弯曲时空中Dirac方程及其解的研究

【作者】 吕嫣；

【导师】 桂元星；

【作者基本信息】 大连理工大学 ， 理论物理， 2004， 硕士

【摘要】 Dirac方程最初是Dirac为解决平直时空中满足Klein-Gordon方程的粒子的负几率与负能问题而建立的。处于弯曲时空(如黑洞附近)的粒子满足的就是含时空曲率的Dirac方程即弯曲时空的Dirac方程。弯曲时空Dirac方程用简洁的形式普适地表示了任一Fermions物质场在任一时空背景所应满足的场方程，是研究宇宙的有力工具。弯曲时空中Dirac方程的解一方面可以解释并预言Dirac粒子在弯曲时空中的行为，另一方面也反映了高度弯曲时空附近天体(如黑洞)的自身演化图象。本文在相关理论的基础上，首先按照Chandrasekhar的方法对一般静态球对称时空中Dirac方程进行分离变量和退耦，得到该类时空下的普通微分形式的Dirac方程。从结果来看，不同的静态球对称时空其径向方程因视界函数的不同而不同，而角向方程则具有相同的的形式。然后以Schwarzschild-de Sitter时空作为背景求解Dirac方程，分别用数值模拟与WKB近似方法求其数值解和半解析解，分析并比较不同的宇宙学常数对散射的影响，结果表明宇宙学常数的值越小，黑洞对粒子的散射越强。更多还原

【Abstract】 Dirac equation has been established in the initial purpose of solving the negative energy and negative probability problems of particles satisfied the Klein-Gordon equation in flat spacetime. The particles located in curved spacetime, for example the region beside a black hole, obey spacetime-curvature-included Dirac equation, i.e. the Dirac equation in curved spacetime. In curved spacetime, the Dirac equation, which expresses the field equation that an arbitrary Fermions material field obeys in an arbitrary background spacetime universally in a neat form, is a powerful method in studying the universe. The solutions of Dirac equation in curved spacetime can on one hand explain and further predict the behavior of Dirac particles in curved spacetime, on the other hand make out the evolvement picture of celestial body (for example the black hole) beside highly curved spacetime. Based on related theories, in this paper, Dirac equations are separated and decoupled in a general static spherical symmetric spacetime according to Chandrasekhar’s method firstly, and the general differential Dirac equations in this spacetime are obtained. The results show that, the radial equations in different spherical symmetric spacetime would be different for different horizon functions, while the angular equations are expressed in a same form. Then the Dirac equations are solved under the background of Schwarzschild-de Sitter spacetime. Two methods, numerical calculation and quantum WKB approximation, are employed to obtain the numerical and semi-analytical solutions of the equations respectively. The influences of different cosmological constant on the scattering of Dirac particles from black hole are analyzed and compared, the result turns out to be that the smaller the value of the cosmological constant is the stronger the scattering of the particles from the black hole is.更多还原