Outgoing and Ingoing Klein-Gordon Waves Near Black Holes
1. Out-Going and In-Going Klein-Gordon Waves Very Near The Blackhole Event
Horizons
Ferdinand Joseph P. Roaa
, Alwielland Q. Bello b
, Engr. Leo Cipriano L. Urbiztondo Jr.c
a
Independent Physics Researcher, 9005 Balingasag, Misamis Oriental
b
Natural Sciences Dept., Bukidnon State University
8700 Malaybalay City, Bukidnon
c
IECEP, Sound Technology Institute of the Philippines
Currently connected as technical consultant/expert for St. Michael College of Caraga (SMCC)
8600 Butuan City, Agusan del Norte
Abstract
In this elementary exercise we consider the Klein-Gordon field in the background of Schwarzschild
space-time metric. Very near the event horizon the radial equation of motion is approximated in form and we
obtain oscillatory solution in the Regge-Wheeler coordinate. The time and radial solutions are then recast in
the outgoing and ingoing coordinates that consequently lead to the outgoing and ingoing waves that have
respectively dissimilar (distinct) analytic properties in the future and past event horizons.
Keywords: Schwarzschild metric, pair production, scalar field, Regge-Wheeler coordinate, event horizons
1. Introduction
This paper is mainly based on our answers
to an exercise presented on page 142 of [1]. The
exercise falls under the topic related to Hawking
radiation although this present document does not
yet tackle the proper details of the cited subject
matter of Hawking radiation. The scope of this
paper only covers the important details in our
solutions to Klein-Gordon field equation against
the background of Schwarzschild space-time
metric [2]. There are basic features of these
solutions that we understand as specifically
relevant to Hawking radiation.
The problem of Hawking radiation was first
explored in the middle of 1970โs in Stephen
Hawkingโs paper [3]. By taking quantum
mechanics into account especially in extreme
proximity to a very strong gravitational field of a
blackhole, Hawking realized that blackholes could
emit particles through pair production happening
so asymptotically close to a very strong
gravitational field. In Hawkingโs results, this
emission of particles is thermal as if blackholes
were hot bodies whose temperatures are
proportional to blackholesโ surface gravities.
In his pioneering approach, Hawking
illustrated this radiation using a scalar field [3].
We shall no longer present here the lengthy
elaboration in his cited 1975 paper. In Hawkingโs
treatment, quantum mechanics was forcefully
implemented in the classical solutions of those
field equations.
2. Immersing a classical field (example, scalar)
in a gravitational field quite complicates
Lagrangians and their resulting equations of
motion because of the presence of non-flat metric
components which represent for gravitational
field. However, a coordinate system can be chosen
so as to recast the equations of motion from which
we can then write their approximate forms
especially so close to event horizons. As earlier
said it is this feature that is so useful in one
approach to the problem of Hawking radiation
2. Klein-Gordon Equation Of Motion In
The Background Of Schwarzschild
Spacetime Metric
We start with the scalar action[4]
๐ = โซ ๐4 ๐ฅ โ (1)
along with a Lagrangian given for a scalar field
โ = โโ๐
1
2
( ๐ ๐๐(๐๐ ๐)( ๐๐ ๐) + 2๐(๐)) (2)
where in the metric signature of positive two (+2)
we take the scalar potential as
๐( ๐) =
1
2
๐2 ๐2 (3)
To get for the equation of motion for the scalar
field, we vary this action with respect to the
variation of the scalar field. This variation we
write as
๐ฟ๐ = โซ ๐๐ ๐ |
๐ฟโ
๐ฟ(๐ ๐ ๐)
๐ฟ๐|
๐ฅ ๐ด
๐
๐ฅ ๐ต
๐
+ โซ ๐4 ๐ฅ (
๐ฟโ
๐ฟ๐
โ๐ ๐
๐๐ (
๐ฟโ
๐ฟ(๐ ๐ ๐)
)) ๐ฟ๐
(4)
This variation is carried out noting that metric
fields ๐ ๐๐ are independent of the variation of ๐ .
