In Curved Spacetime Dirac Equation And Its Solution Of The Study

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.