الخلاصة:
Noncommutative geometry has a very important role in gravitation word because it answers to a lot of questions and explain several phenomena.
With the properties of noncommutative space-time (Seiberg-Witten mapping with Moyal-Weyl ordering), Chiral and gravitational anomalies can be calculated using Fujikawa method in path integral formalism. The result obtained is very important because the non-commutativity gives in this case to space-time a gravitational aspect without a gravitational force where the noncommutativity modifies its Riemannian structure to another one more complicated having a general form of the space-time metric (with torsion and no-metricity).
A contraction of a noncommutative De Sitter SO (4,1) gauge group to the Poincaré (inhomogeneous Lorentz) group ISO (3,1) leads to a deformed gauge fields using the Seiberg-Witten maps, which by explicit calculations gives a deformed metric. This metric (application to FRW space-time) shows several properties of the corresponding black hole. (Calculations and graphs are realized by Maple).
The instability against emission of fermionic particles by the trapping horizon of an evolving black hole is analyzes using the Hamilton-Jacobi method via tunneling effect. This study leads to calculate noncommutative Hawking temperature defined by the surface gravity; this result is purely noncommutative and justified by the graphs showing the variation of Hawking temperature with event horizon and the parameter of noncommutativity.
Finally, Hamilton-Jacobi method via is always useful to study black hole’s radiation via tunneling effect near apparent horizon of Lyra black holes. The correction spending Hawking temperature depends only of the black hole properties (mass) and the structure of Lyra space-time.
