Abstract:
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 which was then substituted into the Klein-Gordon equation. The
variables were separated by multiplying both sides of the equation by , 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 , where are constants and , and
are the solutions of the separate ODEs. The final step was to determine the values of the constants 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 Green's 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.
