Abstract
In the
present work the question connected with the most adequate
mathematical approach for description probable resonant effects in
black holes vicinities is considered. Metrics of Schwarzschild and
Reissner-Nordstrem are analyzed.
Àííîòàöèÿ
 ðàáîòå ðàññìîòðåí âîïðîñ î íàèáîëåå àäåêâàòíîì ìàòåìàòè÷åñêîì ïîäõîäå ê îïèñàíèþ âîçìîæíûõ
ðåçîíàíñíûõ ýôôåêòîâ â îêðåñòíîñòè ÷åðíûõ äûð. Ïðîàíàëèçèðîâàíû ñëó÷àè ìåòðèê Øâàðöøèëüäà è
Ðåéññíåðà-Íîðäñòðåìà.
1.
Introduction
The question connected with falling of a scalar field packet on horizon of
a black hole (and passing across it) was discussed in many works
and from various positions (see, for example, [1 - 10]).
Nevertheless, the treatment as statements of the problems
connected with it, and the conclusions obtained at their solution,
is extremely important both from general theoretical positions,
and from the point of view of a possibility of experimental check
of effects of behaviour of a matter in a vicinity of a black hole,
is still far from full and settling. Therefore authors continue a
cycle of the works, devoted to interaction of non-point object with
the metrics of the distorted space-time in the vicinity of collapsar. In the
present work some qualitative aspects of possible mechanism x-ray
and gamma-ray bursts, connected with "dyadosphere" [11 - 15]
of ëlectromagnetic black hole" are considered. Also quasi-local
processes of dynamics of a scalar (and electromagnetic)
field packet in metrics of Schwarzschild and Reissner-Nordstrem are analyzed.
2.
Non-monochromatic wave packet in the Schwarzschild
metrics
The Schwarzschild metrics (historically
it would be logically to name Hilbert's metrics)
has the form :
|
ds2 = - (1-2M/r) dt2 +dr2 / (1-2M/r)+r2 d W2, |
| (1) |
where:
d W2 = d J2 + sin2 Jd j2 ;
r, J, j are spherical coordinates;
R = 2GM is Schwarzschild radius,
M is a mass of the black hole, G is a gravitational
constant (G=c=1 ).
Klein-Gordon
equation
(the wave equation for a mass field) for a scalar field
f([(r)\vec], t)
in the given metrics is:
|
- |
1
r2 - 2Mr
|
[r2 |
¶
¶t
|
]2 f+ |
¶
¶r
|
([r2 - 2Mr] |
¶
¶r
|
f) + |
|
|
+cosec J |
¶
¶ J
|
(sin J |
¶f
¶J
|
) + [cosec J |
¶
¶j
|
]2 f- m2 r2 f = 0, |
| (2) |
where m is the field parameter ("mass").
For electromagnetic and gravitational
waves to the left-hand side
it
is added term
-j f/r3 , where for
an electromagnetic field (Wheeler equation
[16]) there is j=1 ,
and for a gravitational field there
is j=2
(Regge-Wheeler [17] and Zerilli equations [18]).
The equation (2)
supposes various possibilities
of separation of variables;
usually as angular parts choose
spherical harmonics
Yl[`(m)] (J, j) .
However if we introduce
in consideration more
general than Schwarzschild metrics ones,
here are required the certain updating.
So, it is represented rational
to use following decomposition:
f(r, t, J, j) = R(r) S(J) expi ([`(m)] j- wt) ,
resulting for angular S(J) and radial parts R(r)
to two following equations [19]:
|
|
¶
¶J
|
[sin J |
¶S (J)
¶J
|
] + [lsin J- |
sin J
|
] S (J) = 0, |
| (3) |
|
|
¶
¶r
|
|
æ
è
|
(r2-2Mr) |
¶R (r)
¶r
|
|
ö
ø
|
+ |
æ
è
|
|
w2 r3
r-2M
|
- m2 r2 - l |
ö
ø
|
R (r) = 0, |
| (4) |
where l is
a constant of separation (eigenvalue).
