The Deviation of the Vacuum Refractive Index Induced by a Static Gravitational Field
Xing-Hao Ye, Qiang Lin∗
Department of Physics, Zhejiang University, Hangzhou 310027, China
(Dated: October 29, 2018)
We analyzed the influence of static gravitational field on the vacuum and proposed the concept of inhomoge-
neous vacuum. According to the observational result of the light deflection in solar gravitational field as well as
the corresponding Fermat’s principle in the general relativity, we derived an analytical expression of the refrac-
tive index of vacuum in a static gravitational field. We found that the deviation of the vacuum refractive index
is composed of two parts: one is caused by the time dilation effect, the other is caused by the length contraction
effect. As an application, we simulated the effect of the gravitational lensing through computer programming
and found that the missing central imaging could be interpreted in a reasonable way.
PACS numbers: 42.25.Bs, 42.50.Lc, 04.
I. INTRODUCTION
Vacuum is usually considered as “homogeneous” and
“isotropic”, i.e., vacuum does not differ from place to place,
and the refractive index of vacuum is always equal to 1.
However, the recent theoretical and experimental progresses
demonstrate that such concept of vacuum turns out to be in-
appropriate when there are matters or fields within finite dis-
tance. For example, the vacuum inside a microcavity is modi-
fied due to the existence of the cavity mirrors, which will alter
the zero-point energy inside the cavity and cause an attrac-
tive force between the two mirrors known as Casimir effect
[1, 2], which has been verified experimentally [3, 4]. A sec-
ond example is that, under the influence of electromagnetic
field, vacuum can be polarized, which has led to astonishingly
precise agreement between predicted and observed values of
the electron magnetic moment and Lamb shift, and may in-
fluence the motion of photons [5]. Dupays et al. [6] studied
the propagation of light in the neighborhood of magnetized
neutron stars. They pointed out that the light emitted by back-
ground astronomical objects will be deviated due to the optical
properties of quantum vacuum in the presence of a magnetic
field. Also in [7], Rikken and Rizzo considered the anisotropy
of the optical properties of the vacuum when a static mag-
netic field B0 and a static electric field E0 are simultaneously
applied perpendicular to the direction of light propagation.
They predicted that magnetoelectric birefringence will occur
in vacuum under such conditions. They also demonstrated
that the propagation of light in vacuum becomes anisotropic
with the anisotropy in the refractive index being proportional
to B0 × E0.
The facts that the propagation of light in vacuum can be
modified by applying electromagnetic fields to the vacuum
implies that the vacuum is actually a special kind of opti-
cal medium [5, 6]. This is similar to the Kerr electro-optic
effect and the Faraday magneto-optic effect in nonlinear di-
electric medium. This similarity between the vacuum and the
dielectric medium implies that vacuum must also have its in-
ner structure, which could be influenced by matter or fields as
∗Electronic address:qlin@zju.edu.cn
well. Actually, the structure of quantum vacuum has already
been investigated in quite a number of papers [8, 9, 10].
In this paper, with the analysis of the influence of static
gravitational field on the vacuum, we put forward a new con-
cept that the curved spacetime around a certain matter can be
treated as an optical medium with a graded refractive index.
We suggest that the so-called curved spacetime is a reflec-
tion of the vacuum inhomogeneity caused by the influence of
gravitational matter. Based on this idea, the refractive index
of vacuum is derived. We will also apply this concept to un-
puzzle the problem of the central image missing in almost all
the observed cases of gravitational lensing [11].
II. THE DEVIATION OF THE VACUUM REFRACTIVE
INDEX
According to the astronomical observation, the light prop-
agating through a space with a celestial body nearby will be
deflected. It can be interpreted with the curved spacetime in
general relativity. As a matter of fact, it can also be inter-
preted with the assumption that the vacuum around matter is
inhomogeneous with refractive index deviated from 1. Here
we put forward a theoretical model to describe the refractive
index profile based on the Fermat’s principle for the propaga-
tion of light in a static gravitational field, which was given by
Landau and Lifshitz [12]:
−1/2dl = 0, (1)
where dl is the local length element passed by light and mea-
sured by the observer at position r in the gravitational field, r
is the distance from this element of light to the center of grav-
itational matter M, g00 is a component of the metric tensor
gµν, g
00 dl corresponds to an element of optical path length.
= dt/dτ, where dτ represents the time interval mea-
sured by the local observer for a light ray passing through the
length dl, while dt is the corresponding time measured by the
http://arxiv.org/abs/0704.1173v1
mailto:qlin@zju.edu.cn
observer at infinity. Eq.(1) could then be rewritten as
−1/2dl = δ
nds = 0, (2)
where ds is the length element measured by the observer at
infinity, corresponding to the local length dl.
