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.

INTRODUCTION
Black holes are currently a topic of great interest and remain a significant puzzle for scientists (Renner & Wang, 2021;Polchinski, 2017).Researchers strive to comprehensively understand the phenomena and processes that occur in black holes, including modifying Einstein's field equations in general relativity theory (Straight et al., 2020;Shankaranarayanan & Johnson, 2022).General relativity theory can explain astronomical phenomena, focusing on the structure of massive objects like neutron Stars, black holes, quasars, and the universe's expansion (Yagi & Stein, 2016;Barausse, 2019).
The discrepancy between observational and calculation results generated ideas to alter Einstein's general relativity theory.The first hypothesis changed the right side of the Einstein field equation by suggesting the existence of dark matter.The second idea modified the left side of the Einstein field equation by assuming only matter, like dark matter (Yadav & Verma, 2019) and dark energy (Odintsov & Oikonomou, 2019).
Two variational ideas that could be utilized to develop the modified general relativity theory are Formalism based on the Palatini variation and standard metric variation (Ferraris et al., 1982).
These principles are built on the field equation with the Lagrangian linear in R (Buchdahl, 1970).Brans and Dicke finished the extension of the general relativity theory with the development of the scalar-tensor theory of gravity.One specific example of how gravitational interactions in general relativity theory are connected to the scalar field and the tensor field is the scalar-tensor theory.(Brans & Dicke, 1961).
The () theory, which is a part of the metric or Palatini formalism, and scalartensor theory, introduced the basic ideas of gravity theory (Capozziello & Laurentis, 2011;S. Capozziello et al., 2010).Starting with the spherical symmetry solutions in () theory, one could use the Noether symmetry approach to solve the axial symmetry problem.(Capozziello et al., 2010).One of the simplest modifications to gravity theory is the () theory, which generalizes the scalar Ricci of the Hilbert-Einstein equation to the function () of .(Capozziello et al., 2010).
To construct general relativity theory using a semi-classical framework, the formulation of modified general relativity theory was proposed.The most successful method was determined using the () theory.(Capozziello et al., 2010).It has become a new framework for explaining the interaction of gravity (Faraoni & Capozziello, 2011).
The Klein-Gordon equation is beneficial for describing particles in the relativistic quantum mechanics (Bussey, 2022).This equation appears when the effect of relativity is calculated ( ≈ ).The Klein-Gordon in linear form is a second-order partial differential equation.It is a relativistic wave function that represents the dynamics of elementary particles on a relativistic scale (Joseph, 2020).In this research, we also prove how the solution of the Klein-Gordon equation in modified gravity () theory is consistent with the non-relativistic limit.
A simple function represents the solution of the Klein-Gordon equation in Schwarzschild space-time analytically solved for region 0 ≤  ≤ ∞ (Elizalde, 1988;Qin, 2012).In contrast, the Klein-Gordon equation for time-dependent solved using an asymptotic method for certain angular momentum conditions and proved it for the Schwarzschild radius (Rowan & Stephenson, 1976).Some modifications of general relativity such as (), (), and gravity (, ) as a gravitational modification, were constructed to explain unsolved phenomena like dark matter, inflation, etc. (Nojiri & Odintsov, 2007) and (Multamäki & Vilja, 2006).Capozziello et al. reviewed and introduced the fundamental principles of the theory of gravity, more specifically to the scalar-tensor theory and the () theory (Salvatore Capozziello & Francaviglia, 2008;Capozziello et al., 2012).
The confluent Heun functions provide the angular and radial parts of the Klein-Gordon equation solutions.This study clarified how a charged, rotating black hole's gravitational field (Kerr-Newman space-time) affects a charged, massive scalar field (Vieira et al., 2014).Numerical methods are needed to solve the KG equation in curved space-time (Lehn et al., 2018a;Griffith, 2004).Earlier research that sought exact solutions, which either involved approximative expansions or simplifying assumptions to obtain asymptotic solutions, could have been more conclusive at best.(Lehn et al., 2018a).The Klein-Gordon equation is used for a massless scalar field contained in a Casimir cavity and moves in an equatorial orbit (geodesic) (Sorge, 2014).
The solution of Klein-Gordon equations has also been worked out for other black hole models, which succeeded in numerical solution as a periodic function of a black hole (Pourhassan, 2016).The solution of the relativistic Klein-Gordon equation in curved space-time with a massive field was obtained numerically and then compared to the nonrelativistic Coulomb field solution directly through the interference theory (Lehn et al., 2018b).The solutions explain how the gravitational effect work.The spherically symmetric metric represents it.
From the previous research, we expand the solution of Klein-Gordon in curved space-time in general relativity theory by (Lehn et al., 2018a) to the modified gravity theory using the () theory.Some unsolved phenomena in general relativity, such as dark matter and inflation, encourage me to do it as an alternative theory of gravity.The investigation of a gravitational effect in () theory solving the Klein-Gordon equation has never been studied before.

