This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. These solutions are then expressed in terms of outgoing and ingoing coordinates, which have distinct analytic properties inside and outside the event horizon.

1. Out-Going and In-Going Klein-Gordon Waves Very Near The Blackhole Event Horizons
Roa, F. J. P., Bello, A., Urbiztondo, L.
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:
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 tackled
in the middle of 1970’s in Stephen Hawking’s paper
[3]. By taking quantum mechanics into account
especially in close 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 Hawking’s results,
this emission of particles is thermal as if blackholes
were hot bodies whose temperatures are proportional
to blackholes’ 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. It is although worth remarking that
Hawking treated a field theory quantum mechanically
against a curved space-time background such as the
Schwarzschild metric in a time when there was none
yet a form of quantum gravity. In Hawking’s
treatment, quantum mechanics was forcefully
implemented in the classical solutions of those field
equations.
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. The equations of
motion become highly nonlinear because the
coefficients of the dependent variables become
nonconstant, and become dependent variables
themselves. 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]
𝑆 = ∫ 𝑑4
𝑥 ℒ (1)
along with a Lagrangian given for a scalar field
ℒ = √−𝑔
1
2
( 𝑔 𝜇𝜐
(𝜕𝜇 𝜑)( 𝜕𝜐 𝜑) + 2𝑉(𝜑)) (2)
where in the metric signature of positive two (+2) we
take the scalar potential as
𝑉( 𝜑) =
1
2
𝑀2
𝜑2
(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

2. 𝛿𝑆 = ∫ 𝑑𝜎 𝜇 |
𝛿ℒ
𝛿(𝜕 𝜇 𝜑)
𝛿𝜑|
𝑥 𝐴
𝜇
𝑥 𝐵
𝜇
+ ∫ 𝑑4
𝑥 (
𝛿ℒ
𝛿𝜑
−𝜎 𝜇
𝜕𝜇 (
𝛿ℒ
𝛿(𝜕 𝜇 𝜑)
)) 𝛿𝜑
(4)
This variation is carried out noting that
metric fields 𝑔 𝜇𝜐 are independent of the variation of
𝜑 . In the classical field theory the varied scalar field
must vanish at the two end points A and B, 𝛿𝜑( 𝐴) =
𝛿𝜑( 𝐵) = 0 and by stationary condition, the
variation of this scalar action must vanish that is,
𝛿𝑆 = 0. Thus following these variational (extremal)
conditions, we obtain for the Euler-Lagrange
equation for the classical scalar
𝛿ℒ
𝛿𝜑
− 𝜕𝜇 (
𝛿ℒ
𝛿(𝜕𝜇 𝜑)
) = 0
(5)
Upon the substitution of (2) in (5) we get the
equation of motion for the scalar field in curved
spacetime
1
√−𝑔
𝜕𝜇 [√−𝑔𝑔 𝜇𝜐
(𝜕𝜐 𝜑) ] − 𝑀2
𝜑 = 0
(6)
where we take note of the covariant four-divergence
1
√−𝑔
𝜕𝜇 [√−𝑔𝑔 𝜇𝜐
(𝜕𝜐 𝜑) ] = ∇ 𝜇
( 𝑔 𝜇𝜐
(𝜕𝜐 𝜑) )
(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
𝑑𝑆2
= −𝜂𝑑𝑡2
+ 𝜀𝑑𝑟2
+ 𝑟2
𝑑𝜃2
+ 𝑟2
𝑠𝑖𝑛2
𝜃𝑑𝜙2
𝜂 = 𝜀−1
= 1 −
2𝐺 𝑀𝑞
𝑟
(8)
We note in these that the square of the speed of light
is unity (𝑐2
= 1, Heaviside units) and that 𝑀𝑞 is the
mass of the gravitational body. In addition, our
spacetime is given with a set of spacetime
coordinates 𝑥 𝜇
= (𝑥0
= 𝑡; 𝑥1
= 𝑟; 𝑥2
= 𝜃; 𝑥3
=
𝜙 )
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
𝜑( 𝑥0
, 𝑟, 𝜃, 𝜙) = 𝑇( 𝑡) 𝑅(𝑟)Θ(𝜃)𝜓(𝜙)
(9)
This, given the background spacetime of (8), gives
the following component equations of motion
1
𝜓
𝜕 𝜙
2
𝜓 = −𝜇 𝜙
2
(10.1)
1
Θ
1
𝑠𝑖𝑛𝜃
𝜕 𝜃
[ 𝑠𝑖𝑛𝜃(𝜕 𝜃Θ)] −
𝜇 𝜙
2
𝑠𝑖𝑛2 𝜃
= −𝜇 𝜃( 𝜇 𝜃 + 1)
(10.2)
1
𝑇
𝜕0
2
𝑇 = −𝜔2
(10.3)
1
𝑅
1
𝑟2
𝑑
𝑑𝑟
[ 𝜂𝑟2
𝑑𝑅
𝑑𝑟
] −
𝜇 𝜃
( 𝜇 𝜃 + 1)
𝑟2
= 𝑀2
−
𝜔2
𝜂
(10.4)
Partial differential equations (Pde’s) (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 𝜂.
With the substitution 𝑤 = 𝑐𝑜𝑠𝜃 we
convert (10.2) into a Legendre equation [6].
𝜕 𝑤 ((1 − 𝑤2) 𝜕 𝑤 Θ ) + ( 𝜇 𝜃
( 𝜇 𝜃 + 1) −
𝜇 𝜙
2
1− 𝑤2
)Θ =
0
(10.5)
This equation is solved by the associated [7]
Legendre polynomials
Θ 𝜇 𝜃
𝑘 ( 𝑤) = 𝐴 𝜇 𝜃
𝑘 (1 − 𝑤2) 𝑘/2 𝑑 𝑘
𝑑𝑤 𝑘
𝑃𝜇 𝜃
(𝑤) (10.6)
where 𝑘 = | 𝜇 𝜙| and 𝐴 𝜇 𝜃
𝑘
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 . In this, the
Legendre polynomials 𝑃𝜇 𝜃
(𝑤) are derived by
𝑃𝜇 𝜃
( 𝑤) =
1
2 𝜇 𝜃(𝜇 𝜃! )
𝑑 𝜇 𝜃
𝑑 𝑤 𝜇 𝜃
( 𝑤2
− 1) 𝜇 𝜃
(10.7.1)
These polynomials solve the Legendre equation
given in the following form
(1 − 𝑤2) 𝜕 𝑤
2
𝑃𝜇 𝜃
− 2𝑤 𝜕 𝑤 𝑃𝜇 𝜃
+ 𝜇 𝜃
( 𝜇 𝜃 + 1) 𝑃𝜇 𝜃
=
0
(10.7.2)

