Solution of Klein-Gordon Equation in F(R) Theory of Gravity

Abstract

The theory, as a modification of the general relativity theory, is frequently employed as an alternative theory of gravity and offers a promising avenue for addressing the challenges of formulating a quantum gravity theory. In this study, by applying the separation method of time, radial and angular variables, we derived the general solution of the Klein-Gordon equation in a curved space-time using modified Schwarzschild metric. We modified Ricci scalar form in Einstein’s action principle as a general function of Ricci scalar and formulated the general Schwarzschild metric. The solution of the time function was analytically obtained in exponential form, and the solution of the angular function in terms of Legendre polynomial depends on azimuthal and magnetic quantum numbers. The radial function in terms of a non-linear second-order differential equation was solved by a numerical method using Python. The solutions described the gravitational effect for a light particle on the area gravitationally has a strong interaction, represented by a spherically symmetric metric. For small (in Schwarzschild radius), the results analytically show that the gravitational effect in this region is massive. It follows that even light would be drawn into a black hole and unable to escape. For further research, it is expected to extend the Klein-Gordon equation in relativistic quantum mechanics to modified general relativity theory. This theory offers a different way of looking at the effects of gravity in quantum field theory.