Studies On The Noncommutative Space-Time And The Modified Gravity Theory

After the Seiberg-Witten’s work on the background of the noncommutativity instring theory, people have tried to construct theories on the different kinds of the non-commutative space-time. With the help of the star-product, the Seiberg-Witten map andthe coordinate coherent states approach, we can develop the noncommutative theoryfrom the theory in ordinary space-time. In this thesis, we introduce all these methodsexpressing the noncommutative effects. Based on the analysis of the noncommutativespace-time and the gravity, the noncommutativity of the space-time, if it exists, willaffect the gravity and the related theory anyway. These influences could be direct, orindirect. Due to the different kinds of the influences, we introduce the gravity theory inthe noncommutative space-time in two parts.The noncommutative modified gravity theory is the direct influence of the non-commutativity. In the third chapter, we generalize the Einstein’s gravity in the non-commutative space-time with the symplectic structure. In the ordinary space-time, theclassical Einstein’s gravity can be reformulated from the U (2,2) gauge theory withconstraints to break the symmetry group. For the simplicity of constructing the non-commutative gauge theory, this approach inspires us to adopt the following method:we first construct the U (2,2) gauge theory in the noncommutative space-time, thenwith the help of the Seiberg-Witten map we choose the constraints to break the gaugegroup, and finally we get the noncommutative modified gravity theory. The noncom-mutative space-time with the symplectic structure is the more general case during thevarious noncommutative space-times, and its noncommutative parameters are arbitrarysmooth function of the coordinates under some constraints. The noncommutative mod-ified gravity theory we get here has some different properties with other ones in othernoncommutative space-times, e.g. the first order corrections are not trivial comparedto the canonical noncommutative modified gravity. If we choose the noncommutativeparameters to be constant, the noncommutative modified gravity theory we get here canrecover the canonical noncommutative case. At the same time, under the limit that thenoncommutative parameters tend to zero, the noncommutative corrections disappear.This declares that the classical Einstein’s gravity is the classical limit of the noncom-mutative modified gravity. Another concept of the noncommutative space-time is that the noncommutativ-ity just affect all the other interactions except the gravity. Under this assumption, theLHS of the Einstein’s equation which is the geometric part is not corrected, and thenoncommutativity just corrects the RHS of the Einstein’s equation which is the matter.This method is just the indirect way of the noncommutative influence on the gravity.The coordinate coherent states approach gets the result that the mass and charge ofthe particles are depicted instead of distributing in a point. With this result we cansolve the Einstein’s equation to get the noncommutative inspired black hole solutions.The noncommutative inspired black holes have plenty of different properties from theordinary ones, such as the singularity, the horizons and the thermal properties. Wewant to discuss the thermal properties of the noncommutative inspired black holes bythe tunneling methods. For this purpose, we introduce the several quantum tunnelingmethods of the black holes in detail in chapter5, including the Parikh-Wilczek method,the generalized Parikh-Wilczek method, the Hamilton-Jacobi method and the reformu-lated Hamilton-Jacobi method. When generalizing the Parikh-Wilczek method, we findsome interesting properties，which is that the Hawking temperature of the black holedepends on the charge of the tunneling particles besides the charge of the black hole.The black hole is more likely to emit particles with the opposite charge to that of theblack hole, and its charges increase during the emission. In the chapter6, we dis-cuss the two kinds of the noncommutative inspired black holes—the noncommutativeinspired Schwarzschild black hole and the noncommutative inspired Kerr black hole,with two tunneling methods separately, and get the noncommutative modified Hawkingtemperature and the entropy properties.