Gauss’s Law for gravity and observational evidence reveal no solar lensing in empty vacuum space
E. H. Dowdye, Jr.1
Pure Classical Physics Research, 1850 Lakecrest Drive No. 321, Greenbelt, MD 20770

ABSTRACT
Findings show that the rays of star light are lensed primarily in the plasma rim of the sun and hardly in the vacuum space just slightly above the rim. Since the lower boundary of this vacuum space is only a fraction of a solar radius above the solar plasma rim, it is exposed to virtually the same gravitational ﬁeld. The thin plasma atmosphere of the sun appears to represent an indirect interaction involving an interfering plasma medium between the gravitational ﬁeld of the sun and the rays of star light. The very same light bending equation obtained by General Relativity was derived from classical assumptions of a minimum energy path of a light ray in the plasma rim, exposed to the gravitational gradient ﬁeld of the sun. The resulting calculation was found to be independent of frequency. An intense search of the star ﬁlled skies reveals a clear lack of lensing among the countless numbers of stars, where there are many candidates for gravitational lensing according to the assumptions of General Relativity. Assuming the validity of the light bending rule of General Relativity, the sky should be ﬁlled with images of Einstein rings. Moreover, a lack of evidence for gravitational lensing is clearly revealed in the time resolved images of the rapidly moving stellar objects orbiting about Sagittarius A*.
Subject headings: black hole – gravitational lensing – galaxy center – plasma atmosphere – Gauss’s law

1. Introduction
We shall examine the evidence for gravitational lensing in our region of space near to us, starting with the nearest star to us, our sun. The light bending rule of General Relativity suggests that a direct interaction should take place between the gravitational ﬁeld of the lensing mass and the rays of light from the stars, whether in a vacuum space or in a plasma atmosphere. If the gravitational lensing is observed only at the plasma rim of the sun, it is evident that the past century of observed solar light bending events were due to an indirect interaction between the gravitational ﬁeld of the sun and the rays of star light. This argument is supported by a calculation which derives the very same light bending equation obtained by General Relativity. The equation was derived from the assumptions of a minimum energy path of light
1An Independent Researcher, Member American Physics Society, Founder Pure Classical Physics Research

in a plasma atmosphere exposed to the gravitational gradient ﬁeld of the sun. Appendix A gives a detail calculation and a derivation of the famous light bending equation. We shall take a closer look at the lower boundary of the vacuum space just above the plasma rim of the sun, the nearest star to us, just 8 light-minutes away. We shall examine the nearby stars in our own region of space, less than hundreds of light-years away. There are many cases in the star ﬁlled skies where the lenses and the light sources are by chance co-linearly aligned with the earth based observer, presenting vast opportunities for the observation of Einstein rings. We shall examine the collected images and the astrophysical data of the stars orbiting about Sagittarius A*, a region thought to contain a super massive black hole located at the center of our galaxy, the Milky Way, right in our own back yard, just 26,000 light-years away. Research convincingly reveal:

1

• a lack of evidence for gravitational lensing in the vacuum space, just a fraction of a solar radius above the solar plasma rim; straightforwardly revealed by applying the Gauss’s law of gravity to the solar mass
• a lack of evidence for Einstein rings in a sky of countless numbers of stars, where the candidates for gravitational lenses and the light sources are by good chance co-linearly aligned with the earth based observer
• a lack of evidence for gravitational lensing in the time resolved images of the stars orbiting about presumably a black hole at the site of Sagittarius A*
For nearly a century now, gravitational light bending events have been observed primarily at the rim of the sun during events of solar eclipses. New ﬁndings clearly show that with a straightforward application of Gauss’s law of gravity, an important fundamental of Mathematical Physics, the light bending rule of General Relativity apparently does not apply to the empty vacuum space above the rim of the sun.
2. Some Misapplied Fundamentals on Gravitational Lensing Concepts
An application of Gauss’s law, applied to gravitation as well as to electromagnetism along with the principle of optical reciprocity clearly show that a co-linear alignment of the observer, the lens and the source is unnecessary for an observation of a gravitational light bending eﬀect, as predicted by the light bending rule of General Relativity. The gravitational eﬀect at the surface of an analytical Gaussian sphere due to the presence of a pointlike gravitating mass that is enclosed inside of the sphere depends only on the quantity of mass enclosed. The size or density of the enclosed mass particle is not important. [12], [13] Gauss’s law of gravity [1], [4] is a Mathematical Physics tool that encloses a gravitating mass particle inside of an analytical Gaussian surface which applies directly to the gravitational ﬁeld of the enclosed mass. An analogy to this principle encloses an electrically charged particle inside of a Gaussian surface in application to the electric ﬁeld of the charged particle in the discipline of Electromagnetism [4]. The principle of optical reciprocity [2], [3] simply states