Eq.(2) shows that if we set the scale of length and time at
infinity as a standard scale for the whole gravitational space
and time, the propagation of light then satisfies the standard
representation of Fermat’s principle, with the space — actu-
ally the vacuum — possessing a refractive index given by
= n1n2. (3)
The factor n1 of the refractive index relating to the time
transformation effect dt/dτ can be derived from the Newto-
nian attraction, which contributes partially to the deflection
of light. Considering a photon of relativistic mass m∞ at the
infinity moving down to position r, the work done on to the
photon by the Newtonian gravity is
dr = d(mc2), (4)
where G is the gravitational constant, c is the velocity of light,
M is the mass of a star (say the Sun), r is the distance to the
center of the star. Integrating Eq.(4) gives
mr = m∞e
rc2 , (5)
where mr is the relativistic mass of the photon at position r.
Since the photon energy is E = hν = mc2, where h is the
Planck constant, ν is the photon frequency, then we have m =
hν/c2. Substituting it into Eq.(5) gives
νr = ν∞e
rc2 . (6)
It is just the frequency shift caused by the gravitational force,
which reflects that a clock in a gravitational field runs slower
than that far away from the gravitational center. That is
dτ = e−
rc2 dt, (7)
where dτ denotes the time measured by a clock at position
r, dt is the converted time of dτ, i.e., the time measured by
the clock at infinity. This relation indicates that, if the length
scale is the same, i.e., dl = ds, when an observer at position r
reports a light velocity c1 = dl/dτ, it should be converted by
the observer at infinity into
c′1 =
rc2 dτ
= c1e
rc2 . (8)
FIG. 1: Light deflection caused by a graded refractive index.
This change of light velocity will certainly bring a deflection
to the light propagation. The corresponding refractive index
rc2 . (9)
Let us now consider the deflection angle caused by this
graded refractive index. In Fig.1, the curve AP represents the
light ray, β is the angle between the position vector r and the
tangent at the point P on the ray, ϕ is the deflection angle of
light. Since the refractive index shown in Eq.(9) has spheri-
cal symmetry, i.e., depends only on the distance r for a given
mass M, according to the Fermat’s principle
nds = 0, (10)
where ds = dr
1 + (rα̇)2, α̇ = dα/dr, n = n(r), we have the
corresponding Lagrangian function
L(α, α̇; r) = n(r)
1 + (rα̇)2. (11)
Using the Lagrangian equation
= 0, (12)
we get [13]
nr sin β = constant, (13)
nr sin β = n0r0, (14)
where r0 and n0 represent the radius and refractive index at the
nearest point A respectively.
Since
tan β =
, (15)
associating with Eq.(14) reaches
( nrn0r0 )
2 − 1
. (16)
FIG. 2: Light deflection in solar gravitational field.
For a light ray passing by the Sun as shown in Fig.2, the
total angular displacement of the radius vector r reads
∆α = 2
( nrn0r0 )
2 − 1
, (17)
where r0 represents the nearest distance to the center of the
Sun. Because the gravitational field of the Sun is a weak field,
the value of GM/rc2 is quite small, so substituting Eq.(9) into
Eq.(17) gives a solution of first order approximation
∆α = π +
. (18)
Then the total deflection angle of light caused by the refrac-
tive index n1 in solar gravitational field is
∆ϕ1 = ∆α − π =
. (19)
In fact, this result was obtained early in 1911 by Einstein
[14], who also investigated the effect of red shift and the cor-
responding slowing down of the light velocity in gravitational
field and then figured out the light deflection as shown in
Eq.(19) with the use of Huygens’ principle. Since the ac-
tual total deflection angle of light propagation calculated by
the general relativity [15, 16] and measured by the astronomi-
cal observation [17] is twice that value, we then come to know
that the length transformation effect dl/ds in Eq.(2) must have
the same relation as that of the time transformation effect
dt/dτ expressed in Eq.(9), namely
rc2 . (20)
This relation indicates that a ruler in a gravitational field is
shorter than that far away from the gravitational center. So
when an observer at position r reports a length dl, it should be
converted by the observer at infinity into
ds = e−
rc2 dl. (21)
For a light passing through a length dl, if the time scale is
the same, i.e., dτ = dt , the light velocity c2 = dl/dτ reported
by the observer at position r should then be converted by the
observer at infinity into
c′2 =
rc2 dl
= c2e
rc2 . (22)
FIG. 3: The dependence of the vacuum refractive index n on the
distance r.
