Double Noncommutative Complex Gravitational And 1 +1 Dimensional Jt Gravity Model

In this paper the noncommutative wraped spacetime and related noncommutative models of gravity are studied by using some modern mathematical methods such as the double complex function theory and the noncommutative geometry and so on .First,the double complex symmetric gravitational theory is established and the concrete expression of the static spherically symmetric double complex metrics is derived by introducing the double complex symmetric metric tensors.Thus, the Moffat's results are extended, the hyperbolic complex symmetric gravitational theory is naturally obtained and the famous Schwarzachild solution is contained as the special case. Moreover, in the linearized weak field approximation the hyperbolic complex symmetric gravitational theory can automatically free from the potential problem of negative energy ghost states. Furthermore, the double complex symmetric gravitational action in the double noncommutative spacetime can be constructed by extending the double complex coordinates to the double noncommutative relation satisfied and by doubling the Moyal star product.Then, the action of Jackiw-Teitelboim (JT) gravity in two dimensions is generalized to the double quantum deformed gravity model in two dimensions by use of the double Moyal star product. Not only the two cases of the cosmological constant A > 0 and A < 0 can be studied in an unified way in this paper, but also the double solutions of this model -the fuzzy spheres of de Sitter (dS2) and Anti-de Sitter (AdS2) can be derived simultaneously. Hence the S.Cacciatori'results are included as the special case of this paper. Furthermore, by means of the double analytic continuation and the adaptive parametric transformations we can prove that the commutative and noncommutative double JT models of gravity both have the same spacetime solution-dS2/AdS2 fuzzy sphere. It is easy to find that the solutions corresponding to the double ordinary and hyperbolic noncommutative JT gravity models are respectively the fuzzy spheres AdS2 and...