3. We can combine solutions (10.6) and solutions of
Pde (10.1) to form spherical harmonics[8]
𝑌𝜇 𝜃
𝜇 𝜙
= Θ 𝜇 𝜃
𝑘
𝜓 𝜇 𝜙
(10.8)
The component solution
𝜓 𝜇 𝜙
= 𝐵 𝜙 𝑒 𝑖𝜇 𝜙 𝜙
𝐵 𝜙 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (10.9)
satisfies (10.1) for a given integral number 𝜇 𝜙 whose
square is greater than zero (𝜇 𝜙
2
> 0). Let us note here
that the absolute values of 𝜇 𝜙 are restricted up to a
given integral value of 𝜇 𝜃. That is
𝑘 = | 𝜇 𝜙| = 0, 1, 2, 3, … , 𝜇 𝜃 (10.10)
where
𝜇 𝜃 = 0, 1, 2, 3, … , 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 (10.11)
In an asymptotically flat space (𝑟 → ∞), where the
metric component 𝜂 is unity (𝜂 = 1), the radial
equation of motion Pde(10.4) takes a familiar Bessel
form
1
𝑅
1
𝑟2
𝑑
𝑑𝑟
( 𝑟2
𝑑𝑅
𝑑𝑟
) −
𝜇 𝜃
( 𝜇 𝜃 + 1)
𝑟2
= −𝛼2
𝛼2
= 𝜔2
− 𝑀2
(10.12)
One convenient form of solution to (10.12) that we
choose is
𝑅𝑙 = 𝑄𝑙(1/𝑟) 𝑒 𝛼1 𝑟
(10.13)
𝛼1 = ±𝑖𝛼
𝑙 = 𝜇 𝜃 (integral number)
This satisfies the associated Besselradial equation for
a given integral value of 𝑙.
𝑑2
𝑄𝑙
𝑑 𝑟2
+
2
𝑟
( 𝛼1 𝑟 + 1) 𝑑 𝑄𝑙
𝑑𝑟
+ (
2𝛼1
𝑟
−
𝜇 𝜃
( 𝜇 𝜃+1)
𝑟2
) 𝑄𝑙 =
0
(10.14)
We will only give first two of its solutions.
𝑄𝑙 = 0 =
𝑎0
𝑟
(10.15.1)
𝑄𝑙 = 1 =
𝑎0
𝛼1 𝑟2
( 𝛼1 𝑟 − 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, 𝜂. 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 obtain for the approximate
equation very near the blackhole horizon.
Using Regge-Wheeler coordinate,
𝑟∗
= 𝑟 + 2𝐺𝑀𝑞 𝑙𝑛 (
𝑟
2𝐺 𝑀𝑞
− 1)
∀𝑟 > 𝑟𝐻 (= 2𝐺𝑀𝑞 )
𝜕𝑟
𝜕𝑟∗
= (
𝜕𝑟∗
𝜕𝑟
)
−1
= 𝜂
(11.1)
we recast (10.4) into the form
1
𝑅
𝑑2
𝑅
𝑑 𝑟∗2
+
1
𝑅
2(𝑟 − 2𝐺 𝑀𝑞 )
𝑟2
𝑑𝑅
𝑑𝑟∗
+ 𝜔2
= (
𝜇 𝜃
( 𝜇 𝜃 + 1)
𝑟2
+ 𝑀2) 𝜂
(11.2)
In a region of space so asymptotically close to the
horizon (that is, 𝑟 ≈ 𝑟𝐻), the recast equation of
motion (11.2) can be approximated as
𝑑2
𝑅
𝑑𝑟∗2
+ 𝜔2
𝑅 = 0
(11.3)
This is also since very near the horizon, 𝑟 ≈ 𝑟𝐻. For
our present purposes, we admit only oscillatory
solutions
𝑅( 𝑟∗ ) = 𝑅01 𝑒𝑥𝑝(−𝑖𝜔𝑟∗ ) + 𝑅02 𝑒𝑥𝑝(𝑖𝜔 𝑟∗ )
(11.4)
For (10.3) we obtain also oscillatory solution
𝑇( 𝑡 ) = 𝑇01 𝑒𝑥𝑝(−𝑖𝜔𝑡) + 𝑇02 𝑒𝑥𝑝(𝑖𝜔𝑡)
(11.5)
where 𝑥0
= 𝑡.
Proceeding from such oscillatory solutions
are approximate wave solutions if we are to define
the Ingoing and Outgoing coordinates respectively
𝑢̃ = 𝑡 + 𝑟∗
(11.6.1)
𝑣̃ = 𝑡 − 𝑟∗
(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
Φ(𝑟∗
𝑡)
+
= 𝐴0
+
𝑒𝑥𝑝(−𝑖𝜔𝑣̃) (11.7.1)
while for the Ingoing wave
Φ(𝑟∗
𝑡)
−
= 𝐴0
−
𝑒𝑥𝑝(−𝑖𝜔𝑢̃ ) (11.7.2)
In the contrasting case, we take the limit as
𝑟 → ∞ as very far from the event horizons and in

