On Noncommutative Corrections of Gravitational Energy in Teleparallel Gravity

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 Weitzenb？ck 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. 1. Introduction 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 [1–3]. Thus Snyder was the first to discuss the idea that spatial coordinates could not commutate to each other at small distances which is a change of perspective of tiny scales [4, 5]. It is worth recalling that the concept of noncommutativity itself is not new in Physics; in fact in Quantum Mechanics the uncertain principle, which is a noncommutative relation between coordinates and momenta, plays a fundamental role. Therefore at the beginning, with the pioneer works of Snyder, the idea was to use the noncommutativity between spacetime coordinates to control the ultraviolet divergences into the realm of quantum electrodynamic. Such an approach, however, got into oblivion due to the success achieved by the so-called renormalization process. More recently the interest of the physical community resurfaced with the application of noncommutative geometry in nonabelian theories [6], in gravitation [7–9], in standard model [10–12], and in the problem of the quantic Hall effect [13]. Certainly the discover that the dynamics of an open string can be explained by noncommutative gauge theories at specific limits [14] has contributed to this renewed interest of the scientific community in the topic. From the mathematical point of view, the simplest algebra of the Hermitian operators , whose mean values correspond to observable coordinates, is given by where is an