that the light must take the very same minimum energy path or least time path, in either direction between the source and the observer. This fundamental principle is an essential tool for the understanding of complex lensing systems in Astronomy and Astrophysics [6]. We shall now correctly apply all of these well founded and proven fundamentals to these gravitational lensing problems.

2.1. Gauss’s Law applied to a Point-Like Gravitating Mass

Any gravitational eﬀect on a light ray due to the presence of a gravitating point like mass at the impact parameter R would theoretically depend on the amount of Mass M that is enclosed within the analytical Gaussian sphere of radius R as illustrated in Figure 1. Any gravitational eﬀect that would be noted at the surface of the analytical Gaussian sphere should in principle be totally independent of the radius of the mass particle or the density of the mass that is enclosed within the Gaussian sphere. From Gauss’s Law (Equation 2) equal masses of diﬀerent radii will theoretically have equal gravitational eﬀects at the surface of the Gaussian sphere. The light bending rule

δθ

=

4GM Rc2

(1)

of

General

Relativity

is

essentially

a

localized

1 R

ef-

fect. We are dealing with astronomical distances.

Thus, the bulk of the gravitational eﬀect on the

path of particles of light would takes place along

a segment of the light ray that encloses the im-

pact parameter R. This segment may be only

several orders of magnitude greater than the im-

pact parameter R. Thus, the predominant eﬀect of

the gravitational ﬁeld on the bending of the light

ray would occur along this short segment of the

light path, maximizing at the point where the light

ray is tangent to our analytical Gaussian sphere,

namely, at the impact parameter R.

A very essential tool of Mathematical Physics known as Gauss’s law [1] ,[4],

g · dA = 4πGM

(2)

S

is applied directly to the gravitating masses where the gravitational ﬁeld g is a function only of the mass M enclosed by the spherical Gaussian surface S. The gravitational ﬂux at the surface of

2

the analytical Gaussian sphere is totally indepen-

dent of the radius R of the sphere. [2] The idea

here is that the gravitational ﬁeld at this analyti-

cal Gaussian surface is only a function of the mass

that it encloses. [1], [4] Any mass M , regardless

of the radius of the mass particle that is enclosed

inside of the Gaussian spherical surface of radius

R will contribute exactly the same gravitational

potential at the Gaussian surface. In Figure 1,

the gravitational ﬁeld points inward towards the

center of the mass.

Its

magnitude is

g

=

GM R2

.

In order to calculate the ﬂux of the gravitational

ﬁeld out of the sphere of area A = 4πR2, a mi-

nus sign is introduced. We then have the ﬂux

Φg

=

−gA

=

−(

GM R2

)(4πR2)

=

−4πGM .

Again,

we note that the ﬂux does not depend on the size

of the sphere. It is straightforwardly seen that a

direct application of Gauss’s law to the light bend-

ing rule, Equation 1, coupled with the essential

principle of optical reciprocity (Potton 2004), re-

moves any requirement for a co-linear alignment

of the light source, the point-like gravitating mass

particle (the lens) and the observer for observa-

tion of a gravitational lensing eﬀect as suggested

by General Relativity. [12]

From Equation 2, the ﬂux of the gravitational po-

tential at the surface of the Gaussian spheres, as

illustrated in Figure 1, is the same for all enclosed

mass particles of the same mass M , regardless of

the size of the mass particle. As a result, each

mass particle will produce the very same gravita-

tional light bending eﬀect δθ = δθ1 + δθ2, where δθ1 and δθ2 are the bending eﬀects on the ray of light on approach and on receding the lens, respec-

tively. This of course assumes the validity of Equa-

tion 1. This symmetry requirement suggests that

δθ1 = δθ2. From Equations 1 and 3 it follows that

δθ

=

2δθ1

=

4GM Rc2

and δθ1 = δθ2 =

2GM Rc2

.

