If there is a null gradient field in 1+3 dimensional space-time, we can set up a kind of light-cone coordinate system in the space-time. In such coordinate system, the metric takes a simple form, which is much helpful for simplifying and solving the Einsteins field equation. This light-cone coordinate system has wonderful properties and has been widely used in astrophysics to calculate parameters. In this paper, we give a detailed discussion for the structure of space-time with light-cone coordinate system. We derive the conditions for existence of such coordinate system, and show how to construct the light-cone coordinate system from usual ones, then explain their geometrical and physical meanings by examples.
In this paper, we provide a procedure to solve the eigen solutions of Dirac equation with complicated potential approximately. At first, we solve the eigen solutions of a linear Dirac equation with complete eigen system, which approximately equals to the original equation. Take the eigen functions as base of Hilbert space, and expand the spinor on the bases, we convert the original problem into solution of extremum of an algebraic function on the unit sphere of the coefficients. Then the problem can be easily solved. This is a standard finite element method with strict theory for convergence and effectiveness.
In this paper, we discuss the equation of state for nonlinear spinor gases in the context of cosmology. The mean energy momentum tensor is similar to that of the prefect fluid, but an additional function of state $W$ is introduced to describe the nonlinear potential. The equation of state $w(a)lesssim -1$ in the early universe is calculated, which provides a natural explanation for the negative pressure of dark matter and dark energy. $W$ may be also the main origin of the cosmological constant $Lambda$. So the nonlinear spinor gases may be a candidate for dark matter and dark energy.
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.
In cosmology, the cosmic curvature $K$ and the cosmological constant $Lambda$ are two important parameters, and the values have strong influence on the behavior of the universe. In the context of normal cosmology, under the ordinary assumptions of positive mass-energy and initial negative pressure, we find the initial singularity of the universe is certainly absent and we have $K=1$. This means total spatial structure of the universe should be a 3-dimensional sphere $S^3$. For the cyclic cosmological model, we have $Lambdalesssim 10^{-24} {rm ly}^{-2}$. Obviously, such constraints would be helpful for the researches on the properties of dark matter and dark energy in cosmology.
By constructing the commutative operators chain, we derive the integrable conditions for solving the eigenfunctions of Dirac equation and Schrodinger equation. These commutative relations correspond to the intrinsic symmetry of the physical system, which are equivalent to the original partial differential equation can be solved by separation of variables. Detailed calculation shows that, only a few cases can be completely solved by separation of variables. In general cases, we have to solve the Dirac equation and Schrodinger equation by effective perturbation or approximation methods, especially in the cases including nonlinear potential or self interactive potentials.
In this paper, we drive and simplify some important equations and relations for an evolving star with spherical symmetry, and then give some simple analysis for their properties and implications. In the light-cone coordinate system, these equations and relations have a normal and neat form which is much accessible than the usual Einstein field equation. So they may be helpful for students to study general relativity and for researchers to do further discussion.
Most fully developed galaxies have a vivid spiral structure, but the formation and evolution of the spiral structure are still an enigma in astrophysics. In this paper, according to the standard Newtonian gravitational theory and some observational facts, we derive an idealized model for spiral galaxy, and give a natural explanation to the spiral structure. We solve some analytic solutions to a spiral galaxy, and obtain manifest relations between density and speed. From the solution we get some interesting results: (I) The spiral pattern is a stationary or static structure of density wave, and the barred galaxy globally rotate around an axis at tiny angular speed. (II) All stars in the disc of a barred spiral galaxy move in almost circular orbits. (III) In the spiral arms, the speed of stars takes minimum and the stellar density takes maximum. (IV) The mass-energy density of the dark halo is compensatory for that of the disc, namely, it takes minimum in the spiral arms. This phenomenon might reflect the complicated stream lines of the dark halo.
The gravitational collapse of a star is a warmly discussed but still puzzling problem, which not only involves the dynamics of the gases, but also the subtle coordinate transformation. In this letter, we give some more detailed investigation on this problem, and reach the results: (I). The comoving coordinate system for the stellar system is only compatible with the zero-pressure free falling particles. (II). For the free falling dust, there are three kind of solutions respectively corresponding to the oscillating, the critical and the open trajectories. The solution of Oppenheimer and Snyder is the critical case. (III). All solutions are exactly derived. There is a new kind singularity in the solution, but its origin is unclear.
The multipole moments method is not only an aid to understand the deformation of the space-time, but also an effective tool to solve the approximate solutions of the Einstein field equation. However, The usual multipole moments are recursively defined by a sequence of symmetric and trace-free tensors, which are inconvenient for practical resolution. In this paper, we develop a simple procedure to generate the series solutions, and propose a method to identify the free parameters by taking the Schwarzschild metric as a standard ruler. Some well known examples are analyzed and compared with the series solutions.