This change of light velocity will also bring a deflection to
the light propagation. The refractive index corresponding to
this kind of deflection is
rc2 , (23)
which also causes a deflection angle of light
∆ϕ2 =
. (24)
Therefore, the total deflection angle of light in solar gravita-
tional field is
∆ϕ = ∆ϕ1 + ∆ϕ2 =
. (25)
The above result shows that, if the two refraction effects are
considered simultaneously, then the gravitational space — ac-
tually the vacuum in the gravitational field — can be regarded
as an optical medium with a total refractive index given by
n = n1n2 = e
rc2 . (26)
n is composed of two factors: n1 — related with the time trans-
formation or “curved time”; n2 — related with the space trans-
formation or “curved space”. So the curved spacetime of gen-
eral relativity is reflected in the synthesized refractive index
n, which is also a reflection of the inhomogeneity of the vac-
uum, showing that the vacuum near the matter is influenced
more than that far away from the matter.
The above expression of n shows that the refractive index
of the vacuum at the infinity from the gravitational matter is
1, i.e., the usual refractive index of vacuum. The closer of the
position to the center of matter M, the higher the refractive
index of the vacuum. The relation between n and r is depicted
in Fig.3, where 2GM/c2 is taken as the unit of r. For exam-
ples, the corresponding radii for the surface of the Sun in solar
gravitational field and the surface of the Earth in earth gravita-
tional field are 2.36× 105 and 7.20× 108 respectively — both
are far beyond the r-axis illustrated in Fig.3.
The deviation of the vacuum refractive index from the usual
value 1 is given by
∆n = n − 1 = e
rc2 − 1. (27)
FIG. 4: The two images I1, I2 of a gravitational lens.
In weak field it becomes
. (28)
In order to provide the readers with a quantity impression,
let us give two examples. For the solar gravitational field
(M = 1.99 × 1030kg), the deviation of n on the surface of
the Sun (r = 6.96 × 108m) is 4.24 × 10−6. For the earth grav-
itational field (M = 5.98 × 1024kg), the deviation of n on the
surface of the Earth (r = 6.38 × 106m) is only 1.39 × 10−9,
which is so small that it can hardly be observed in usual ex-
periments. Nevertheless, for a massive celestial body such as
a heavy star, a galaxy or a cluster of galaxies, the deviation is
not only observable, but also important and useful in gravita-
tional astronomy.
III. APPLICATIONS
The deflection of light by massive bodies leads to the effect
of gravitational lensing. Formerly, this effect should be calcu-
lated complicatedly with the general relativity [18]. Once we
have introduced the concept of graded vacuum refractive in-
dex and obtained its relation with mass M and position r, the
problem of gravitational lensing could then be treated easily
with the conventional optical method.
Considering a source S and a lens L of mass M, the light
emitted from S is bent due to the gravitational field of the
lens. The bent light could be figured out through Eq.(13) and
Eq.(26). Drawing the extension line of the light from the ob-
server O, the apparent (observed) position of the source image
I could then be found out. The result is shown in Fig.4.
This method could also be applied in studying the central
imaging. In doing this, the vacuum refractive index profile
inside the lensing body should be considered as well.
Noticing that Eq.(26) could be virtually rewritten as
n = e−
c2 , (29)
where Pr represents the gravitational potential at position r
from the center of the lens.
As a model for discussion, we suppose a lens (for example,
a galaxy or a cluster of galaxies) of radius R with a density
FIG. 5: A ray tracing result for the central imaging.
distribution
ρ = ρ0[1 − (
)k], (30)
where ρ0 is the central density of the lens, 0 6 r 6 R, k > 0.
The density ρ decreases with the distance r from the center of
mass; the decreasing varies with the parameter k. This model
gives the distribution of gravitational potential as
Po = −4πρ0G
3(3 + k)
Pi = −4πρ0G{
2(2 + k)
R2 − [
(2 + k)(3 + k)
)k]r2},(31)
for outside (r > R0) and inside (r 6 R0) the gravitational lens
respectively.
The vacuum refractive index profile outside and inside the
gravitational lens then reads
no = exp
8πρ0G
3(3 + k)
ni = exp
8πρ0G
2(2 + k)
R2 − [
(2 + k)(3 + k)
)k]r2}
.(32)
Fig.5 shows a ray tracing result for the imaging of a gravita-
tional lens with the above described vacuum refractive index
profile. In the figure, only three paths of ray (the three thick
lines) could pass through the observer O, forming the upper,
lower and central images respectively. From the figure, we
find that, under the same conditions, the larger the distance
OL from the observer to the lens, the closer the central imag-
ing light to the center of the lens. If the source S and the
observer O are counterchanged, it could also be known from
the figure that, the larger the distance S L from the source to
the lens, the closer the central imaging light to the center of
the lens. In addition, through the change of the lens mass
M = 43πR
3ρ0k/(3 + k), we also find that, when the mass M
increases, the distance from the central imaging light to the
center of the lens decreases (Fig.6, where the mass ratio of the
lenses corresponding to the four central imaging rays from
bottom to top is 2 : 3 : 4 : 5 ).
