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In this paper, a noncommutative gravitational theory is constructed byapplying Moyal deformation quantization and the Seiberg-Witten map toteleparallel gravity, a classical gravitational theory, as a gauge theory oflocal translational symmetry. Since our model is based on teleparallel gravity,it is an extremely simple noncommutative gravitational theory. We also clearlydivide the role of the products, such that the metric is responsible for therule of the inner product (which is calculated by taking the sum over thesubscripts) and the Moyal product is responsible for tensor and fieldnoncommutativity. This solves problems related to the order of the products andthe relationship between the metric and the Moyal product. Furthermore, weanalyze the cosmic evolution of the very early universe and the spacetimefeatures around black holes using the constructed noncommutative gravitationaltheory, and find that gravity acts repulsively in the extreme region where itsquantum effects become prominent.

Gravity in Extreme Regions Based on Noncommutative Quantization of Teleparallel Gravity