In the classical field theory the varied scalar field
must vanish at the two end points A and B,
๐ฟ๐( ๐ด) = ๐ฟ๐( ๐ต) = 0 and by stationary
condition, the variation of this scalar action must
vanish that is, ๐ฟ๐ = 0. Thus following these
variational (extremal) conditions, we obtain for the
Euler-Lagrange equation for the classical scalar
๐ฟโ
๐ฟ๐
โ ๐๐ (
๐ฟโ
๐ฟ(๐๐ ๐)
) = 0
(5)
Upon the substitution of (2) in (5) we get the
equation of motion for the scalar field in curved
spacetime
1
โโ๐
๐๐[โโ๐๐ ๐๐(๐๐ ๐) ] โ ๐2 ๐ = 0
(6)
where we take note of the covariant four-
divergence
1
โโ๐
๐๐ [โโ๐๐ ๐๐(๐๐ ๐) ] = โ ๐( ๐ ๐๐(๐๐ ๐) )
(7)
that is given with metric compatible connections.
In the metric signature of positive two the
fundamental line element in the background of
Schwarzschild spacetime metric is given by
๐๐2 = โ๐๐๐ก2 + ๐๐๐2 + ๐2 ๐๐2 + ๐2 ๐ ๐๐2 ๐๐๐2
๐ = ๐โ1 = 1 โ
2๐บ๐ ๐
๐
(8)
We note in these that the square of the speed of
light is unity (๐2 = 1, Heaviside units) and that
๐ ๐ is the mass of the gravitational body. In
addition, our spacetime is given with a set of
spacetime coordinates ๐ฅ ๐ = (๐ฅ0 = ๐ก; ๐ฅ1 =
๐; ๐ฅ2 = ๐; ๐ฅ3 = ๐ )
Our convenient solution to think of is in
product form so as to easily facilitate the
separation of variables. Such product solution is in
the form
๐( ๐ฅ0, ๐, ๐, ๐) = ๐( ๐ก) ๐ (๐)ฮ(๐)๐(๐)
(9)
This, given the background spacetime of (8), gives
the following component equations of motion
1
๐
๐ ๐
2
๐ = โ๐ ๐
2
(10.1)
1
ฮ
1
๐ ๐๐๐
๐๐[ ๐ ๐๐๐(๐๐ฮ)] โ
๐ ๐
2
๐ ๐๐2 ๐
= โ๐ ๐( ๐ ๐ + 1)
(10.2)
1
๐
๐0
2
๐ = โ๐2 (10.3)
3. 1
๐
1
๐2
๐
๐๐
[ ๐๐2
๐๐
๐๐
] โ
๐ ๐( ๐ ๐ + 1)
๐2 = ๐2 โ
๐2
๐
(10.4)
Partial differential equations (Pdeโs)
(10.1) and (10.3) can be solved by ordinary
method such as separation of variables. Depending
on the signs of the constants in these equations, the
respective solutions can take oscillatory forms.
Pde (10.2) is of Hypergeometric type[5] so as
(10.4), which is complicated by the presence a
non-flat metric component ๐.
With the substitution ๐ค = ๐๐๐ ๐ we
convert (10.2) into a Legendre equation [6].
