  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.
