Klein-Gordon Equation in the Gravitational Field of a Charged Point Source D.A. Georgieva, S.V. Dimitrov, P.P. Fiziev, T.L. Boyadjiev Gravity, Astrophysics and Strings, Kiten’ 05

A point particle solution of Einstein-Maxwell field equations has the form: where luminosity variable, radial coordinate when where finite luminosity of the point source

We use the metric coefficient g as an independent variable instead of the radial coordinate or luminosity variable. This gives the following dependence of the luminosity variable on g: with where classical radius,the parameter connects the classical radius with the Schwarzschild radius via

The wave functionof a massive non-charged scalar field interacting in the gravitational field (1) of a point-like source satisfies the following Klein-Gordon equation: is the D’Alambert operator. in case of spherical symmetry, described by metric (1). angular part of the Laplace- Beltrami operator. Then, in case of spherical symmetry the KGE has the form:

The angular part of the wave function can be explicitly derived in terms of spherical harmonics eigenfunctions of the angular part of the Laplace- Beltrami operator, i.e. Separation of variables Then, for the wave functionwe obtain the equation

Due to invariance with respect to time translations one has: so that the radial function is a solution of a second order Ordinary Differential Equation – the radial KGE: Substitutingwe get The final form of KGE

We change the variables with the tortoise coordinate u is a dimensional variable. From formula (2) we get for g(u) : Introducing becomes (dimensionless) the KGE with a potential (3)

g(u 0 )  k   The function g(u) is implicitly defined as a solution of the following Cauchy problem: where u 0 is an arbitrary constant and k is the gravitational mass defect of the point source. The function g(u) can be given implicitly by: g(u): u  u 0  F (g) – F ( k  ). The function F(u) depends on the values of 

Case  The Cauchy problem has two singular points: at g  (regular) and (irregular). Case  There are two irregular singularities: g  and g .

Case  > 1: In this case one has three singular points g , g   and g . The first two are regular. They corresponds to the event horizon  M and the classical radius  Q  / M respectively. The singular point g  is essentially irregular and corresponds to  and g must satisfy:

The general solution of equation (3) depends on two arbitrary constants and the eigenvalue , P l (u) = P l (u;C 1 ;C 2 ;  ). These three parameters can be defined from the boundary conditions for the problem and the normalization condition that the wave function satisfies. - the wave function is zero at the place where the source is positioned. - Comes from the asymptotic behavior at infinity. - L 2 normalization condition.

g(u 0 )  k  

We use a algorithm based on the Continuous Analogue of Newton’s Method. Let y note the couple (P l (u), ), where  . 1) – we introduce a formal evolution parameter i.e. we mark 2) – we suppose thatand whenwhere y 0 is a given initial approximation sufficiently close to the exact solution

3) – let us put where dot denotes the derivative by t. 4) – applying CANM to the spectral problem we obtain the following system for Z(u) and  (4)

5) – the solution of the problem (4) is sought in the form: where v(u) is a new unknown function. Substituting the expansion for Z (u) in (4) we get that v(u) is a solution of the linear boundary value problem: 6) – if we know v(u) the quantity  can be calculated from the equality (5)

7) – if y 0 is a given initial approximation, at each iteration k = 0,1, 2,... the next approximation to the exact solution is obtained by formulas – a given discretization of “time” t. 8) – the linear boundary value problems (5) are solved at each step using a hermitian spline-collocation scheme of fourth order of approximation.

Case:  < 1

Case:  = 1

Case:  > 1