For the actual condition of gravitational imaging, the dis-
tances OL, S L and the mass M are all astronomical figures;
FIG. 6: Tracing the central imaging rays for lenses of different mass.
therefore, the light of central imaging is extremely close to
the center of the lens. However, for a lensing body with a
density increasing towards the center, it is possible that there
are barrier matters near the center which will destroy the for-
mation of the central image. Besides, the relatively longer
inner path of the central imaging light adds the possibility of
light being held back by the lens matters on the way. These
and some other factors such as the relative faintness of the
central imaging light and the possibly higher brightness of the
lens core itself, all decrease the possibility of central imaging
being actually observed. This analysis is firmly supported by
the fact that the number of observed images is not “odd” as
expected by the existed theories but “even” in almost all cases
of gravitational lensing [11].
IV. CONCLUSIONS
We have proposed the concept of inhomogeneous vacuum
with graded refractive index based on the analysis of the influ-
ence of static gravitational field on the vacuum. we derived the
expression of this refractive index analytically. By using this
expression, we investigated the effect of gravitational lensing
in a conventional optical way and provided a reasonable inter-
pretation for the problem of central image missing.
The result indicates that, the concept of inhomogeneous
vacuum is mathematically equivalent to the curved spacetime
in the general relativity; therefore, an effective and convenient
alternative method (i.e., optical method) could be established
to solve the so complicated problems in gravitational astron-
omy. Physically, under such point of view, the motion of light
in gravitational space is a motion of light wave in a quantum
vacuum with graded refractive index. And as we know that,
in conventional optics, the Fermat’s principle says that the op-
tical path between two given points is an extremum. This is
also equivalent to the theorem in the general relativity that a
particle always moves along a geodesic line in a curved space-
time.
Acknowledgments
We wish to acknowledge the supports from the Na-
tional Key Project for Fundamental Research (grant no.
2006CB921403), the National Hi-tech project (grant no. 2006
AA06A204) and the Zhejiang Provincial Qian-Jiang-Ren-Cai
Project of China (grant no. 2006R10025).
[1] H. Gies and K. Klingmüller, Phys. Rev. Lett. 96, 220401
(2006).
[2] T. Emig et al., Phys. Rev. Lett. 96, 080403 (2006).
[3] S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997).
[4] H. B. Chan et al., Science 291, 1941 (2001).
[5] N. Ahmadi and M. Nouri-Zonoz, Phys. Rev. D. 74, 044034
(2006).
[6] A. Dupays et al., Phys. Rev. Lett. 94, 161101 (2005).
[7] G. L. J. A. Rikken and C. Rizzo, Phys. Rev. A. 63, 012107
(2000) ; 67, 015801 (2003).
[8] A. Armoni, A. Gorsky, and M. Shifman, Phys. Rev. D. 72,
105001 (2005).
[9] A. Barroso et al., Phys. Rev. D. 74, 085016 (2006) .
[10] K. R. Dienes, E. Dudas, and T. Gherghetta, Phys. Rev. D. 72,
026005 (2005).
[11] J. N. Winn, D. Rusin, and C. S. Kochanek, Nature 427, 613
(2004).
[12] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields
(Pergamon Press, New York, 1975).
[13] M. Born and E. Wolf, Principles of Optics (7th edition) (Cam-
bridge University Press, Cambridge, 1999).
[14] H. A. Lorentz, A. Einstein, and H. Minkowski, The Principle of
Relativity (Methuen, London, 1923).
[15] H. C. Ohanian, Gravitation and Spacetime (W. W. Norton and
Company Inc., New York, 1976).
[16] S. Weinberg, Gravitation and Cosmology (John Wiley and
Sons, New York, 1972).
[17] E. B. Fomaleont and R. A. Sramek, Phys. Rev. Lett. 36, 1475
(1976).
[18] S. Mollerach and E. Roulet, Gravitational Lensing and Mi-
crolensing (World Scientific, New Jersey, 2002).

  We analyzed the influence of static gravitational field on the vacuum and
proposed the concept of inhomogeneous vacuum. According to the observational
result of the light deflection in solar gravitational field as well as the
corresponding Fermat's principle in the general relativity, we derived an
analytical expression of the refractive index of vacuum in a static
gravitational field. We found that the deviation of the vacuum refractive index
is composed of two parts: one is caused by the time dilation effect, the other
is caused by the length contraction effect. As an application, we simulated the
effect of the gravitational lensing through computer programming and found that
the missing central imaging could be interpreted in a reasonable way.

