Wave Equation And Its Properties In Curved Spacetime

The best way to explore the characteristics of spacetime is to consider adding a perturbation field in spacetime.The key to perturbation is to solve the wave equation of the field.In 1962,Newman and Penrose introduced a spin coefficient in the gravitational field,which led to a re-examination of the separation of bivariable partial differential equations over hundreds of years of history.In this paper,we use the Newman and Penrose formalism to study the wave equations of various massless spin fields in static spherically symmetric spacetime.Based on the decoupling of wave equations in the sipnor form of various massless spin particle,this paper finds that they can be expressed in a single equation.The equation can be used to separate variables,in the arbitrary spherically symmetric spacetime,the solution ofthe angular equation is a spin-spherical harmonic function,the metric function only affects the radial equation.Using the uniform equation of the spin field we obtained,we discuss the wave equations of the spin field in the Schwarzschild,Schwarzschild-de Sitter,de Sitter,and spherical symmetry Rindler spacetime.In the Schwarzschild spacetime,the radial equation of spin field is a hidden confluent Heun equation;in Schwarzschild-de Sitter spacetime,the radial equation of spin field is a hidden Heun equation;while in de Sitter and Rindler spacetime,the radial equation of the spin field is a hidden hypergeometric equation.This shows that the wave equations of massless spin fields in these spacetimes can be accurately solved.Our results lay the foundation for studying the perturbation of this type of black hole.