๐ ๐ค((1 โ ๐ค2) ๐ ๐คฮ )
+ ( ๐ ๐( ๐ ๐ + 1) โ
๐ ๐
2
1 โ ๐ค2
)ฮ
= 0
(10.5)
This equation is solved by the associated [7]
Legendre polynomials
ฮ ๐ ๐
๐ ( ๐ค) = ๐ด ๐ ๐
๐ (1 โ ๐ค2) ๐/2 ๐ ๐
๐๐ค ๐
๐๐ ๐
(๐ค) (10.6)
where ๐ = | ๐ ๐|and ๐ด ๐ ๐
๐ = ๐๐๐๐ ๐ก๐๐๐ก. In this, the
Legendre polynomials ๐๐ ๐
(๐ค) are derived by
๐๐ ๐
( ๐ค) =
1
2 ๐ ๐(๐ ๐! )
๐ ๐ ๐
๐๐ค ๐ ๐
( ๐ค2 โ 1) ๐ ๐
(10.7.1)
These polynomials solve the Legendre equation
given in the following form
(1 โ ๐ค2) ๐ ๐ค
2 ๐๐ ๐
โ 2๐ค๐ ๐ค ๐๐ ๐
+ ๐ ๐( ๐ ๐ + 1) ๐๐ ๐
= 0
(10.7.2)
We can combine solutions (10.6) and
solutions of Pde (10.1) to form spherical
harmonics[8]
๐๐ ๐
๐ ๐
= ฮ ๐ ๐
๐ ๐ ๐ ๐
(10.8)
The component solution
๐ ๐ ๐
= ๐ต ๐ ๐ ๐๐ ๐ ๐
๐ต ๐ = ๐๐๐๐ ๐ก๐๐๐ก (10.9)
satisfies (10.1) for a given integral number ๐ ๐
whose square is greater than zero (๐ ๐
2
> 0). Let us
note here that the absolute values of ๐ ๐ are
restricted up to a given integral value of ๐ ๐. That
is
๐ = | ๐ ๐| = 0, 1,2,3, โฆ, ๐ ๐ (10.10)
where
๐ ๐ = 0, 1,2, 3,โฆ , ๐๐๐ก๐๐๐๐๐ (10.11)
In an asymptotically flat space (or that is
the Minkowskiโs spacetime), where the metric
component ๐ is unity (๐ = 1), the radial equation
of motion Pde(10.4) takes a familiar Bessel form
1
๐
1
๐2
๐
๐๐
( ๐2
๐๐
๐๐
) โ
๐ ๐( ๐ ๐ + 1)
๐2 = โ๐ผ2
๐ผ2 = ๐2 โ ๐2 (10.12)
One convenient form of solution to (10.12) that we
choose is
๐ ๐ = ๐๐(1/๐) ๐ ๐ผ1 ๐ (10.13)
๐ผ1 = ยฑ๐๐ผ
๐ = ๐ ๐ (integral number)
This satisfies the associated Besselradial equation
for a given integral value of ๐.
๐2 ๐๐
๐๐2 +
2
๐
( ๐ผ1 ๐ + 1)
๐๐๐
๐๐
+ (
2๐ผ1
๐
โ
๐ ๐( ๐ ๐ + 1)
๐2
) ๐๐ = 0
(10.14)
We will only give first two of its solutions.
๐๐ = 0 =
๐0
๐
(10.15.1)
๐๐ = 1 =
๐0
๐ผ1 ๐2
( ๐ผ1 ๐ โ 1)
(10.15.2)
3. The Time And Radial Equations of
Motion And Their Solutions Given In
Outgoing And Ingoing Coordinates
In the presence of strong gravitational
field we are back with the radial equation of
motion that is given by (10.4), where gravity takes
effect through the metric tensor component, ๐. The
authors of this draft have not dwelt on the
solutions to the said equation in terms of the radial
coordinate. We believe that such equation is quite
complicated to obtain for solutions in closed form.
However, this radial equation of motion can be
recast in an alternative radial coordinate so as to
write this equation in a form from which we can
4. obtain for the approximate equation very near the
blackhole horizon.
Using Regge-Wheeler coordinate,
๐โ = ๐ + 2๐บ๐ ๐ ๐๐(
๐
2๐บ๐ ๐
โ 1)
โ๐ > ๐๐ป (= 2๐บ๐ ๐ )
๐๐
๐๐โ = (
๐๐โ
๐๐
)
โ1
= ๐
(11.1)
we recast (10.4) into the form
1
๐
๐2 ๐
๐๐โ2 +
1
๐
2(๐ โ 2๐บ๐ ๐)
๐2
๐๐
๐๐โ + ๐2
= (
๐ ๐( ๐ ๐ + 1)
๐2 + ๐2) ๐
(11.2)
In a region of space so asymptotically close to the
horizon (that is, ๐ โ ๐๐ป), the recast equation of
motion (11.2) can be approximated as
๐2 ๐
๐๐โ2 + ๐2 ๐ = 0
(11.3)
This is also since very near the horizon, ๐ โ ๐๐ป.