The general solution Sl[`(m)], 0 (J)
the
equations for an angular part (3) is spheroidal angular function
with spin weight s=0 [4]. If we change of independent (r ® x )
and dependent variables (R (r) ® F (x) ) in the equation (8)
|
r=2M (1-x), R (r) = R0 (r) F (x), |
|
|
R0 (r) º (r/M-2) 2i wM exp(iM |
Ö
|
w2 - m2
|
(r/M-1)), |
|
then (8) may be re-writing as:
|
|
d2 F (x)
dx2
|
+ A (x) |
dF (x)
dx
|
+ B (x) F (x) = 0 |
| (5) |
|
A (x) = (b+ |
g
x
|
+ |
d
x-1
|
), B (x) = |
abx - h
x (x-1)
|
, |
| (6) |
|
a = -(1+2i wM) + |
i (2 w2 M - m2 M)
|
, |
| (7) |
|
b = 4iM |
Ö
|
w2 -m2
|
, g = 1+4i wM, d = 1, |
| (8) |
|
h = l- 2iM |
Ö
|
w2 - m2
|
- 2i wM +8wM2 |
Ö
|
w2 - m2
|
-4M (2 w2 M - m2 M). |
| (9) |
The equation (5)
belongs to a class confluent Heun
equations (CHE) [20].
It is necessary to note that the
spheroidal equation (3) is special case CHE (if the term
corresponding b in (5) is equal to zero)).
The point x=0 corresponds to event horizon r=2M , and the
point x=1 is singularity r=0. The standard solution of the
equation (5) in a vicinity x=0 (at |x | < 1 , w ¹ i/4M, 2i/4M... ) (so-called "Frobenius solution") has the form:
|
(1) F (x) º Hc(a) (b, a, g, d,h;x) = |
¥
å
k=0
|
c{a }k xk, |
|
|
Hc(a)(b, a, g, d, h; x=0) = 1, |
| (10) |
where
factors
c{a }k are defined from a recurrent formula
|
-(k+1) (k + g) c{a }k+1 + (k ((k + g-1 +d- b) +h) c{a }k- |
|
|
+ b(k-1 + a) c{a }k-1 = 0, c{a }0 = 1, c{a}-1 = 0. |
| (11) |
The second linearly independent
solution of the equation (5) in the vicinity x=0 is
the local solution in a
vicinity of infinity ( x = ¥), so called
"Thome solution" [21] :
|
(2)F (x) º Hc(r) (b, a, g, d, h; x) = |
¥
å
k=0
|
c{r }k xa- k, |
|
|
|
lim
x ® -¥
|
Hc(r) (b, a, g, d, h; x) = 1, |
| (12) |
|
- b(k+1) c{r }k+1 + ((k + a) (k - b-d- g+1)-h) c{r }k- |
|
|
-(k + a- g)(k-1 + a) c{r }k-1 = 0, c{r }0=1, c{r }-1 = 0. |
| (13) |
Legitimacy of such
choice is caused by that second Frobenius solution is not
linearly independent with first one (Hc(a))
for all w: if w = 0 then characteristic exponent 1-g = 0.
Further, it is necessary to note, that (2)F(x) is the
recessive solution only if Re(-b) > 0 , i.e. if |w| < m .
In a vicinity of w = m both solutions possess the
structure with singularities of different type. The first solution (1)F (x)
has singularity due to factors c{a }k (at |w| > m ),
and the second one (2)F(x) has singularity due to the
term xa (at | w| < m ). Thus, if we consider the central two-point connection problem
with given asymptotics of solutions (its correspond to "falling" and
"reflected" waves in terms
quantum mechanics) near to regular (event horizon)
and irregular points (in the infinity), then general solution
will
represent parametric resonant function.