4. this limiting case, from (11.2) we obtain another
approximate radial equation
𝑑2
𝑅
𝑑𝑟∗2
+ (𝜔2
− 𝑀2
)𝑅 = 0 (11.8.1)
given for waves very far from the event horizons.
If we choose to have the scalar field as massless 𝑀 =
0, then (11.8.1) can take identical form as that of
(11.3).
𝑑2
𝑅
𝑑𝑟∗2
+ 𝜔2
𝑅 = 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 𝜂 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 𝛾+
, while the ingoing waves
along the infalling null path 𝛾−
. Respectively, these
paths are given by
𝛾+
: 𝜒 − 𝜂 = −𝑎+
(12.1.1)
𝛾−
: 𝜒 + 𝜂 = 𝑎−
(12.1.2)
Since our radial coordinate is within the interval 𝑟𝐻 <
𝑟 < ∞, 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 𝐻+
, future null
infinity ℑ+
, past null infinity ℑ−
and past event
horizon 𝐻−
. We approximate these boundaries as
straight lines
𝐻+
: 𝜒 − 𝜂 = −𝜋 (12.1.3)
𝐻−
: 𝜒 + 𝜂 = −𝜋 (12.1.4)
ℑ+
: 𝜒 + 𝜂 = 𝜋 (12.1.5)
ℑ−
: 𝜒 − 𝜂 = 𝜋 (12.1.6)
These are obtained from the given changes of
coordinates
𝜒 + 𝜂 = 2𝑢̃′, tan 𝑢̃′ = 𝑢̃ (12.2.1)
and
𝜒 − 𝜂 = −2𝑣̃ ′ , tan 𝑣̃′ = 𝑣̃ (12.2.2)
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
𝑡 = 𝑐𝑜𝑛𝑠𝑡 − 𝑟∗
. As we have noted very near the
horizon, the Regge-Wheeler coordinate is pushed off
to negative infinity (𝑟∗
→ −∞) and as a
consequence, the coordinate time t approaches
(positive) infinity ( 𝑡 → ∞). The future event
horizon is where all 𝑟∗
→ −∞ and 𝑡 → ∞. So,
along 𝑢̃ = 𝑐𝑜𝑛𝑠𝑡, very near the horizon (𝑟 ≈ 𝑟𝐻),
the infalling wave hits 𝐻+
as it moves with positive
infinite values of coordinate time t – into an infinite
coordinate future.
In the limit as 𝑟∗
→ −∞ and 𝑡 → ∞, the
out-going coordinate takes on positive infinite values,
𝑣̃ = ∞ so that the outgoing wave is not defined on
the future event horizon.

5. 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 𝑡 = 𝑐𝑜𝑛𝑠𝑡 + 𝑟∗
. 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 𝑟∗
→ −∞ and
𝑡 → − ∞. That is to say then, along 𝑣̃ = 𝑐𝑜𝑛𝑠𝑡, very
near the horizon, the out-going wave hits the past
event horizon as it moves with negative infinite
values of coordinate time t – into an infinite
coordinate past.
We also have to note that as 𝑟∗
→ −∞ and
𝑡 → − ∞, the infalling coordinate takes on negative
infinite values so that the infalling wave is not
defined on the past event horizon.
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5. The Fourier Components Of The
Outgoing Waves
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6. Concluding Remarks
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7. Acknowledgment
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8. References
[1]Townsend, P. K., Blackholes – 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, 199—220 (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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