This

says that the total contribution of the light bend-

ing eﬀect due to the gravitating point-like mass

particle on any given inﬁnitely long light ray is the-

oretically divided equally at the impact parameter

R, separating the approaching segment and the

receding segment of the optical path. A conﬁrma-

tion of this will be clearly seen in the next section

with application of the principle of reciprocity and

a derivation of the equation of the Einstein ring,

illustrating the symmetry requirement of General

Relativity.

Fig. 1.— Gauss’s Law applied to Equal Gravitating Masses of Diﬀerent Radii Enclosed
Fig. 2.— Fundamental Principle of Optical Reciprocity Illustrated on a Lensed Light Ray

3

2.2. Principle of Optical Reciprocity applied to the Lensed Light Ray
In any space, the principle of reciprocity [2],[3], a very fundamental principle of optics, must hold as illustrated in Figure 2. The principle simply states that light moving along a preferred optical path, from a source to an experimenting observer, must take the very same optical path from the hypothetical light source of the observer back to the distant real light source. As a consequence of this fundamental principle, any additional source placed along a preferred optical path will all appear to the observer as a point-like light source sitting on top of the image of the real light source. As a consequence of this principle, in the case of an image of an Einstein ring of a single real point-like light source, any additional light sources placed on a preferred optical path of the real light source would appear as point light sources sitting on the image of the Einstein ring of the distant real light source.
The total gravitational light bending eﬀect acting on the light ray upon approach and upon receding a point-like gravitating mass is give by

δθ = δθ | + δθ | = 4GM 1 approaching Rc the lens

2 receding
the lens

2

(3)

In this example, for simplicity, the gravitating mass M is chosen to be positioned at the midpoint on the line joining the observer and the light source for the simpliﬁed special case DL : DSL : DS = 1 : 1 : 2. [5] This simpliﬁed special case is illustrated in Figure 3.
The astronomical distance DL is the distance from the observer to the lens, DSL is the distance from the lens to the source and DS is the distance from the observer to the source. This is a simpliﬁed special case, where DL = DSL, presented in most academic textbooks. It is readily seen that the axis of symmetry for a given light ray is perpendicular to the line joining the source and the observer only for the special case where the lens is positioned exactly at the midpoint between the source and the observer. The vast astronomical distances between the stars present much larger impact parameters on a much grander scale to the light passing by the stars. In the plasma free space far above the rim of the stars, an indirect interaction certainly could not take place. It is for this

reason, the star ﬁlled sky reveals a clear lack of gravitational lensing among the countless numbers of stars.

2.3. Symmetry Requirement Demonstrated on derivation of Einstein Ring Equation
From symmetry we have

δθ1

=

δθ2

=

2GM Rc2

(4)

Again, the astronomical distance DL is the dis-

tance from the observer to the lens. Since we are

dealing with very small angles, from Figure 3, the

deﬂection of the light ray due to the gravitational

eﬀect on approach to the gravitating mass is just

simply δθ1 =

R DL

=

2GM Rc2

wherefrom

R2 DL

=

2GM c2

and

R2 DL 2

=

2GM DL c2

= δθ12.

Solving this for the radius of the impact parameter

of the light ray and thus the radius of the Einstein

Ring expressed in units of radians we have

δθ1 =

2GM DL c2

(5)

which is the radius of the Einstein ring in units of radians for a lens place exactly midway between the source and the observer. This is a special case, where DL = DSL. (See Appendix B for the general case (DL = DSL)) Note that the gravitational bending eﬀect on the light ray for the approach segment alone is exactly equal to the radius of the solved Einstein ring expressed in radians and is given as

2GM

δθ1 = Rc2

(6)

This eﬀect is exactly one half of the total accumulative gravitational eﬀect acting on the light ray for the approach and receding segments. [12] From symmetry requirement, the integral gravitational eﬀect on a light ray upon approach to a gravitating mass positioned exactly at the midpoint of a line joining the source and the observer, must equal that of the integral gravitational eﬀect on the light ray upon receding the gravitating mass

δθ | = δθ | 1 approaching

2 receding

the lens

the lens

(7)

4

as suggested by Equation 4 and the laws of conservation of energy and of momentum. [13] The accumulative gravitational eﬀect along the light ray must sum the total eﬀects of gravity acting on the light ray for both the approach and receding segments of any ray of light passing by a point-like gravitating mass. [12], [13] The total light bending eﬀect is therefore

δθ

=

4GM Rc2

(8)