METHODS
The solutions of the Klein-Gordon equations in several gravitational fields and their consequences are fundamental to discuss.It is important to note that, in principle, the physics of these things may be understood by looking at how scalar fields behave in black hole backgrounds.The Klein-Gordon equation must thus be solved for both natural and complex areas, and associated phenomena like the radiation of scalar particles must be investigated.
In this research, we generalize the Klein-Gordon equation in curved space-time for a light particle on the area that gravitationally has a strong interaction.A spherically symmetric metric represents it.First, we extended Einstein's field equation in general relativity to modified general relativity through () theory.Second, the metric for a static spherical solution in Schwarzschild space-time was constructed using ℎ () theory of general relativity.It is known as modified Schwarzschild space-time.From Einstein's action principle as a general function of Ricci scalar (), the general Schwarzschild metric was formulated.Third, we solved the Klein-Gordon equation by substituting the modified Schwarzschild metric by using the method of separation of variables.Lastly, the general solution of Klein-Gordon was derived in exact calculation by separating time, radial, and angular variables, and the numerical solution of the radial equation is also presented.The radial function in terms of a non-linear second-order differential equation was solved by a numerical method using Python.

𝑓(𝑅) Theory of General Relativity
() theory of general relativity is one modification of gravity theory first proposed by Hans Adolph Buchdahl in 1970.Generalization of Einstein-Hilbert's action to become a general function of as follows where  () is the matter Lagrangian, and () is a function of the Ricci scalar.We obtained the Einstein field equation in general relativity theory by setting () = .

Klein-Gordon Equation on Modified Schwarzschild Metric
The relativistic energy of free mass  is (Bjorken et al., 1966), (Romadani & Rani, 2020)  2  = ( 2  2 +  2  4 ) (15) in quantum mechanics,  and  are operators where  is expressed by ℏ   , and  is expressed by −ℏ and substitute to Equation (15) becomes a simple formulation of Equation ( 16) is written by which is known as the Klein-Gordon equation in the Minkowskian metric.The general equation of the Klein-Gordon equation in tensor form is (    +  2 ) = 0 (18) with   =     .Equation ( 19) becomes the covariant metric   and contravariant metric   in modified Schwarzschild is written by (Romadani, 2015) The Klein-Gordon equation on spherically symmetric metrics in Equation ( 19) can be derived becomes where √ =  2  .To solve Equation ( 22), we defined (, , , ) = ()()(, ) (23) by using the separation of variables with substitute Equation ( 23) to ( 22), we found that

Solution for Time Function
From Equation ( 24), time variable  is written as 25) the solution of Equation ( 26) satisfies the boundary limit, and the solution is

Solution for Angular Function
The equation of the angular part from Equation ( 24 and   () is the Legendre polynomial satisfy the Rodrigues formula the angular equation for variable  is written by with the solution () =   +  − (33) because the solution has to cover the latter by allowing  to run negative, we found that () =   (34) the constants factor in front absorbs that into  where 0 ≤  ≤ 2.Equation ( 35) has to require ( + 2) = () so  must be an integer  = 0, ±1, ±2, ….The solution for angular function    (, ) is formulated in Table 1.

Numerical Solutions for Radial Function
Here, we see the numerical solution of the radial equation in Equation (36).We assume  =  =  = 1 to predict the solution of () for  = 0,  = 1, and  = 2. Using Python, we found the numerical solution for the radial function in Tabel 2.
The numerical results for the radial function in Table 2 have been plotted in Figure 1, and we can see that the increasing azimuthal quantum number  is followed by the increasing slope of () (Bjorken et al., 1966).Figure 1 shows the evolution of radial wave function as relativistic effects in () are increased with varying Schwrazschild radius.We numerically evaluate the expression of the radial wave function given by Equation (36).As the Schwarzschild radius increases (for small ), the wave function moves closer to the sphere (Griffith, 2004).For  → 0, the solution of the radial wave function in () reduced to the general relativity (Lehn et al., 2018a).However, the authors have previously demonstrated a gravitational effect similar to the Klein-Gordon equation without a  parameter (standard general relativity).The radial wave solution behaves as a damped oscillator, and in fact, it is a bound solution that satisfies the boundary condition of being zero at infinity.
In Figure 2, the numerical results of radial solution are compared in both modified Schwarzschild in () theory and Schwarzschild metric for  = 0 and  = 1.In approximation, for  = 0 in Equation (32), we get the standard radial function of the Klein-Gordon equation in Schwarzschild metric (Cruz-Dombriz et al., 2009).In other words, the red lines in Figure 2 equal the blue lines.It shows that the Klein-Gordon equation in () theory has a more general form rather than the standard Klein-Gordon equation.In Figure 2, we compared the numerical results of radial wave function between () theory and general relativity theory for  = 0 and  = 1.We can say that the behavior of radial wave function in () theory is consistent with general relativity, especially in a small radius of .We have shown that solutions of the Klein-Gordon equation in () theory can be computed with ordinary numerical methods and that the results are consistent with the non-relativistic limit.In general relativity theory for  fixed, we can see that increasing  also increases the value of ().Still, in (), the value of () decreases because for  greater, the radial part of Equation ( 36

Table 2 .
The numerical results of the radial equation for  = 0,  = 1, and  = 2