Journal of Modern Physics, 2012, 3, 1966-1971
http://dx.doi.org/10.4236/jmp.2012.312245 Published Online December 2012 (http://www.SciRP.org/journal/jmp)
An Eccentric Derivation of the Gravitational-Lens Effect
Yoonsoo Bach Park, Il-Tong Cheon
1Faculty of Science, Korea Advanced Institute of Science and Technology, Daejeon, South Korea
2Korean Academy of Science and Technology, Seongnam, South Korea
Received September 15, 2012; revised October 28, 2012; accepted November 5, 2012
ABSTRACT
The gravitational-lens effect is interpreted in the framework of the Newtonian mechanics. Regarding the photon of
energy h

as a corpuscle with a tiny mass of 2
h
mc


. We calculate it’s path bended by the gravitational force near
the surface of the sun. Effects of dark matter have also been evaluated.
Keywords: Gravitational-Lens Effect; Curved Space; Dark Matter
1. Introduction
The Gravitational-Lens Effect (GLE) provides one of the
powerful methods to investigate physical properties of
stellar objects. This effect is actually the result derived
by the General Theory of Relativity (GTR) and, therefore,
dynamical understanding is almost impossible without
basic knowledge of the GTR, which is usually taught in
the graduate course of university. Nevertheless, it might
be instructive to derive the GLE on the basis of the
Newtonian mechanics which is founded rather upon our
experiential events. Indeed, the bending of the light path
is originated in the space-time structure induced by the
gravitational field, and, therefore, it may not be under-
standable in the framework of the classical mechanics.
Since the light (photon) does not carry any mass, the
bending of its path cannot be taken place by the gravita-
tional force in the classical Newtonian mechanics.
However, if the photon is interpreted as a corpuscle with
a mass, 2
“”E mc


E, where

is the photon energy
and the corpuscle is assumed to move always at the
speed of light, , without any variation of cm

, the path
of the photon can be bended by the gravitation of the
reference stellar object even in the classical Newtonian
dynamics. Along this line, the bending angle of the light
passing through near the solar surface shall be calculated
and our method of calculation will be extended to the
system enveloped in dark matter halo.
2. Derivation of the Gravitational-Lens
Effect
Let us consider the gravitational-lens effect caused by the
sun. When the photon is regarded as a corpuscle with a
mass m

, the gravitational force of the sun acting on it
yields
2,
GM m
fr

 
G
(1)
where is the gravitational constant,
M
 is the solar
mass and is the distance between the sun and the
corpuscle. The motion of the corpuscle can be obtained
by solving the Newtonian equation of motion [1]. Its
orbit is actually hyperbolic.
r
Instead of taking such a treatment, let us try to derive
the hyperbolic orbit of the corpuscle on the basis of the
geometric property of the conic section. The eccentricity
of the hyperbola is generally defined as
,
I
P
F
e
I
I
P
H

I
PF
(2)
where and
I
I are distances of an arbitrary
point
PH
I
P on the hyperbola from the focus and from the
directrix,
I
Q, respectively, as is shown in Figure 1. One
of two focuses in the hyperbola is assighned by
F
at
which the sun is located. In other words, the trace of the
point
I
PePF which satisfies the relation (2) draws a
hyperbola with the eccentricity . Let I be


r

where the angle

runs clockwise around the focus
F
,
then
I
I
PH

cos ,
II
PHsRr
is given as
(3)

  

RAF
s
where o

 is the radius of the sun and
I
o
BA

.
Accordingly, Equation (2) yields



.
cos
r
esR r





(4)
C
opyright © 2012 SciRes. JMP

Y. B. PARK, I.-T. CHEON 1967
YI QI QO
HI
PI
OI BI AO
X
F
I
θ
HO
Figure 1. Hyperbolic orbit of a corpuscle. F: focus, : the
origin, : directrix, : tangent line. The sun is at F.
I
O
o
Q
rR 
0
I
Q
Since at

