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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
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