The Deviation of the Vacuum Refractive Index Induced by a Static Gravitational Field
Xing-Hao Ye, Qiang Lin∗
Department of Physics, Zhejiang University, Hangzhou 310027, China
(Dated: October 29, 2018)
We analyzed the influence of static gravitational field on the vacuum and proposed the concept of inhomoge-
neous vacuum. According to the observational result of the light deflection in solar gravitational field as well as
the corresponding Fermat’s principle in the general relativity, we derived an analytical expression of the refrac-
tive index of vacuum in a static gravitational field. We found that the deviation of the vacuum refractive index
is composed of two parts: one is caused by the time dilation effect, the other is caused by the length contraction
effect. As an application, we simulated the effect of the gravitational lensing through computer programming
and found that the missing central imaging could be interpreted in a reasonable way.
PACS numbers: 42.25.Bs, 42.50.Lc, 04.
I. INTRODUCTION
Vacuum is usually considered as “homogeneous” and
“isotropic”, i.e., vacuum does not differ from place to place,
and the refractive index of vacuum is always equal to 1.
However, the recent theoretical and experimental progresses
demonstrate that such concept of vacuum turns out to be in-
appropriate when there are matters or fields within finite dis-
tance. For example, the vacuum inside a microcavity is modi-
fied due to the existence of the cavity mirrors, which will alter
the zero-point energy inside the cavity and cause an attrac-
tive force between the two mirrors known as Casimir effect
[1, 2], which has been verified experimentally [3, 4]. A sec-
ond example is that, under the influence of electromagnetic
field, vacuum can be polarized, which has led to astonishingly
precise agreement between predicted and observed values of
the electron magnetic moment and Lamb shift, and may in-
fluence the motion of photons [5]. Dupays et al. [6] studied
the propagation of light in the neighborhood of magnetized
neutron stars. They pointed out that the light emitted by back-
ground astronomical objects will be deviated due to the optical
properties of quantum vacuum in the presence of a magnetic
field. Also in [7], Rikken and Rizzo considered the anisotropy
of the optical properties of the vacuum when a static mag-
netic field B0 and a static electric field E0 are simultaneously
applied perpendicular to the direction of light propagation.
They predicted that magnetoelectric birefringence will occur
in vacuum under such conditions. They also demonstrated
that the propagation of light in vacuum becomes anisotropic
with the anisotropy in the refractive index being proportional
to B0 × E0.
The facts that the propagation of light in vacuum can be
modified by applying electromagnetic fields to the vacuum
implies that the vacuum is actually a special kind of opti-
cal medium [5, 6]. This is similar to the Kerr electro-optic
effect and the Faraday magneto-optic effect in nonlinear di-
electric medium. This similarity between the vacuum and the
dielectric medium implies that vacuum must also have its in-
ner structure, which could be influenced by matter or fields as
∗Electronic address:qlin@zju.edu.cn
well. Actually, the structure of quantum vacuum has already
been investigated in quite a number of papers [8, 9, 10].
In this paper, with the analysis of the influence of static
gravitational field on the vacuum, we put forward a new con-
cept that the curved spacetime around a certain matter can be
treated as an optical medium with a graded refractive index.
We suggest that the so-called curved spacetime is a reflec-
tion of the vacuum inhomogeneity caused by the influence of
gravitational matter. Based on this idea, the refractive index
of vacuum is derived. We will also apply this concept to un-
puzzle the problem of the central image missing in almost all
the observed cases of gravitational lensing [11].
II. THE DEVIATION OF THE VACUUM REFRACTIVE
INDEX
According to the astronomical observation, the light prop-
agating through a space with a celestial body nearby will be
deflected. It can be interpreted with the curved spacetime in
general relativity. As a matter of fact, it can also be inter-
preted with the assumption that the vacuum around matter is
inhomogeneous with refractive index deviated from 1. Here
we put forward a theoretical model to describe the refractive
index profile based on the Fermat’s principle for the propaga-
tion of light in a static gravitational field, which was given by
Landau and Lifshitz [12]:
−1/2dl = 0, (1)
where dl is the local length element passed by light and mea-
sured by the observer at position r in the gravitational field, r
is the distance from this element of light to the center of grav-
itational matter M, g00 is a component of the metric tensor
gµν, g
00 dl corresponds to an element of optical path length.