0


, we have
.
R
se  (5)
Thus, by solving Equation (4) with respect to


r

,
the equation of the hyperbola can be obtained in the form
 
1,
1cos
Re
e
r 




m
(6)
which is exactly identical to the result obtained by
solving directly the Newtonian equation of motion [1].
For the particle with its mass,

moving with the
velocity , the angular momentum, v
,Lmrv


E
(7)
is always conserved and the total energy of the system,
, is
2
2
mv
E


2
2,
2
kLk
rr
mr

 
kGMm
(8)
where

. The solution of Equation (8) with
respect to positive 1
r is
12
2
2
2
.
EL
Lmk




 




vc
2
1
11
mk
r




 (9)
For

, the angular momentum and the total en-
ergy of the system at

0
rR
e
can be obtained by
replacing by  in Equations (7) and (8). Then,
reciprocal of the first parenthesis in Equation (9) is equal
to the numerator of Equation (6), and, therefore, we find
the value of the eccentricity as
2
1.
Rc
eGM 


(10)

This result can also be obtained from the second
parenthesis in Equation (9). It is easily proved to be
2
Rc
GM


0
. And it corresponds to the denominator of Equa-
tion (6) at


, which is . Then, we obtain the
same result as that in Equation (10).

1e
On the other hand, the equation of a hyperbola in
rectangular coordinates whose origin is located at the
center of two focuses can be expressed as
22
22
1.
xy
ab


2a
OA
a
(11)
A difference between distances of an arbitrary point on
the hyperbola from each focus is always equal to .
This is a basic substance of the hyperbola. From Equa-
tion (11), we know immediately that the distance o
in Figure 1 is . And, thus, the distance between two
focuses is


2aR

,
. Let

11
x
y be the coordinates
of a crossing point between the hyperbola and the
straight line perpendicular to the X-axis at the focus


,0x1. Then, making use of the substance of the
hyperbola as well as the Pythagoras theorem, we can
easily find
2
12
R
yRa 

. (12)
In addition, from Equation (6), it is obvious that

1
π1
2
yr Re 


 



ae
.
Making this result equal to Equation (12) gives

aR

, which is actually value of 1
x
and, conse-
quently,
.
1
R
ae 
 (13)

Substituting values of 1
x
ae and

11yRe 
in Equation (11), one can find
1
1
e
bR e


. (14)
Since the equation of the asymptote of the hyperbola is
22
22
0,
xy
ab (15)



Y. B. PARK, I.-T. CHEON 1968
the angle of the asymptote,

, is found as
2
arctan 1.earctan b
a




 (16)
Thus, the bending of the light path, i.e. the gravi-
tational-lens effect, is
2
rctan 1.e
30
2 10kg,
kg,
5
4.710,e
20.878,
2π2π2a    (17)
With values, 
and we ob-
tain
5
710km, M
2032
6.710km /s


R
5
310 km/sc G
(18)
and


 (19)
which is in agreement with the result obtained by J. G.
von Soldner using a completely different formalization in
1801 [2,3]. However, this result is unfortunately just a
half of that obtained in the GTR [4]. The missing factor 2
may be referred to as “relativistic factor”.
3. Newtonian Derivation of the Relativistic
Factor 2
In context of the concept that the structure of the space is
associated with the gravitational field, it is convinced that
the space around the sun must be curved by its gravita-
tion. Moreover, it should be remarked that the light has
nature to travel always along the edge of the space
whatever it is straight or curved. The curve I in Figure 2
has been the path of the corpuscle moving under opera-
tion of the gravitation. However, if it is interpreted as the
edge of the curved space induced by the gravitational
field, the photon will travel along this curve I without
any influence of the force.
Let us now calculate the path of the corpuscle moving
in this curved space when the gravitational force acts
directly on it.
As is shown in Figure 2, the directrix
I
Q in the
regular space is now bended into
N
Q by the same
amount of curvature as the edge of the surface of the
curved space I. Namely,
N
Q is the translated hyperbola
of the curve I. Then, the arbitrary point
I
P on the hy-
perbola I is shifted to
N
P. Thus, the curve N should be
the orbit of the corpuscle when the gravitational force
directly acts on it in the curved space. All points,
N
P,
I
P, o
H
, A
H
and
I
H
are on the horizontal line. As
the straight line of directrix,
I
Q is bended into the curve
N
Q the distance
I
I is shifted to PH
N
A, i.e. PH
I
I. When the distance NA
HPH P
I
o is expressed by
, i.e. Io , we have, of course, AI
PH
uPH u
H
Hu

, and,
therefore,
N
I ought to be . Furthermore, the
distance oI
PP u
A
B is expressed by
s
, the distance
I
A
should also be , i.e.
PH
sIA PH s