It is necessary to note,
that interpretation
of this fact
can be based on the introduction of än abnormal dispersion" analog concept
for
the metrics
considered as the medium
of a scalar field dynamics.
Amplitude of a field
|
F(x) ~ Hc(r)(b(w,w0), a(w,w0), g(w), 1,h(w,w0);x), w0 º m, |
|
in this case is an analog
a role of displacement
of an electric charge
in the standard theory of dispersive media. It is possible to enter
also a parameter of refraction
and permettivity of the metrics
for a scalar field. As it has
above mentioned, the equation
(2) for an electromagnetic
field (at j=1 , Wheeler case)
is led to CHE so the approach is quite universal.
Continuation of the solution äcross the horizon" (in a vicinity
of a singular point x=1) demands the application corresponding s
-homotopic transformation [22] of CHE leading to
following pair of basic Heun equation solutions:
|
(1) |
~
F
|
(x) = (x-1)1 - d Hc(a) (b, - a+ d-1, g, 2 - d, h - g(1 - d); x) = |
|
|
= Hc(a) (b, - a, g, 1, h; x), |
| (14) |
|
(2) |
~
F
|
(x) = (x-1)1 - d Hc(r) (b, -a+ d-1, g, 2 - d, h - g(1 - d); x) = |
|
|
= Hc(r) (b, - a, g, 1, h; x). |
| (15) |
Thus, the general solution of the equation (5) äbove"
and ünder horizon" (0 < x < 1 ) represents a composition of
confluent Heun functions Hc(a) (ängular") and Hc(r)
("radial"):
|
R-region: C1 Hc(a) (b, a, g,1, h; x) + C2 Hc(r) (b, a, g, 1, h; x), |
| (16) |
|
F-region: |
~
C
|
1
|
Hc(a) (b,- a, g, 1, h; x) + |
~
C
|
2
|
Hc(r) (b, -a, g, 1, h; x). |
| (17) |
Considering the solution
in F-region (as analytical continuation of the solution of the
equation in R-region), it is possible to show (following
[23]) that reflection of horizon R is not equal
to zero so interpretation of the general solution as superpositions
of the falling and reflected waves is lawful.
There is no influence
the decomposition method of the solution of the
Klein-Gordon equation over
resonance existence. Decompose of the equation of Klein-Gordon
(2) with use of a method of "phase shift"
[24]:
|
f(r, t, J, j) = |
å
l[`(m)]
|
|
ó
õ
|
¥
-¥
|
r-1 fw,l (r) Yl[`(m)] (J, j) exp(-i wt) d w, |
| (18) |
|
f w, l (r) = vexp(i wr* + i g (r)), r* = r+2M ln|r/2M-1 |, r = 1-2M/r, |
|
|
g (r) = (- |
|
) ln(1 - r) + |
¥
å
n = -1
|
(an+bn r) (4 r-4 r2) n, |
|
where an, bn
are some factors, which obvious kind is defined by substitution
of the solution in the initial equation
(with corresponding recurrent formulae). The phase g(r) is defined
by the same way;
so as there exists next condition:
|
b1 = |
1
|
|
æ
è
|
2 |
Ö
|
w2 - m2
|
a1 + |
|
|
|
1
4
|
( |
|
) 2 - |
1
4
|
+i (a1 + |
1
2
|
- |
|
) |
ö
ø
|
-2a1, |
| (19) |
then the resonance
effect at w @ m
will be realize in the same way, as
well as at decomposition with the Heun functions. In a
complex conjugated functions to the augmented waves fwl (r) together with the last ones form pair linear
independent basic solutions, and analytical continuation under
horizon gives effect of reflection directed by [23].
Nevertheless, obtaining the solution is in such a way possible
only for additional assumptions and don't extend on other
metrics and consequently can serve only in the didactic purposes.
