High-dimensional Axisymmetric Gravitational Discussed

In this thesis, we have mainly finished the research on high-dimensional axisymmetric Kaluza-Klein theory and quantum entropies of extremal dilaton black hole. Chapter 1 giyes rise to a simple review of the development history of the Kaluza-Klein gravitational theory.Chapter 2 is devoted to an introduction to supergravity theory. Geometrically, high-dimensional spacetime is a homogenious fibre bundle, and the high-dimensional metric may-be reduced to the four-dimensional one. In the five-dimensional Kaluza-Klein theory, the fifth dimension has the S~1 topology, so fields may be Fourier expanded. Considering the physical requirement, in general, only zero-mode is studied. In the standard Kaluza-Klein theory, the physical laws don't depend on the fifth coordinate, namely satisfy the cylinder condition. Eleven-dimensional supergravity has been playing an important role, in which the compactification can occur as people have expected, the seven dimensions being compactified and the four dimensions being left.In Chapter 3, we mainly review some typical research finished by Dobi-asch and Maison, Gibbons and Wiltshire, and Gibbons and Maeda, as well as Chodos; Detweiler, Gross and Perry. Dobiasch and Maison discussed the generally, spherically symmetric problem and put forward the model with arbitrary coupling constant. Based on Dobiasch and Maisons' work, Gibbons and Wiltshire discussed the five-dimensional spherically symmetric black hole. And, Gibbons and Maeda gave the black hole solution in the high-dimensional gravitational model. Chodos and Detweiler obtained the five-dimensional solution with electrostatic field, while Gross and Perry discussed the five-dimensional solutions with magnetic field, in which Kerr-Taub-Bolt is an axisymmetric one with magnetic monopole.Chapter 4 shows our some research results oil the five-dimensional ax-isymmetrk: Kluza-Klein theory and EMD theory, as well as black hole thermodynamics. We propose a five-dimensional axisymmetric metric with a real parameter 7/ which determines the signature. For this metric, the field equations can be reduced to the Ernst equation with the parameter. We obtain the TS-like solutions, which give the usual TS solutions for 7/ = — 1 and yield the five-dimensional axisymmetric spacetime solutions with electrostatic field. By calculating directly, we also obtain the solution with magnetic monopole. which modifies the Kerr-Taub-Bolt solution. We demonstrate that in the electrostatic case, the low-energy string theory and the five-dimensional axisymmetric Kluza-Klein theory possess the same field equations. By the SO(2,r/) transformation, the Chodos-Detweiler solution with 77 is given, for which both the parameter k and the mass-to-charge are confined.In Chapter 5, we first discuss the entropies of the scalar and spinor fields in the extremal dilaton black hole backgrounds. It is shown that at the Harking temperature the entropies of quantum fields are some constants associated with the coupling constant, which gives a correction to the Harking's result. Lastly, we give some prospects on astrophysics, black hole physics and EMD theory.