The principle of optical reciprocity simply states that any light ray or a photon of light must take the very same path, along the same minimum energy path, in either direction between the source and the observer as depicted in Figure 2.
Using the light bending rule of General Relativity, it is theoretically demonstrated and graphically illustrated in Figure 4 that all observers of varying distances from a gravitating mass or lens should see an Einstein ring. Only the mid-ﬁeld observer depicted in Figure 4 will derive Equation 6 which gives exactly the same value as that given by Equation 5 for a simpliﬁed special case, where DL = DSL, which is presented in most academic textbooks. Appendix B gives the general case where DL is not necessarily equal to DSL. In the general case, the lens may not be placed exactly midway between the light source and the observer. The near-ﬁeld and the far-ﬁeld observers depicted in Figure 4 will also derive Einstein ring equations with coeﬃcients corresponding to their unique geometries. Each observer has distinct sets of lensed light rays, each of the lensed light rays with their corresponding axis of symmetry. As illustrated, the near-ﬁeld observer would see the largest, most lensed Einstein ring. The far-ﬁeld observer would see the smallest, least lensed Einstein ring. From symmetry, the axis of symmetry of the lensed curve would lean towards the nearﬁeld observer and away from the far-ﬁeld observer. The axis of symmetry would be perpendicular to the line joining the mid-ﬁeld observer and the light source only in the case where the lens is exactly midway between the observer and the source.
The lensed light rays depicted in Figure 4 all belong to the very same family of equations derived from the light bending rule (Equation 1) of General Relativity. Any light ray that is gravitationally bent by a point-like gravitating mass, as pre-

dicted by General Relativity, will always have an axis of symmetry which would be perpendicular to the line joining the source and the observer only when the lens is positioned exactly at the midpoint on the line joining the observer and the source. All observers should see, according to General Relativity, an Einstein ring. Figure 4 depicts the geometry of the lens rays as a function of the position of the observer relative to the lens and the source. [12]
2.4. Condition for Observing of an Einstein Ring using a Lens of 1 Solar Mass and 1 Solar Radius

Table 1: Astrophysical Data of the Sun

Solar M ass

M

1.99 · 1030Kg

Solar Radius

R

6.96 · 108m

G Constant

G

6.67 · 10−11m3s2/Kg

V elocity of Light c

2.99793 · 108ms−1

δθ (rad) δθ (deg) Radius of Sun

4GM Rc2
4GM Rc2
R(deg)

8.4952 · 10−6 rad 0.0004867 deg 0.275deg

F ocal Length

R(deg) δθ(deg)

565.0

We shall use the collected astrophysical data of the

sun in Table 1 for this task. Using this data we

ﬁnd that for a stellar system of 1 solar mass and 1

solar radius, the light bending Equation (8) yields

an angles of 8.4952 · 10−6 radians. This angle is

0.0004867 degrees or 1.752 arcsec. The diameter

of the solar disk is observed to be 0.55 degrees, a

radius of 0.275 degrees. If the radius of the solar

disk were compared with the angle of solar light

bending of the plasma rim (in degrees), we would

have

a

factor

of

R(deg) δθ(deg)

=

565.0.

This

means

that

in order to observe an Einstein ring of a distant

stellar light source due the plasma rim of the sun,

the observer would have to back away from the sun

for at least 565 astronomical units(AU’s). This is

the focal length of the plasma rim lensing

system of the sun. It is the distance required for

the parallel rays of star light to converge to a point

after being deﬂected by the solar plasma rim.

Note that if the observer were to back oﬀ to a dis-

5

Fig. 3.— Symmetry Requirement of the Accumulative Lensing Eﬀect Illustrated
Fig. 4.— The near-ﬁeld, mid-ﬁeld and far-ﬁeld observers and illustration of their corresponding axes of symmetry.

tance greater than 565 AU’s then the light rays from the plasma lens would come to a focus before reaching the observer and the Einstein ring would not be detected. Because of the vast distances between the stars in the night skies, any light bending of the light rays, according to General Relativity, would have to have larger impact parameters, extending out well above the plasma rim of the lensing stars into the empty vacuum space where there is apparently no gravitational lensing eﬀects taking place at all. Since there is no evidence for Einstein rings among the star-ﬁlled skies, this is apparently the case. Consequently, if gravitational lensing in empty vacuum space did not occur, then there can be no observations of Einstein Rings due to a co-linearly alignment of the widely separated stellar objects in the night skies.

