Rigorous Solution Of The Einstein Field Equations

This thesis mainly investigates the exact solutions of Einstein's field equa-tions. The exact solutions of the Einstein's field equations play an important role in the theory of general relativity and cosmology. Although many interesting and important solutions have been obtained, there are still many fundamental but open problems, such as " if there exists a time-periodic solution to the vacuum Einstein's field equations", " if there exists naked singularity in the universe" etc. In this thesis, we construct several kinds of exact solutions of Einstein's field equations including time-periodic solutions of vacuum Einstein's field equations which have geometric singularity or physical singularity, and the solution with naked singularity. Besides, we also investigate the exact solutions of the matter field equations and obtain some new results. The thesis is organized as follows.In Chapter 1, we introduce some basic knowledge on the Einstein's field equations and some important results of the exact solutions of Einstein's field equations. The main results obtained in this thesis are also stated.In Chapter 2, we investigate the time-periodic solutions of vacuum Ein-stein's field equations which have geometric singularity. Firstly, we get a general expression of a new kind of exact solutions of vacuum Einstein's field equa-tions. Then according to the general expression, we construct three kinds of new time-periodic solutions of the vacuum Einstein's field equations, they are the regular time-periodic solution with vanishing Riemann curvature tensor, the regular time-periodic solution with finite Riemann curvature tensor and the time-periodic solution with geometric singularities, respectively. However, the norm of the Riemann curvature tensors of all these solutions vanishes, therefore these solutions essentially describe some regular time-periodic space-times, these space-times contain some geometric singularities, but no physical singularities. Finally, we analyze the geometric singularity for the solutions and investigate some new physical phenomena.In Chapter 3, we study the time-periodic solutions of vacuum Einstein's field equations which have physical singularity. We construct a new time-periodic so-lution by solving the vacuum Einstein's field equation and prove that the solution has physical singularity by calculation of the Riemann curvature tensors and the norm of the Riemann curvature tensors. Besides, we give some new solutions by Jacobi elliptic functions.In Chapter 4, we investigate some other new exact solutions of vacuum Ein-stein's field equations. Firstly, we give another class of exact solutions of vacuum Einstein's field equations which have geometric singularity inspired by Chapter 2, and give two interesting examples whose Riemann curvature takes infinity at the time coordinate. Moreover, we prove that this class of solutions are not Minkowski, and essentially different from other solutions. Secondly, we construct a diagonal but not Schwarzschild solution, and prove that this solution possesses some physical singularities, moreover, the singularity is naked, i.e, no event hori-zon exists around the singularity. Besides, none of the Weyl scalars related to this solution vanishes. Finally, we get a family of solutions to the vacuum Einstein's field equations with cosmological constant. According to the general form of this family of solutions, we construct successfully the time-periodic solutions.In Chapter 5, we construct the exact solutions of Einstein's field equations with matter field. On one hand, we get a new family of non-static plane sym-metric solutions for the radiating gravitational field and construct a space-time-periodic solution which describes a very interesting physical space-time. On the other hand, based on a metric which is a more general form of Ori's, we solve the Einstein's equations with the energy-momentum tensors for electromagnetic field and construct a new time-machine solution.In Appendix, we give an introduction of the mathematical knowledge in the general theory of relativity, especially some knowledge about differential geome-try.