. In accordance with
Y
I
Q
O
Q
N
Q
I
I
P
I
H
A
H
I
O
I
B
I
A
O
X
F
θ
N
H
O
P
N
N
o N
Q
I
Q
Figure 2. Paths of the corpuscle. Q: tangent line, :
directrix in the curved space, : original directrix.
Equation (2), the eccentricity should be
.
N
N
A
PF
ePH 
N
PF

(20)
When is expressed as
N
N
rNA PH

and

I
ANI
PHPP

is considered, we have




=,
1cos
2
NN NN
IA NI
NN
NN N
rr
ePHP Psu
r
RRr
e





(21)

 




where R se  can be obtained from the second line of
Equation (21) because
N
o
PA0 for N, and, then,



0rAFR   0u
0
N
NNo  and . To lead the final
expression in Equation (21), we have also used that for



2cos.
N
oNIIoNN N
PHPPPHu Rr

  
 (22)


Solving Equation (21) with respect to
N
N
r

, we
obtain

1
2.
1cos
2
NN
e
R
re

N








(23)

Y. B. PARK, I.-T. CHEON 1969
Replacing 2
e by
N
e, i.e.
,
2
Ne
e

(24)
we obtain the standard expression of the hyerbola

1.
1cos
N
NN
N
N
Re
e
r





 (25)
This equation can also directly be derived in the re-
gular space where the eccentricity,
N
e is given by the
definition equivalent to those written in Equations (2)
and (20) as
,
N
NNN
PF
ePH  (26)
in which the point
N
H
is shown in Figure 3. When
distances oI
A
B and
I
N
BB are expressed by
s
and
x
, i.e. oI
A
Bs and IN BBx

, distances
N
I
PH and
N
N

cos ,
N N




cos .
NN N



0
N
PH are described as
NI N
PHs Rr 
 (27)
and
NNIN NI
PHBB PH
xs R r



 (28)
At

, Equation (26) yields
,
N
R
e
x
s




0
NN
rR  

(29)
because of and, thus,
11
,
N
ee





xR
 (30)
where R
se  is used. Therefore,




11
.
cos
NN
N
N
NN
NN N
N
r
e
RsR
ee
r
RRr
e
cos
NN N
r








 







(31)
Solving Equation (31) with respect to


N
N
r

, we
find the result exactly identical to Equation (25).
Our result in Equation (24) explicates that the eccen-
tricity,
N
ee
5
2.3510 .
, of the hyperbolic orbit of the corpuscle in the
curved space yields just a half of the eccentricity, ,
obtained in the regular space, when the gravitational
force is directly acting on it, i.e.
Q
I
N
e (32)
Y
N
W
N
H
A
H
I
O
N
X

I
N
Q

Y
I
W
I
L
N
H
N
P
N
F
B
I
B
N
A
O
θ
N
I N
I
QN Q

e
O
I
Figure 3. Paths of the corpuscle. W and W: asymp-
totes of the hyperbolae I and N. and : directrices
of the hyperbolae I and N, respectively.
Thus, the Gravitational-lens effect in this case can be
derived from the formula, Equation (17), provided is
replaced by
N
e,
2
2π2arctan1.
NN
e 


21.755
N
(33)
The numerical value is, now, found as

.

(34)

This value is twice of 2

value obtained in Equation
(19) and agrees with the result obtained by Einstein [4] in
the GTR.
The GTR holds the concept that the photon travels
along the surface of the curved space even without inter-
acting with the source of gravitation yielding the curva-
ture in the space, while the physical world described here
is that the path of the corpuscle moving along the surface
of the curved space originated from the gravitation is
forced to bend by the gravitational force acting directly
on it. Although the space curvature in our case is just a
half of the result obtained by the GTR, the corpuscle
eventually keeps moving along the same path as that
predicted by the GTR.
4. The Gravitational-Lens Effect by the
Dark Matter Halo
If the dark matter exists in surrounding of the galaxy as a
halo, the gravitational-lens effect will be induced by it. In

Y. B. PARK, I.-T. CHEON 1970
order to simplify the calculation, let us consider a
spherical galaxy with the mass G11
510
M
M
10R
100R
 and
the radius G ly completely covered by the dark
matter (DM) with a thickness DMG
4
RR