For our present purposes, we admit only
oscillatory solutions
๐ ( ๐โ ) = ๐ 01 ๐๐ฅ๐(โ๐๐๐โ) + ๐ 02 ๐๐ฅ๐(๐๐๐โ)
(11.4)
For (10.3) we obtain also oscillatory solution
๐( ๐ก ) = ๐01 ๐๐ฅ๐(โ๐๐๐ก) + ๐02 ๐๐ฅ๐(๐๐๐ก)
(11.5)
where ๐ฅ0 = ๐ก.
Proceeding from such oscillatory solutions
are approximate wave solutions if we are to define
the Ingoing and Outgoing coordinates respectively
๐ขฬ = ๐ก + ๐โ (11.6.1)
๐ฃฬ = ๐ก โ ๐โ (11.6.2)
These are the Eddington-Finkelstein coordinates
and via these coordinates we can combine the
solutions above into approximate wave solutions.
For the Outgoing wave we have
ฮฆ(๐โ ๐ก)
+
= ๐ด0
+
๐๐ฅ๐(โ๐๐๐ฃฬ ) (11.7.1)
while for the Ingoing wave
ฮฆ(๐โ ๐ก)
โ
= ๐ด0
โ
๐๐ฅ๐(โ๐๐๐ขฬ ) (11.7.2)
In the contrasting case, we take the limit
as ๐ โ โ as very far from the event horizons and
in this limiting case, from (11.2) we obtain another
approximate radial equation
๐2 ๐
๐๐โ2
+ (๐2 โ ๐2)๐ = 0 (11.8.1)
given for waves very far from the event horizons.
If we choose to have the scalar field as massless
๐ = 0, then (11.8.1) can take identical form as
that of (11.3).
๐2 ๐
๐๐โ2
+ ๐2 ๐ = 0 (11.8.2)
Let us note that in the previous case (for waves
very near the event horizons), it does not matter
whether we have a massless scalar or not since the
mass term in (11.2) drops off because of the
vanishing metric component ๐ very near the event
horizons. That is, very near the event horizons,
effectively we have a massless scalar that
corresponds to a massless scalar field very far
from the said horizons.
4. Analytic Properties Of The Wave
Solutions
As we have noted in the previous section,
very near the event horizon the scalar field is
effectively massless and very far from the said
horizon, there corresponds the same radial
equation of motion for a massless scalar field. For
our present discussion purposes we would only
have to take the crude approximation that we have
the same out-going solution (11.7.1) for the two
cases of waves very near the horizon and waves
very far from the horizon. That is, we have to
assume that we have the same out-going waves
propagating very near the horizon that have
reached very far from the horizon.
Because the given waves are massless, we
will assume that the out-going waves travel along
the out-going null path ๐พ+, while the ingoing
waves along the infalling null path ๐พโ.
Respectively, these paths are given by
๐พ+ : ๐ โ ๐ = โ๐+ (12.1.1)
๐พโ: ๐ + ๐ = ๐โ (12.1.2)
5. Since our radial coordinate is within the interval
๐๐ป < ๐ < โ, the waves under concern here
belong to region I of the Carter-Penrose diagram.