Historically, the eﬀect of light bending has been noted only at the solar rim, the thin plasma of the sun’s atmosphere due to an indirect interaction between the rays of star light and the gravitational ﬁeld of the sun. Appendix A gives a detail calculation for the bending of light rays in a plasma atmosphere exposed to a gravitational gradient ﬁeld. It is easily demonstrated using the mathematical Physics of the Gauss law of gravity that the light bending eﬀect of the sun varies as 1/R at various radii R of analytical Gaussian surfaces, concentric to the center of the sun as suggested by General Relativity. The integrated eﬀect of the solar lensing eﬀect is essentially a 1/R eﬀect which is predominant along a segment of the light path that encloses the impact parameter, R, where the light ray is tangent to a spherical Gaussian surface at the impact parameter R. Since the astronomical distances are extremely large, for all practical purposes, the integrated 1/R eﬀect of the light bending occurs along a segment of the light ray that is extremely short compared to astronomical distances.

Theoretically, the light bending along the segment

due to the gravity of a point-like mass on ap-

proach and on receding are virtually equal and

divided at the impact parameter R where the re-

lation δθ |1 approaching = δθ |2 receding still holds even

the lens

the lens

though the Earth based observer is relatively close

to the sun. [12] Remarkably, as it may seem, how-

ever, historically the solar light bending eﬀect has

been observed only at the solar rim, namely, the

6

light refracting plasma of the solar atmosphere. It is too widely taught solar light bending is observed at the rim to maximize the eﬀect of detection. This has contributed obviously to misapplication of the important fundamentals. We note again, that the thickness of the thin plasma shell of the sun, frequently referred to as the solar rim, is very negligible in comparison to the solar radius R. Table 2 brieﬂy summarizes the observations of gravitational light bending as a function of the distance above the thin solar plasma rim. It is important to note that the bulk of the observed solar light bending events were recorded during solar eclipses. The moon has provided a near perfect masking of the solar disk, allowing only the thin plasma rim of the sun to be exposed for the astrophysical observations.

Table 2: The Observed and Predicted Gravita-

tional Lensing at Distances h (in units of RSUN )

above the Solar Plasma Rim Distance h Acceleration Observed Predicted by

above Rim 1/r2 Effect Lensing Relativity

(RSUN )

(gSUN )

(arcsec)

(arcsec)

1.0

0.25

none

0.88

0.5

0.44

none

1.17

0.2

0.69

none

1.45

0.1

0.83

negligible

1.59

0

1.00

1.75

1.75

Assuming the validity of the light bending rule of General Relativity, the current technical means of the astronomical techniques should have easily allowed observations of solar light bending of stellar light rays at diﬀerent solar radii, using the Gauss law of gravity applied to analytical spherical surfaces, namely at the radius of 2R, 3R and even beyond 4R, where R is one solar radius. For instance, at the analytical Gaussian surface of radius 2R, the predicted light bending eﬀect of General Relativity would have been an easily detectable effect of one half the eﬀect of 1.75 arcsec noted at the solar rim; at the surface of radius 3R, an effect of one third the eﬀect at the solar rim, etc., etc. The equatorial radius RSUN is approximately 695,000 km. The thickness of the solar rim is been recorded to be less than 20,000 km; less than 3 percent of the solar radius R. From this, we can easily see that a gravitational lensing eﬀect in vacuum space several solar radii above the solar plasm rim

should be a very noticeable eﬀect to the modern astronomical means.
2.5. The Fundamentals applied to the Orbit of S2 about Sagittarius A*
The past decades of intense observations using modern astronomical techniques in Astrophysics alone reveal an obvious lack of evidence for lensing eﬀects on collected emissions from stellar sources orbiting about Sagittarius A*, believed to be a super massive black hole located at the galactic center of our Milky Way. This is most obviously revealed in the time resolved images collected since 1992 on the rapidly moving stars orbiting about Sagittarius A*. [7], [8], [10], [11] The space in the immediate vicinity of a black hole is by deﬁnition an extremely good vacuum. The evidence for this is clearly seen in the highly elliptical orbital paths of the stars orbiting about the galactic core mass. The presence of material media near the galactic core mass would conceivably perturb the motion of the stellar object s16 which has been observed to move with a good fraction of the velocity of light. The presence of any media other than a good vacuum would have caused the fast moving stellar object s16 to rapidly disintegrate. Astrophysical observations reveal that s16 has a velocity approaching 3 percent of the velocity of light when passing to within a periastron distance corresponding to 60 astronomical units from Sagittarius A*, perceived to be a massive black hole. This gives convincing evidence that the space in this region has to be, without a doubt, an extremely good vacuum, no chance of an indirect interaction between the gravity of the super massive galactic core mass and the light emitted by these rapidly moving stars orbiting this mass.
Application of the light bending rule of Equation 4 together with considerations of Gauss’s Law and the principle of optical reciprocity to the data of the observed orbit the researchers in [11] has shown, it is clear that some gravitational lensing eﬀects should have been detected in the time resolved images of the orbit of S2, owing to the current level of todays observational means.
Selected points of the observed orbit of S2 and the corresponding predicted lensing of the orbit, based on the light bending rule of General Relativity, were tabulated in Reference [12]. From these calculations the predicted magnitude of the lensing

