THE SEPARATION OF VARIABLES METHOD FOR SOLVING THE KLEIN-GORDON EQUATION IN CURVED SPACETIME
Hashim Gad Elseed, Mohammed Hashim Albashir, Hajhamed Diab Aljaly, Naga Abdalaziz
This study was carried out with the aim of finding a solution to the Klein-Gordon equation in a curved spacetime using the separation of variables method. The first step involved transforming the equation into a form that was suitable for separation of variables. This was achieved by using a Fourier transform to separate the time and spatial variables. Next, a separable solution was assumed in the form of â–ˆ(Ï†(t,x,y,z)=T(t)X(x)Y(y)Z(z),)which was then substituted into the Klein-Gordon equation. The variables were separated by multiplying both sides of the equation by â–ˆ(X(x)Y(y)Z(z)), resulting in four separate ordinary differential equations (ODEs). These ODEs were solved using standard methods such as separation of variables or characteristic equations. The general solution for the Klein-Gordon equation was found by combining the solutions of the four separate ODEs into a single solution of the form â–ˆ(Ï†(t,x,y,z)=âˆ‘c_n T_n (t)X_n (x)Y_n (y)Z_n (z)), where c_n are constants and â–ˆ(T_n (t),X_n (x),Y_n (y)), and â–ˆ(Z_n (z))are the solutions of the separate ODEs. The final step was to determine the values of the constants â–ˆ(c_n ) that satisfied the boundary conditions for the Klein-Gordon equation. This was done by using methods such as the method of eigenfunctions or the method of Greens functions. The results of this study showed that the separation of variables method is an effective way to solve the Klein-Gordon equation in a curved spacetime. These findings have important implications for our understanding of quantum field theory in curved spacetime and provide a basis for further research in this area.
Keywords: Klein-Gordon equation, Separation of variables method, Curved spacetime, Quantum field theory, Ordinary differential equations
Journal Name :
EPRA International Journal of Multidisciplinary Research (IJMR)
Published on : 2023-02-05