Klein-Gordon Equation in

the Gravitational Fieldof

a Charged Point Source

D.A. Georgieva, S.V. Dimitrov, P.P. Fiziev, T.L. Boyadjiev

Gravity, Astrophysics and Strings, Kiten’ 05

1,sin 22222122 dddggdtds

A point particle solution of Einstein-Maxwell field equations has the form:

where

r luminosity variable,,2

1)(2

2

QM

g

222 zyxr radial coordinate

.00

r ,,0 ,,0 rwhen 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 ong:

)2(,)1(211 ggf

,

gfg cl

with

MG 2

M

Q

.2

G

cl

M

Qcl

2

where classical radius, the parameter

connects the classical radius with the Schwarzschild radiusvia

,,,,,,, 2 tmt

22, sin

1sin

sin

1

.011 2

,22

2221 mgg tt

,,,tThe wave function of a massive non-charged scalarfield interacting in the gravitational field (1) of a point-like source satisfies the following Klein-Gordon equation:

,det,2121

ggggg is the D’Alambert operator.

,22

2221 gg tt 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:

.,1,:, ,,,, zzz llllll YllYY

.0

112

222

21

llltt

llmgg

,,,,,, , zlll Ytt

,, zllY

The angular part of the wave function can be explicitly derivedin terms of spherical harmonics

,, zllY eigenfunctions of the angular part of the Laplace-Beltrami operator, i.e.

Separation ofvariables

Then, for the wave function ,tl we obtain the equation

)(, liEt

l Ret

l

l

PR )(

011

222

2

ll P

d

dgllmgEP

d

dg

.01

2222

2

ll R

llmgER

d

dg

d

dg

Due to invariance with respect to time translations one has:

so that the radial function )(lR is a solution of a second order

Ordinary Differential Equation – the radial KGE:

Substituting we get

The final form of KGE

,* dug

dd cl

:*

.

2dg

ggf

gfdu

clcl mE ,

,22

2

llll PPuw

du

Pd

.112

1)( 232

22

ggfgfllggwl

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(u0) ,

111112

22

2

g

ggdu

dg

The function g(u) is implicitly defined as a solution of thefollowing Cauchy problem:

where u0 is an arbitrary constant and is the gravitational mass defect of the point source.

The function g(u) can be given implicitly by:

g(u): u u0 F (g) – F ().

The function F(u) depends on the values of

21222

21222

)(11

)(11ln

12

1ln1

2

2

222

2

gf

gfgf

gfgF

gCase

The Cauchy problem has two singular points: at g (regular) and (irregular).

Case

There are two irregular singularities: g andg .

.1

ln12

2g

g

gg

ggF

1

1arctan

1

2ln

1122

2

2

2

2

gf

gf

g

gfgF

.

CaseCase > 1:> 1:In this case one has three singular points g ,g and g . The first two are regular.They corresponds to the event horizon Mand the classical radius Q/ M respectively.The singular point g is essentially irregular and corresponds to

and g must satisfy: .1, 2 gcl

The general solution of equation (3) depends on two arbitrary constants and the eigenvalue , Pl(u) = Pl (u;C1;C2;). These three parameters can be defined from the boundary conditionsfor the problem and the normalization condition that the wavefunction satisfies.

00 uPl

0lP

010

2

u

l uPdu

- the wave function is zero at the place where the source is positioned.

- Comes from the asymptotic behavior at infinity.

- L2 normalization condition.

,,, 02

2

2

uuPPuwdu

Pdlll

l

,12

1)( 32

22

ufufufllgugwuw ll

...2,1,0,,,)1(211

lugugfuf

,0,00 ll PuP ,010

2

u

l uduP

,

111112

22

2

g

ggdu

dg g(u0)

We use a algorithm based on the Continuous Analogue of Newton’s Method.

Let y note the couple (Pl (u), ), where .

,,0 t

.,, ttuPty l

0,0,0 0 uPyy l *yty

,t

., *** uPy l

1) – we introduce a formal evolution parameter

i.e. we mark

2) – we suppose that and

when where y0 is a given initial approximation

sufficiently close to the exact solution

,,, ZPy l

00

.12

,,

,

2

00

u

l

u

l

ll

lllll

duPduZP

PZuPuZ

PuwPPZuwZ

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)

,uvuPuZ l

.0,0

,)(

0

vuv

uPvuwv ll

.12

1

00

2

1

u

l

u

l duPduvP

5) – the solution of the problem (4) is sought in the form:

where v(u) is a new unknown function. Substituting theexpansion for Z (u) in (4) we get that v(u) is a solution ofthe linear boundary value problem:

6) – if we know v(u) the quantity can be calculated from the equality

(5)

.,1 1,1, kkkkkkkklkkl vPP

1,1, , klklP

k

7) – if y0 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

Case: > 1

Case: > 1