In this paper, a noncommutative gravitational theory is constructed by applying Moyal deformation quantization and the Seiberg-Witten map to teleparallel gravity, a classical gravitational theory, as a gauge theory of local translational symmetry. Since our model is based on teleparallel gravity, it is an extremely simple noncommutative gravitational theory. We also clearly divide the role of the products, such that the metric is responsible for the rule of the inner product (which is calculated by taking the sum over the subscripts) and the Moyal product is responsible for tensor and field noncommutativity. This solves problems related to the order of the products and the relationship between the metric and the Moyal product. Furthermore, we analyze the cosmic evolution of the very early universe and the spacetime features around black holes using the constructed noncommutative gravitational theory, and find that gravity acts repulsively in the extreme region where its quantum effects become prominent.
A noncommutative gravitational theory constructed by applying Moyal deformation quantization and the Seiberg–Witten map to teleparallel gravity shows that gravity acts repulsively in an extreme region where its quantum effects become prominent. The problems involved with the order of products and the relationship between the metric and the Moyal product can be solved by assigning their roles, such that the former is responsible for the rule of the inner product and the latter is in charge of tensor and field noncommutativity. As a result, it is found that the cosmic evolution of the very early stage of the universe was modified as if the noncommutative effect works similarly to a cosmological constant. Furthermore it was clarified that the degree of gravitational redshift of a spherically symmetric space with central mass becomes smaller, which means the attractive force is effectively weakened. Such consequences indicate that in the noncommutative spacetime, the gravitational interaction is softened compared to the classical gravity.
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