In this paper, we discuss the system of Friedman-Robertson-Walker (FRW) metric coupling with massive nonlinear dark spinors in detail, where the thermodynamic movement of spinors is also taken into account. The results show that, the nonlinear potential of the spinor field can provide a tiny negative pressure, which resists the universe to become singular. The solution is oscillating in time and closed in space, which approximately takes the form: gµν =R 2 (1 − δ cos t) 2 diag(1, −1, − sin 2 r, − sin 2 r sin 2 θ), with R = (1 ∼ 2) × 10 12 light year, and δ = 0.96 ∼ 0.99. The present time is about t ∼ 18 • .

In this paper, we propose a new approach to the relativistic quantum mechanics for many-body, which is a self-consistent system constructed by juxtaposed but mutually coupled nonlinear Dirac's equations. The classical approximation of this approach provides the exact Newtonian dynamics for many-body, and the nonrelativistic approximation gives the complete Schrödinger equation for many-body.

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In this paper, we strictly establish classical concepts and relations according to a Dirac equation with scalar, vector and nonlinear potentials. To calculate classical parameters for moving spinor, the local Lorentz transformations for parameters are derived. The calculation shows that different kinds of potentials result in different energy-speed relations, and the energy-speed relations for these potentials are derived in detail. The usual mass-energy relation E = mc 2 holds only for the linear spinor.The energy-speed relations can be used as fingerprints to identify the interactive potentials of a particle by elaborated experiments. The analysis and results of this paper can also provide some natural explanations for the foundation of quantum mechanics, and clarify some long-standing puzzles in the theory.

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In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate. The time coordinate can be preseted according to practical evolving process and keep synchronous with the evolution of the realistic world. In this coordinate system, it is convenient to express the physical laws and to calculate physical variables with clear geometrical meaning. We call it "natural coordinate system". The constructing method for the natural coordinate system is concretely provided, and its physical and geometrical meanings are discussed in detail. In NCS we make classical approximation of spinor equation to get Newtonian mechanics, and then make weak field approximation of Einstein's equation and low speed approximation of particles moving in the space-time. From the analysis and examples we find it is a nice coordinate system to describe the realistic curved space-time, and is helpful to understand the nature of space-time.

The energy momentum tensor of perfect fluid is a simplified but successful model in astrophysics. In this paper, assuming the particles driven by gravity and moving along geodesics, we derived the functions of state in detail. The results show that, these functions have a little correction for the usual thermodynamics. The new functions naturally satisfy the causal condition and consist with relativity. For the self-potentials of the particles, we introduce an extra function W, which acts like negative pressure and can be used to describe dark matter. The results are helpful to understand the relation and interaction between space-time and matter.

The special theory of relativity is the foundation of modern physics, but its unusual postulate of invariant vacuum speed of light results in a number of plausible paradoxes. This situation leads to radical criticisms and suspicions against the theory of relativity. In this paper, from the perspective that the relativity is nothing but a geometry, we give a uniform resolution to some famous and typical paradoxes such as the ladder paradox, the Ehrenfest's rotational disc paradox. The discussion shows that all the paradoxes are caused by misinterpretation of concepts. We misused the global simultaneity and the principle of relativity. As a geometry of Minkowski space-time, special relativity can never result in a logical contradiction.