According to results of work [25], if we accept as basic functions
the spherical harmonics and pseudo-flat waves:
|
f(r, t,J, j) = |
å
l[`(m)]
|
|
ó
õ
|
¥
-¥
|
f(r, w, l, |
-
m
|
) Y l[`(m)](J, j) exp(-i wt) d w, |
| (18a) |
then the solution of the Klein-Gordon equation for Schwarzschild metric
is the combination of pair linearly independent
solutions with the preassigned asymptotics:
|
f(r, w, l) = cl (-w) yl (r, w) + cl(w) yl (r, -w), |
| (20) |
|
|
lim
r ®2M
|
yl (r, w) ~ (|r-2M |/2M) -2i wM, |
|
|
~ exp(i r |
Ö
|
w2 - m2
|
+i M ln r (2 w2 - m2) / |
Ö
|
w2 - m2
|
) / ( |
Ö
|
w2 - m2
|
r i l+1). |
|
Accordingly, for Green's function G (r, r¢, w) , being
the solution of the Klein-Gordon equation with the right
hand d4 (X-X¢) (X = (r, t) ):
|
(-¶m gmn |
Ö
|
-g
|
¶n +m2 |
Ö
|
-g
|
) G(X, X¢) = d4 (X-X¢) |
|
we obtain representation in the
form of:
|
G (X, X¢) = |
å
l[`(m)]
|
|
ó
õ
|
¥
-¥
|
Gl (r, r ¢, w) Yl[`(m)](J, j) Y*l[`(m)] (J¢, j¢) exp(-i w(t-t¢)) d w, |
| (21) |
where Gl (r, r¢, w)
is the kernel satisfying Klein-Gordon
equation with the right hand d(r-r¢).
Research of the
obtained Green function by means of its representation with the
use of
basic solutions (20) has allowed to establish property of
stability of vacuum in a vicinity of Schwarzschild horizon: there
is no flux of particles at infinity in conditions of stability of
a black hole (öut"-conditions on horizon of the absolute past,
ïn"-conditions on horizon of the absolute future). However it
contradicts to Hawking result [26,2] about existence of thermal
radiation of a black hole with temperature kT=1/8 pM .
Apparently, this result is caused by acceptance
of aprioristic assumptions of asymptotics of solutions
of the Klein-Gordon equation
on the event horizon.
It is represented reasonable to use
as basic functions above-stated confluent Heun functions:
with its application an event horizon may be the generator of particles
and property of stability of vacuum in its
vicinity because of structure of the common solutions
(16) and (17) is according to requirements of Hawking theory.
Thus, naturally there is a question on an opportunity
of generation of particles not only
in the vicinity of event horizon, but also in some
"macroscopical" its vicinity caused by resonances
of solutions
w @ m .
3. Reissner-Nordstrem metrics and physical processes in the dyadosphere
Studying of gamma-bursts
and attempts of an explanation of
their mechanisms have led to creation of
several new theories on a joint of astrophysics,
the theory of elementary particles and cosmology.
Major candidates for roles of the
theories explaining all set of the
phenomena observable during gamma-bursts
are the mechanisms considering
occurrence with shock-wave processes in
a barion
matter
in vicinity
of black holes (of external and internal genesis),
Compton processes in magnetized
plasma and generation of electron-positron pairs in "dyadosphere"
of an ëlectromagnetic black hole" (EMBH) with the
electric charge Q [12-14].
From the point of view of developed
above the approach based on
use of basic functions with "natural" asymptotics,
it is appropriately
consider the problem on its generalization on the
case of Reissner-Nordtream metrics,
describing space-time in a vicinity
of an electromagnetic black hole.
Besides, this approach
is adequate applied in enough big region of
"classical" dyadosphere,
and can give its additional description.