= dt/dτ, where dτ represents the time interval mea-
sured by the local observer for a light ray passing through the
length dl, while dt is the corresponding time measured by the
http://arxiv.org/abs/0704.1173v1
mailto:qlin@zju.edu.cn
observer at infinity. Eq.(1) could then be rewritten as
−1/2dl = δ
nds = 0, (2)
where ds is the length element measured by the observer at
infinity, corresponding to the local length dl.
Eq.(2) shows that if we set the scale of length and time at
infinity as a standard scale for the whole gravitational space
and time, the propagation of light then satisfies the standard
representation of Fermat’s principle, with the space — actu-
ally the vacuum — possessing a refractive index given by
= n1n2. (3)
The factor n1 of the refractive index relating to the time
transformation effect dt/dτ can be derived from the Newto-
nian attraction, which contributes partially to the deflection
of light. Considering a photon of relativistic mass m∞ at the
infinity moving down to position r, the work done on to the
photon by the Newtonian gravity is
dr = d(mc2), (4)
where G is the gravitational constant, c is the velocity of light,
M is the mass of a star (say the Sun), r is the distance to the
center of the star. Integrating Eq.(4) gives
mr = m∞e
rc2 , (5)
where mr is the relativistic mass of the photon at position r.
Since the photon energy is E = hν = mc2, where h is the
Planck constant, ν is the photon frequency, then we have m =
hν/c2. Substituting it into Eq.(5) gives
νr = ν∞e
rc2 . (6)
It is just the frequency shift caused by the gravitational force,
which reflects that a clock in a gravitational field runs slower
than that far away from the gravitational center. That is
dτ = e−
rc2 dt, (7)
where dτ denotes the time measured by a clock at position
r, dt is the converted time of dτ, i.e., the time measured by
the clock at infinity. This relation indicates that, if the length
scale is the same, i.e., dl = ds, when an observer at position r
reports a light velocity c1 = dl/dτ, it should be converted by
the observer at infinity into
c′1 =
rc2 dτ
= c1e
rc2 . (8)
FIG. 1: Light deflection caused by a graded refractive index.
This change of light velocity will certainly bring a deflection
to the light propagation. The corresponding refractive index
rc2 . (9)
Let us now consider the deflection angle caused by this
graded refractive index. In Fig.1, the curve AP represents the
light ray, β is the angle between the position vector r and the
tangent at the point P on the ray, ϕ is the deflection angle of
light. Since the refractive index shown in Eq.(9) has spheri-
cal symmetry, i.e., depends only on the distance r for a given
mass M, according to the Fermat’s principle
nds = 0, (10)
where ds = dr
1 + (rα̇)2, α̇ = dα/dr, n = n(r), we have the
corresponding Lagrangian function
L(α, α̇; r) = n(r)
1 + (rα̇)2. (11)
Using the Lagrangian equation
= 0, (12)
we get [13]
nr sin β = constant, (13)
nr sin β = n0r0, (14)
where r0 and n0 represent the radius and refractive index at the
nearest point A respectively.
Since
tan β =
, (15)
associating with Eq.(14) reaches
( nrn0r0 )
2 − 1
. (16)
FIG. 2: Light deflection in solar gravitational field.
For a light ray passing by the Sun as shown in Fig.2, the
total angular displacement of the radius vector r reads
∆α = 2
( nrn0r0 )
2 − 1
, (17)
where r0 represents the nearest distance to the center of the
Sun. Because the gravitational field of the Sun is a weak field,
the value of GM/rc2 is quite small, so substituting Eq.(9) into
Eq.(17) gives a solution of first order approximation
∆α = π +
. (18)
Then the total deflection angle of light caused by the refrac-
tive index n1 in solar gravitational field is
∆ϕ1 = ∆α − π =
. (19)
In fact, this result was obtained early in 1911 by Einstein
[14], who also investigated the effect of red shift and the cor-
responding slowing down of the light velocity in gravitational
field and then figured out the light deflection as shown in
Eq.(19) with the use of Huygens’ principle. Since the ac-
tual total deflection angle of light propagation calculated by
the general relativity [15, 16] and measured by the astronomi-
cal observation [17] is twice that value, we then come to know
that the length transformation effect dl/ds in Eq.(2) must have
the same relation as that of the time transformation effect
dt/dτ expressed in Eq.(9), namely
rc2 . (20)
This relation indicates that a ruler in a gravitational field is
shorter than that far away from the gravitational center. So
when an observer at position r reports a length dl, it should be
converted by the observer at infinity into
ds = e−
rc2 dl. (21)
For a light passing through a length dl, if the time scale is
the same, i.e., dτ = dt , the light velocity c2 = dl/dτ reported
by the observer at position r should then be converted by the
observer at infinity into
c′2 =
rc2 dl
= c2e
rc2 . (22)
FIG. 3: The dependence of the vacuum refractive index n on the
distance r.