(Fig.1)
In Figure 1, we have only drawn region I
as bounded by future event horizon ๐ป+, future null
infinity โ+, past null infinity โโ and past event
horizon ๐ปโ. We approximate these boundaries as
straight lines
๐ป+: ๐ โ ๐ = โ๐ (12.1.3)
๐ปโ: ๐ + ๐ = โ๐ (12.1.4)
โ+: ๐ + ๐ = ๐ (12.1.5)
โโ: ๐ โ ๐ = ๐ (12.1.6)
These are obtained from the given changes of
coordinates
๐ + ๐ = 2๐ขฬโฒ, tan ๐ขฬโฒ = ๐ขฬ (12.2.1)
and
๐ โ ๐ = โ2๐ฃฬ โฒ , tan ๐ฃฬ โฒ = ๐ฃฬ (12.2.2)
(Remark: Let us not confuse ๐ in here with the 0th
component of the metric tensor in (8). In this
section this eta is just a vertical coordinate on the
Carter-Penrose graph.)
Let us take the infalling wave along an
infalling null path given for a constant (of non-
infinite values) infalling coordinate so we may
write ๐ก = ๐๐๐๐ ๐ก โ ๐โ. As we have noted very
near the horizon, the Regge-Wheeler coordinate is
pushed off to negative infinity (๐โ โ โโ) and as
a consequence, the coordinate time t approaches
(positive) infinity ( ๐ก โ โ). The future event
horizon is where all ๐โ โ โโ and ๐ก โ โ. So,
along ๐ขฬ = ๐๐๐๐ ๐ก, very near the horizon (๐ โ ๐๐ป),
the infalling wave hits ๐ป+ as it moves with
positive infinite values of coordinate time t โ into
an infinite coordinate future.
In the limit as ๐โ โ โโ and ๐ก โ โ, the
out-going coordinate takes on positive infinite
values, ๐ฃฬ = โ so that the outgoing wave is not
defined on the future event horizon.
In the case for the out-going wave let us
take it along an out-going null path with constant
out-going coordinate. So we may write the
coordinate time t as ๐ก = ๐๐๐๐ ๐ก+ ๐โ. As this wave
propagates very near the event horizon where the
Regge-Wheeler coordinate is pushed off to
negative infinity consequently, coordinate t
approaches to negative infinity. Let us note then
that past event horizon is where all space-time
points have ๐โ โ โโ and ๐ก โ โ โ. That is to
say then, along ๐ฃฬ = ๐๐๐๐ ๐ก, very near the horizon,
the out-going wave hits the past event horizon as it
moves with negative infinite values of coordinate
time t โ into an infinite coordinate past.
We also have to note that as ๐โ โ โโ
and ๐ก โ โ โ, the infalling coordinate takes on
negative infinite values so that the infalling wave
is not defined on the past event horizon.
5. Fourier Components OfThe Out-going
Waves
We shall skip the detailed consideration
that leads to the appropriate parametrization of the
wave solutions as discussed in [1]. Such
consideration applies parallel transport to arrive at
the required outgoing wave solution ultimately
parametrized in terms of the infalling coordinate.
In our convenience we have deviated a
little from [1] and take a parametrization in the
given form
๐+ = 2๐ฃฬ โฒ(๐ขฬ) (13.1.1)
6. This differs from what is arrived at [on pp. 126 of
[1]] in the numerical factor 2GM as given in this
draft. So in this draft we write
๐ฃฬ( ๐ขฬ) = โ2๐บ๐๐๐(โ๐ขฬ) (13.1.2)
โโ < ๐ขฬ < 0
In the limit as ๐ขฬ โ โโ, we have ๐ขฬ โฒ =
โ๐/2, ๐โ = โ๐, ๐พโ โ ๐ปโ, while ๐ฃฬ = โโ,
๐ฃฬ โฒ = โ๐/2 . Consequently, ๐+ = โ๐ and ๐พ+ โ
โโ. As the outgoing null path comes so
asymptotically close to the line of the past null
infinity all outgoing waves along this path can
only be viewed in the infinite past since ๐ฃฬ = ๐ก โ
๐โ = โโ, ๐ก โ โ โ, ๐โ โ โ, while as the
infalling null path comes so infinitely close to the
line of past event horizon, all infalling waves
along the infalling null path can only be viewed in
the infinite past since ๐ขฬ = ๐ก + ๐โ = โโ, ๐ก โ
โ โ, ๐โ โ โโ, ๐ โ ๐๐ป.