According to the dyadosphere theory,
there is region above EMBH horizon
|
rEMBH º 1.47 ·105 m(1 + |
Ö
|
1 - x2
|
) < r < r ds º 1.12 ·108 |
Ö
|
mx
|
cm, |
|
|
m = M/M Sun > 3.2, x = Q/Qmax £ 1, |
|
which it is possible to present
in the form of set of concentric condensers with thickness d0 << MG/c2 and charge surface density s(r) = Q/4 pr2 . If to assume d0 = (h/2p)/me c , rate
of generation of pairs e+-e- during polarization of
Schwinger-Heisenberg
vacuum [28,29] represented in the form:
|
|
dN
dt
|
= |
1
4 pc
|
( |
4e s
(h/2p)
|
) 2 exp(-psc / s) 4 pr2 ( |
(h/2p)
me c
|
), sc = Ec/4 p = m2e c3/4 p(h/2p) e. |
|
After the rise of "double layers" of electron-positron
plasma the evolution of dyadosphere
and PEM (pair-elecromagnetic pulse) origin
is described by the various theoretical
models based on relativistic hydrodynamics [15,30]. Application of
the mathematical approach based on the theory of confluent Heun equation,
in case of EMBH is caused by essential similarity of
Schwarzschild and Reissner-Nordstrem metrics
and corresponding
Klein-Gordon equations solutions.
In the vicinity of EMBH horizon it is possible to write the equation of type (2) with
an additional member -f/r3 (generalization of the Wheeler equation)
for electromagnetic
waves; after of some its transformations it is possible to lead
to CHE form.
Dispersion of electromagnetic waves on the metrics in a
dyadosphere is very complicated phenomenon;
there are a possibility, when dependence of a
parameter of refraction of the metrics will
correspond to a case of existence of a zone of
öptical less dense medium",
so for short-wave radiation
(x-rays and the gamma-rays) arises full reflection.
If we consider dynamics of a
scalar field of electromagnetic oscillators with which it
is possible to simulate double layers in the dyadosphere, then after
separation of variables in the Klein-Gordon equation for a radial part gets
solution (5), but parameters in it, naturally, already others:
|
a = - (1+i (2 wM-qQ)) +i (2 wM-qQ) / |
Ö
|
w2-m2
|
, |
|
|
b = 4i (M-Q2/2M) |
Ö
|
w2-m2
|
, g = 1+2i (2 wM-qQ), d = 1, |
|
|
h = l-b/2-i (2 wM-qQ) + 4 (M-Q2/2M) |
Ö
|
w2-m2
|
(2wM-qQ) - |
|
|
-4 ((2 wM-qQ) w-m2M) (M-Q2/2M.) |
|
Nevertheless, an origin of a parametrical resonance probably
and in this case. How it is possible to interpret the resonant
phenomena in dyadosphere and corresponding disassembled earlier in
a small vicinity of horizon asymptotic effects of behaviour
of common CHE solution? For the exact quantitative analysis
introduction in consideration of effects of quantum
electrodynamics and precision details of the theory of
polarization of vacuum is necessary. However already now it is
possible to assume, that the behaviour electron-positron
plasma will be defined by essentially nonlinear effects connected
with dynamic structure of a scalar electromagnetic field,
connected with generation in "condenser layers" of dyadosphere
e+ - e--pairs. Consideration and the comparative analysis of
effects in the dyadosphere demand the use of bifurcation theory
and development of special codes for numerical
modeling processes at the gamma-bursts, using it.
We shall note, that the theory of gamma-bursts is
extremely roughly developing section of the astrophysics
essentially influencing already settled concepts. Rather
interesting in this plan its communication with the theory of
black holes is represented. For example, in work [31] the
hypothesis of compact objects of new type (ßtars of dark energy"),
most advantageously explaining many effects observable at
gamma-bursts is put forward. It is interesting, that here there
is a certain coordination with results of work [32]. It is
necessary to note, that conclusions rather (in particular) the
structures of the horizon made by means of considered above
mathematical device somewhat also demand the certain updatings of
concept of horizon and influence of its topology on external
processes.
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