This change of light velocity will also bring a deflection to
the light propagation. The refractive index corresponding to
this kind of deflection is
rc2 , (23)
which also causes a deflection angle of light
∆ϕ2 =
. (24)
Therefore, the total deflection angle of light in solar gravita-
tional field is
∆ϕ = ∆ϕ1 + ∆ϕ2 =
. (25)
The above result shows that, if the two refraction effects are
considered simultaneously, then the gravitational space — ac-
tually the vacuum in the gravitational field — can be regarded
as an optical medium with a total refractive index given by
n = n1n2 = e
rc2 . (26)
n is composed of two factors: n1 — related with the time trans-
formation or “curved time”; n2 — related with the space trans-
formation or “curved space”. So the curved spacetime of gen-
eral relativity is reflected in the synthesized refractive index
n, which is also a reflection of the inhomogeneity of the vac-
uum, showing that the vacuum near the matter is influenced
more than that far away from the matter.
The above expression of n shows that the refractive index
of the vacuum at the infinity from the gravitational matter is
1, i.e., the usual refractive index of vacuum. The closer of the
position to the center of matter M, the higher the refractive
index of the vacuum. The relation between n and r is depicted
in Fig.3, where 2GM/c2 is taken as the unit of r. For exam-
ples, the corresponding radii for the surface of the Sun in solar
gravitational field and the surface of the Earth in earth gravita-
tional field are 2.36× 105 and 7.20× 108 respectively — both
are far beyond the r-axis illustrated in Fig.3.
The deviation of the vacuum refractive index from the usual
value 1 is given by
∆n = n − 1 = e
rc2 − 1. (27)
FIG. 4: The two images I1, I2 of a gravitational lens.
In weak field it becomes
. (28)
In order to provide the readers with a quantity impression,
let us give two examples. For the solar gravitational field
(M = 1.99 × 1030kg), the deviation of n on the surface of
the Sun (r = 6.96 × 108m) is 4.24 × 10−6. For the earth grav-
itational field (M = 5.98 × 1024kg), the deviation of n on the
surface of the Earth (r = 6.38 × 106m) is only 1.39 × 10−9,
which is so small that it can hardly be observed in usual ex-
periments. Nevertheless, for a massive celestial body such as
a heavy star, a galaxy or a cluster of galaxies, the deviation is
not only observable, but also important and useful in gravita-
tional astronomy.
III. APPLICATIONS
The deflection of light by massive bodies leads to the effect
of gravitational lensing. Formerly, this effect should be calcu-
lated complicatedly with the general relativity [18]. Once we
have introduced the concept of graded vacuum refractive in-
dex and obtained its relation with mass M and position r, the
problem of gravitational lensing could then be treated easily
with the conventional optical method.
Considering a source S and a lens L of mass M, the light
emitted from S is bent due to the gravitational field of the
lens. The bent light could be figured out through Eq.(13) and
Eq.(26). Drawing the extension line of the light from the ob-
server O, the apparent (observed) position of the source image
I could then be found out. The result is shown in Fig.4.
This method could also be applied in studying the central
imaging. In doing this, the vacuum refractive index profile
inside the lensing body should be considered as well.
Noticing that Eq.(26) could be virtually rewritten as
n = e−
c2 , (29)
where Pr represents the gravitational potential at position r
from the center of the lens.
As a model for discussion, we suppose a lens (for example,
a galaxy or a cluster of galaxies) of radius R with a density
FIG. 5: A ray tracing result for the central imaging.
distribution
ρ = ρ0[1 − (
)k], (30)
where ρ0 is the central density of the lens, 0 6 r 6 R, k > 0.
The density ρ decreases with the distance r from the center of
mass; the decreasing varies with the parameter k. This model
gives the distribution of gravitational potential as
Po = −4πρ0G
3(3 + k)
Pi = −4πρ0G{
2(2 + k)
R2 − [
(2 + k)(3 + k)
)k]r2},(31)
for outside (r > R0) and inside (r 6 R0) the gravitational lens
respectively.