For the case as ๐ขฬ โ 0 we have ๐ขฬโฒ = 0,
๐โ = 0 and ๐พโ is now along the line ๐ + ๐ = 0.
In this limiting case, ๐ฃฬ = โ so that ๐ฃฬ โฒ = ๐/2 and
consequently, ๐+ = ๐ and ๐พ+ โ ๐ป+. What this
means is that waves along the outgoing null path
as this path comes so very close to the line of
future event horizon can only be viewed in the
infinite future given ๐ฃฬ = ๐ก โ ๐โ = โ, ๐ก โ โ
and ๐โ โ โโ.
Following from the parametrization of the
out-going coordinate in terms of the infalling
coordinate, the outgoing wave becomes a
parametrized function of the infalling coordinate.
Such parametrized function must satisfy the
following boundary conditions
ฮฆ( ๐โ ๐ก)
+
[๐ฃฬ( ๐ขฬ)] = ๐ด0
+
๐๐ฅ๐(๐2๐๐บ๐๐๐(โ๐ขฬ) )
โโ < ๐ขฬ < 0
ฮฆ( ๐โ ๐ก)
+
[๐ฃฬ( ๐ขฬ)] = 0, ๐ขฬ > 0
(13.2.1)
In here, we shall consider only the Fourier
integration involving positive frequency modes,
๐โฒ > 0.
ฮฆฬ( ๐โ ๐ก)
+ ( ๐, ๐โฒ) = โซ ๐๐ขฬ
โ
โโ
ฮฆ( ๐โ ๐ก)
+
(๐ขฬ, ๐)๐โ๐๐โฒ ๐ขฬ
= ๐ด0
+
โซ ๐๐ขฬ ๐๐ฅ๐(๐2๐๐บ๐๐๐(โ๐ขฬ) ) ๐โ๐๐โฒ ๐ขฬ0
โโ
(13.2.2)
This integration is facilitated with the aid of close
contour integration on a complex z-plane, where
for a point P(x, y) in a rectangular coordinate
system there corresponds a complex number, ๐ง =
๐ฅ + ๐๐ฆ. The said close contour is given by
โฎ ๐๐ง ๐( ๐ง) ๐โ๐๐โฒ๐ง = โซ ๐๐ง ๐( ๐ง) ๐โ๐๐โฒ๐ง
๐๐ดฮ
+ โซ ๐๐ง ๐( ๐ง) ๐โ๐๐โฒ๐ง
๐ด๐ต
+ โซ ๐๐ง ๐( ๐ง) ๐โ๐๐โฒ๐ง
๐ต๐
(13.2.3)
We may choose this close integral in the
left upper quadrant to involve the points O(0, 0) at
the origin of the rectangular coordinate system
with a corresponding complex number z = 0 +
iO = 0, A(-R, 0) with z = โR + i0 = โR and
B(0, R) for the complex number z = 0 + iR =
iR. There are three integral paths connecting these
points: straight line along the (real) x axis from O
to A, a quarter circle from A to B and straight line
along (imaginary) y axis from B back to point of
origin.
Briefly, we may examine for the
singularities of the function ๐( ๐ง) =
exp(๐2๐๐บ๐๐๐(โ๐ง)) to get for the basic reason
why we can choose โ๐ฅ > 0 to be the branch cut
and why we are having the close integral in the
upper left quadrant.
๐( ๐ง) = exp(๐2๐๐บ๐๐๐(โ๐ง)) (13.2.4)
๐ง = ๐ฅ + ๐๐ฆ
At y = 0, where z = x, we have
๐( ๐ง = ๐ฅ) = exp(๐2๐๐บ๐๐๐(โ๐ฅ))
(13.2.5.1)
Note in this case that ๐( ๐ง = ๐ฅ) is singular that is,
not defined on โ๐ฅ > 0 since
๐๐(โ๐ฅ) (13.2.5.2)
is undefined on โ๐ฅ > 0. So in this case we shall
think of โ๐ฅ > 0 as the branch cut region.
