , the deformed metricĝ µν (x, Θ) is not diagonal even if the commutative (nondeformed) one has this property. Therefore, we can conclude that the noncommutativity modifies the structure of the gravitational field.
For the Schwarzschild solution we have: The function A(r) is dimensionless, but its derivatives A ′ , A ′′ and A ′′′ have respectively the dimensions L −1 , L −2 and L −3 . As a consequence, all the components of the deformed metriĉ g µν (x, Θ) in (4.7) have the correct dimensions. Now, if we introduce (4.8) into (4.7), then we obtain the deformed Schwarzschild metric. Its non-zero components are: We can evaluate then the contributions of these corrections to the tests of General Relativity. For example, if we consider the red shift of the light propagating in a gravitational field [20], then we obtain for the case of the Sun: where R is the radius of the Sun. Because for the Sun we have α = 2 G M c 2 = 2.95 · 10 3 m and R = 6.955 · 10 8 m, then we obtain from (4.10): The noncommutativity correction has a value which is much too small, compared to the value which results from General Relativity, and the precision of the measurement is not sufficient to put a reasonable bound on the noncommutativity parameter.

Concluding remarks
Using the Seiberg-Witten map we have determined the noncommutativity corrections to the Schwarzschild solution up to the second order in the parameters Θ µν . Following Ref.
[7], we reviewed first a de Sitter gauge theory of gravitation over a spherical symmetric commutative Minkowski space-time. Then, a deformation of the gravitational field has been constructed along the Ref. [8] by gauging the noncommutative de Sitter SO(4, 1) group and using Seiberg-Witten map. The corresponding space-time is also of Minkowski type but endowed now with spherical noncommutative coordinates. We determined the deformed gauge fields up to the second order in the noncommutativity parameters Θ µν . The deformed gravitational gauge potentials (tetrad fields)ê a µ (x, Θ) have been obtained by contraction of the noncommutative gauge group SO(4, 1) to the Poincaré (inhomogeneous Lorentz) group ISO(3, 1). As an application, we have calculated these potentials for the case of the Schwarzschild solution and defined the corresponding deformed metricĝ µν (x, Θ). The corrections appear only in the second order of the expansion in Θ, i.e. there are no first order correction terms. For the calculations we used an analytical program conceived for the GRTensor II package of the Maple platform.
We have considered also the red shift test in the noncommutative theory and determined the value of the relative displacement ∆ λ λ for the case of Sun. The result shows that the correction is too small to have observable effects.
Having found the Schwarzschild solution for a noncommutative theory of gravity we have been breaking new ground towards approaching the black-hole physics on noncommutative space-time [21].