1.01
ly,
i.e. the radius of the sphere including the DM is
D
MG
. The density of the dark matter is assumed
to be as large as 10 times that of the galaxy. (i) Without
any dark matter, the GLE of the corpuscle passing by the
galaxy surface is found as 26.11
RR


 byqua-
tions (9), (24) and (33), provided
using E
M
 and  are
aced by G
R
repl
M
andG. (iihen the corpuscle
passes beside the surface of the dark matter, the GLE is
obtained as 27.8
R) W
9

 , which
is calculated by replac-
ing
M
 aR byGDM
nd MM and
D
M
R inua-
tion (9). (iii) The light can penetrate into the DM without
scattered and sneak by the surface of the galaxy. The
path of the corpuscle in the region outside the DM can be
calculated with Equations (9), (24) and (33) by assuming
that the total mass, GDM MM the sphere of
the radius G R. The orbits definitely a hyperbola. On
the other hand, the path of the corpuscle inside the DM is
able to obtain by a numerical method explained below.
Let us consider a
Eq
de
corpuscle which starts from the
ga
, is in
i
si
laxy surface, namely 0

 and G R, and dashes in
the dark matter until runnay frothe DM region.
Let the corpuscle be located at

r
ing awm

in a certain
maxy oment after it started from the galsurface, i.e. a
distance, r, from and an angle,

, around the galaxy
center. According to the Gaussian heorem, the gravita-
tional potential outside a sphere can be determined by the
total mass inside the sphere. Thus, the total mass inside a
sphere with the radius r participates in formation of a
hyperbolic orbit of the corpuscle at that moment. Remark
that the location of the corpuscle,

r
T

, is situated actu-
ally at the common point on the sphere surface and the
hyperbolic orbit. Since the corpuscle migrates in the DM,
it forms every moment a different sphere with a different
radius and, accordingly, the total mass inside the sphere
will change simultaneously. Namely, the hyperbola to
which the corpuscle on the surface of the sphere formed
in each moment belongs is not the same one anymore as
formed in the previous moment. The trace of all these
points which the corpuscle occupied every moment
yields a curved line. It may not be a simple hyperbola but
a slightly deviated one. Such a physical process might be
explicitly described with following equations. The total
mass of the system is 1
G
MM


 with

3
1DM
GG
r
R










1,
DM
G










(35)

M
Rr   where

r for GD R

is the distance of
uscle of ththe e center e galaxy when it is
located at the angle
corpfrom th

around the focus. The eccentricity
at this position is
22
Rc 111,
22
GG
G
Rc
ee
GM GM




 

 (36)
and, then, for


GDM
Rr R  ,



1.
1cos11cos
G
ee
Re e



r 





(37)
Notice that

is also a function of

G
r
R

. Generally,
it is possible to solve Equation (37) analytically with
respect to


G
R
r

. However, it is not easy and, thus, we
solve Equation (37) numerically with respect to the value
of


r

one-by-one for each angle

during the cor-
puscle passes through the DM regionThe orbit of the
corpuscle found in this manner shows slightly deviated
from a simple hyperbola, particularly in the region of the
dark matter. Actually, the region of the DM in which the
light is moving holds only within 16 degrees around the
galaxy center, i.e. 88 
.



.
Finally, our result found for the GLE is 27.97




,
whi
5. Conclusions
that the GLE could be derived in the
hich is slightly larger than that of the case (ii). Ts
result is obviously understandable because of that the
closer the orbit of the corpuscle is to the gravitational
center, the larger the GLE becomes.
It has been shown
framework of Newtonian mechanics if the photon were
regarded as a corpuscle with the tiny effective mass,
2
c

h
m

, which was based on the quantum theory and
the Special T
optically
vi
informations of the dark
m
6. Acknowledgements
ut under the KAST Mentor
heory of Relativity. The factor 2 arising
from the GTR has been derived by introducing the
concept of the curved space due to the gravitational field
and its normal reaction. Of course, all these procedures
would be expediential. However, it would help senior
high school and undergraduate university students to
comprehend the physical structure of the GLE.
Since existence of the dark matter is not
sible because it does not interact with the light, the
GLE is known as a powerful method to observe it. The
present work investigates the GLE by a spherical galaxy
completely covered by the dark matter. The results are
definitely in detectable range.
It is true that one can obtain
atter by investigating the GLE.
This work has been carried o

Y. B. PARK, I.-T. CHEON 1971
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