7. On the other hand at x = 0, where z = iy,
we get
๐( ๐ง = ๐๐ฆ) = exp(๐๐๐บ๐) exp(๐2๐๐บ๐๐๐๐ฆ)
(13.2.6)
This is singular on โ๐ฆ < 0 because clearly
๐๐๐ฆ (13.2.7)
is not defined for all y less than zero. So in this
case, โ๐ฆ < 0 is also a branch cut here.
Proceeding from (13.2.3), we obtain the
result for such close contour integration (๐โฒ > 0).
โซ ๐๐ฅ exp(๐2๐๐บ๐๐๐(โ๐ฅ)) ๐โ๐๐โฒ๐ฅ0
โโ =
โ๐exp(๐๐๐บ๐) โซ ๐๐ฆ exp(๐2๐๐บ๐๐๐๐ฆ) ๐ ๐โฒ๐ฆโ
0
(13.3.1)
In this we apply for ๐ฅ = ๐๐ฆ, while retaining the
real infinite integral limits. This only proves that
โซ ๐๐ฆ exp(๐2๐๐บ๐๐๐๐ฆ) ๐ ๐โฒ๐ฆโ
0 =
โ โซ ๐๐ฆ exp(๐2๐๐บ๐๐๐๐ฆ) ๐ ๐โฒ๐ฆ0
โโ (13.3.2)
We substitute this in (13.3.1) to get
โซ ๐๐ฅ exp(๐2๐๐บ๐๐๐(โ๐ฅ)) ๐โ๐๐โฒ๐ฅ0
โโ =
๐exp(๐๐๐บ๐) โซ ๐๐ฆ exp(๐2๐๐บ๐๐๐๐ฆ) ๐ ๐โฒ๐ฆ0
โโ
(13.3.3)
then apply ๐ฆ = ๐๐ฅ = (โ๐)(โ๐ฅ) once more while
retaining the real infinite integral limits in order to
recover for ln(โ๐ฅ).
โซ ๐๐ฅ exp(๐2๐๐บ๐๐๐(โ๐ฅ)) ๐โ๐๐โฒ๐ฅ0
โโ =
โexp(2๐๐๐บ๐)โซ ๐๐ฅ exp(๐2๐๐บ๐๐๐(โ๐ฅ)) ๐ ๐๐โฒ๐ฅ0
โโ
(13.4)
6. The Fourier Components Of The
Outgoing Waves
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7. Concluding Remarks
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8. Acknowledgment
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9. References
[1]Townsend, P. K., Blackholes โ Lecture Notes,
http://xxx.lanl.gov/abs/gr-qc/9707012
[2]Carroll, S. M., Lecture Notes On General
Relativity, arXiv:gr-qc/9712019
[3]S. W. Hawking, Particle Creation by Black
Holes, Commun. math. Phys. 43, 199โ220 (1975)
[4]Ohanian, H. C. Gravitation and Spacetime,
New York:W. W. Norton & Company Inc.
Copyright 1976
[5]Bedient, P. E., Rainville, E. D., Elementary
Differential Equations, seventh edition, Macmillan
Publishing Company, 1989, New York, New
York, USA
[6]Abramowitz, M., Stegun, I. A., Handbook of
Mathematical Functions With Formulas, Graphs,
and Mathematical Tables,
http://www.math.ucla.edu/~cbm/aands/,
http://th.physik.uni-
frankfurt.de/~scherer/AbramovitzStegun/
[7]Pennisi, L., L., Gordon, L.,I., Lasher, S.,
Elements of Complex Variables, second edition,
Holt, Rinehart and Winston 1963
[8]Merzbacher, E., Quantum Mechanics, second
edition, 1961, 1970, John Wiley & Sons, Inc. New
York, USA
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