The vacuum refractive index profile outside and inside the
gravitational lens then reads
no = exp
8πρ0G
3(3 + k)
ni = exp
8πρ0G
2(2 + k)
R2 − [
(2 + k)(3 + k)
)k]r2}
.(32)
Fig.5 shows a ray tracing result for the imaging of a gravita-
tional lens with the above described vacuum refractive index
profile. In the figure, only three paths of ray (the three thick
lines) could pass through the observer O, forming the upper,
lower and central images respectively. From the figure, we
find that, under the same conditions, the larger the distance
OL from the observer to the lens, the closer the central imag-
ing light to the center of the lens. If the source S and the
observer O are counterchanged, it could also be known from
the figure that, the larger the distance S L from the source to
the lens, the closer the central imaging light to the center of
the lens. In addition, through the change of the lens mass
M = 43πR
3ρ0k/(3 + k), we also find that, when the mass M
increases, the distance from the central imaging light to the
center of the lens decreases (Fig.6, where the mass ratio of the
lenses corresponding to the four central imaging rays from
bottom to top is 2 : 3 : 4 : 5 ).
For the actual condition of gravitational imaging, the dis-
tances OL, S L and the mass M are all astronomical figures;
FIG. 6: Tracing the central imaging rays for lenses of different mass.
therefore, the light of central imaging is extremely close to
the center of the lens. However, for a lensing body with a
density increasing towards the center, it is possible that there
are barrier matters near the center which will destroy the for-
mation of the central image. Besides, the relatively longer
inner path of the central imaging light adds the possibility of
light being held back by the lens matters on the way. These
and some other factors such as the relative faintness of the
central imaging light and the possibly higher brightness of the
lens core itself, all decrease the possibility of central imaging
being actually observed. This analysis is firmly supported by
the fact that the number of observed images is not “odd” as
expected by the existed theories but “even” in almost all cases
of gravitational lensing [11].
IV. CONCLUSIONS
We have proposed the concept of inhomogeneous vacuum
with graded refractive index based on the analysis of the influ-
ence of static gravitational field on the vacuum. we derived the
expression of this refractive index analytically. By using this
expression, we investigated the effect of gravitational lensing
in a conventional optical way and provided a reasonable inter-
pretation for the problem of central image missing.
The result indicates that, the concept of inhomogeneous
vacuum is mathematically equivalent to the curved spacetime
in the general relativity; therefore, an effective and convenient
alternative method (i.e., optical method) could be established
to solve the so complicated problems in gravitational astron-
omy. Physically, under such point of view, the motion of light
in gravitational space is a motion of light wave in a quantum
vacuum with graded refractive index. And as we know that,
in conventional optics, the Fermat’s principle says that the op-
tical path between two given points is an extremum. This is
also equivalent to the theorem in the general relativity that a
particle always moves along a geodesic line in a curved space-
time.
Acknowledgments
We wish to acknowledge the supports from the Na-
tional Key Project for Fundamental Research (grant no.
2006CB921403), the National Hi-tech project (grant no. 2006
AA06A204) and the Zhejiang Provincial Qian-Jiang-Ren-Cai
Project of China (grant no. 2006R10025).
[1] H. Gies and K. Klingmüller, Phys. Rev. Lett. 96, 220401
(2006).
[2] T. Emig et al., Phys. Rev. Lett. 96, 080403 (2006).
[3] S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997).
[4] H. B. Chan et al., Science 291, 1941 (2001).
[5] N. Ahmadi and M. Nouri-Zonoz, Phys. Rev. D. 74, 044034
(2006).
[6] A. Dupays et al., Phys. Rev. Lett. 94, 161101 (2005).
[7] G. L. J. A. Rikken and C. Rizzo, Phys. Rev. A. 63, 012107
(2000) ; 67, 015801 (2003).
[8] A. Armoni, A. Gorsky, and M. Shifman, Phys. Rev. D. 72,
105001 (2005).
[9] A. Barroso et al., Phys. Rev. D. 74, 085016 (2006) .
[10] K. R. Dienes, E. Dudas, and T. Gherghetta, Phys. Rev. D. 72,
026005 (2005).
[11] J. N. Winn, D. Rusin, and C. S. Kochanek, Nature 427, 613
(2004).
[12] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields
(Pergamon Press, New York, 1975).
[13] M. Born and E. Wolf, Principles of Optics (7th edition) (Cam-
bridge University Press, Cambridge, 1999).
[14] H. A. Lorentz, A. Einstein, and H. Minkowski, The Principle of
Relativity (Methuen, London, 1923).
[15] H. C. Ohanian, Gravitation and Spacetime (W. W. Norton and
Company Inc., New York, 1976).
[16] S. Weinberg, Gravitation and Cosmology (John Wiley and
Sons, New York, 1972).
[17] E. B. Fomaleont and R. A. Sramek, Phys. Rev. Lett. 36, 1475
(1976).
[18] S. Mollerach and E. Roulet, Gravitational Lensing and Mi-
crolensing (World Scientific, New Jersey, 2002).
