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We use the theory of teleparallelism equivalent to general relativity based on noncommutative spacetime coordinates. In this context, we write the corrections of the Schwarzschild solution. We propose the existence of a Weitzenböck spacetime that matches the corrected metric tensor. As an important result, we find the corrections of the gravitational energy in the realm of teleparallel gravity due to the noncommutativity of spacetime. Then we interpret such corrections as a manifestation of quantum theory in gravitational field.

The notion of noncommutative spatial coordinates arose with Heisenberg, who wrote a letter to Peierls, in 1930, about the existence of an uncertain relation between coordinates in space-time as a possible solution to avoid the singularities in the self-energy terms of pontual particles. Based on such an advice, Peierls applied those ideas on the analysis of the Landau problem which can be described by an electric charge moving into a plane under the influence of a perpendicular magnetic field. Since then, Peierls commented about it with Pauli, who included Oppenheimer in the discussion. Oppenheimer presented the ideas to Hartrand Snyder, his former Ph.D. student [

From the mathematical point of view, the simplest algebra of the Hermitian operators

An alternative theory of gravitation is the so-called teleparallel gravity which was introduced by Einstein, as an attempt to unify gravity and electromagnetic field [

The expressions for the energy momentum and angular momentum of the gravitational field, in the context of the Teleparallelism Equivalent to General Relativity (TEGR), are invariant under transformations of the coordinates of the three-dimensional spacelike surface; they are also dependent on the frame of reference, as is to be expected. They have been applied consistently over the years for many different systems [

Therefore we have two successful theories described above and a natural forward step is to combine both of them. Then here our aim is to study the Teleparallelism Equivalent to General Relativity in the noncommutative spacetime context. In Section

The Teleparallelism Equivalent to General Relativity (TEGR) is constructed out of tetrad fields (instead of a metric tensor) in the Weitzenböck (or Cartan) spacetime, in which it is possible to have distant (or absolute) parallelism [

Let us start with a manifold endowed with a Cartan connection [

The curvature tensor obtained from

Performing a variational derivative of the Lagrangian density with respect to

The field equations can be rewritten as

In this section we will start with Schwarzschild spacetime [

Now we have to write another tetrad field for the new components of the metric tensor, but still adapted to a rest frame, which is the referential frame we would like to analyze the problem of the gravitational energy. This means that we suppose that a relation exists between the spacetime which is defined by the corrected metric tensor and a Weitzenböck spacetime. Indeed we are assuming that the Lorentz symmetry still holds. We, in this sense, assume the following correspondence

In this work we start with Schwarzschild spacetime, and then we give the corrections due to the noncommutativity of spacetime. Here it is introduced by replacing the normal product between tetrads by the Moyal product, rather than applying such a procedure in lagrangian density. This approach is well known in the literature to predict some noncommutative corrections in the metric tensor. The new metric tensor leads to a new tetrad field which is used to calculate the gravitational energy of spacetime. It is well known that the energy of Schwarzschild spacetime is equal to