This document summarizes the solutions to the Klein-Gordon equation of motion for a scalar field in the background of a Schwarzschild spacetime metric near a black hole. Very near the event horizon, the radial equation is approximated as an oscillatory solution in the Regge-Wheeler coordinate. These solutions are then expressed as outgoing and ingoing waves using Eddington-Finkelstein coordinates. While the outgoing waves have regular behavior at the future event horizon, the ingoing waves hit the future event horizon in finite coordinate time as the radial coordinate approaches negative infinity.

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 paper 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].
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 extreme
proximity to a very strong gravitational field of a
blackhole, Hawking realized that blackholes could
emit particles through pair production happening
so infinitely 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. We
shall no longer present here the lengthy
elaboration in his cited 1975 paper. 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

2. motion because of the presence of non-flat metric
components which represent for gravitational
field. The equations of motion become highly
nonlinear. 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.
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.
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 .
The varied scalar field must vanish at the two end
points A and B, 𝛿𝜑( 𝐴) = 𝛿𝜑( 𝐵) = 0 and by
stationary condition, 𝛿𝑆 = 0, 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 −
2𝐺𝑀 𝑞
𝑟
(8)
We note in these that the square of the speed of
light is unity (𝑐2 = 1, Heaviside units) and that
𝑀 𝑞 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.
As to the discussion purposes of this
present paper we are only interested in the

3. solutions of (10.3) and (10.4) and skip knowing
those of (10.1) and (10.2).
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 with the radial equation of motion
(10.4), where gravity takes effect through the
metric tensor component, 𝜂. This equation 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 + 2𝐺𝑀 𝑞 𝑙𝑛(
𝑟
2𝐺𝑀 𝑞
− 1)
∀𝑟 > 𝑟𝐻 (= 2𝐺𝑀 𝑞 )
𝜕𝑟
𝜕𝑟∗ = (
𝜕𝑟∗
𝜕𝑟
)
−1
= 𝜂
(11.1)
we recast (10.4) into the form
1
𝑅
𝑑2 𝑅
𝑑𝑟∗2 +
1
𝑅
2(𝑟 − 2𝐺𝑀 𝑞)
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)
We admit only oscillatory solutions
𝑅( 𝑟∗ ) = 𝑅01 𝑒𝑥𝑝(−𝑖𝜔𝑟∗) + 𝑅02 𝑒𝑥𝑝(𝑖𝜔𝑟∗)
(11.4)
For (10.3) we also obtain 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 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 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 in the previous case (for waves very
near the event horizons) that 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

4. 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)
As we note very near the horizon, the
Regge-Wheeler coordinate is pushed off to
negative infinity (𝑟∗ → −∞) and that future event
horizon is where all 𝑟∗ → −∞ and 𝑡 → ∞. Then
along 𝑢̃ = 𝑐𝑜𝑛𝑠𝑡, very near the horizon the
infalling wave hits 𝐻+ as it moves into an infinite
coordinate future. While 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.
Past event horizon is where all space-time
points have 𝑟∗ → −∞ and 𝑡 → − ∞ so that out-
going wave along an out-going null path with
constant out-going coordinate hits the past event
horizon as it moves 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.
5. Concluding Remarks
There are still some discussions to be
made but since this paper has been limited in
scope the necessary additional discussions are
reserved in papers to come. Such additional
discussions shall include the parametrized forms
of the wave solutions that will enable us to obtain
for their Fourier components. These components
shall then serve in the scalar field operators to be
defined along with Bogoliubov coefficients
leading to the arrival of Planck distribution for
black body radiation at a given Hawking
temperature.
6. 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