7

eﬀect was found to be orders of magnitude greater than the observed radial separation between the S2 source and the position of the galactic center mass. With current technical mean, this predicted lensing should have been a very noticeable eﬀect. To date, clear evidence of a gravitational lensing eﬀect based on the light bending rules of General Relativity is yet to be revealed in the time resolved images of the stellar objects orbiting about Sagittarius A*, a region under intense astrophysical observations since 1992.
3. Discussion & Conclusions
Historically, the light bending eﬀect has been observed predominately at the thin plasma rim of the sun. A direct interaction between the sun’s gravity and the rays of star light in empty vacuum space just a fraction of a solar radius above the solar plasma rim, where there is no plasma atmosphere, is yet to be observed. The stellar sky presents vast opportunities to modern Astronomy and Astrophysics to allow for the detection of gravitational lensing events, as predicted by General Relativity, due to the large numbers of stellar objects that just happen to be co-linearly aligned with the earth based observers; again of course assuming the validity of the light bending rule of General Relativity. Because of the vast astronomical distances between the stars, the gravitational lensing eﬀect would have to take place in deep space, at impact parameters far above the plasma rim of the lensing stars. If this were indeed the case and the light bending rule of General Relativity involved a direct interaction between the gravitational ﬁeld of the stars and the rays of light in vacuum space, then the entire celestial sky should be ﬁlled with images of Einstein rings.
References
[1] Arfken, G., Hans Weber, ”Mathematical Methods for Physicist”, Academic Press, pp. 7779 (1995)
[2] Born, M., Wolf, E., Principles of Optics, Pergamon Press, London - New York, 71, 100 104 (1975)
[3] Potton, R. J., ”Reciprocity in Optics”, Institute of Physics Publishing, Rep. Prog. Phys. 67, pp. 717-754 (2004)
[4] Jackson, J. D., Classical Electrodynamics, 3rd. ed., John Wiley & Sons, Inc., pp. 27-29

(1999)
[5] Blandford, R., Narayan, R., ”Cosmological applications of gravitational lensing”, Annual Review of Astronomy and Astrophysics, 30, 331 (1992)
[6] Carroll, B. W., Ostlie, D. A., ”An Introduction to Modern Astrophysics”. Addison-Wesley Publishing Co. (1996)
[7] Genzel, R., Schoedel, R., Ott, T., Eckart, A, Alexander, T., Lacombe, F., Rouan, D., Aschenbach, B.,Near-infrared ﬂares from accreting gas around the supermassive black hole at the Galactic Centre, Nature, Vol. 425, pp. 934-937 (2003)
[8] Melia, F., Falcke, H., ”The supermassive black hole at the Galactic center”. Anny. Rev. Astrophys, 39:309-52 (2001)
[9] Melia, F., ”The Black Hole at the center of Our Galaxy”, Princeton University Press, Princeton (2003)
[10] Narayan, R., ”Black holes: Sparks of interest”, Nature, Vol. 425, 6961, pp. 908-909 (2003)
[11] Schoedel R. et al., ”A star in a 15.2-year orbit around the supermassive black hole at the centre of the Milky Way.” Nature, 419, pp. 694 696 (2002)
[12] Dowdye, E. H., ”Time resolved images from the center of the Galaxy appear to counter General Relativity”, AN, Vol. 328, 2, pp. 186-191 (2007)
[13] Dowdye, E. H., ”Extinction Shift Principle: A Pure Classical Alternative to General and Special Relativity”, Physics Essays, Vol. 20, 56, pp. 13A - 14A (2007)

8

A. Appendix: Bending of Light Rays in the Solar Plasma Rim as function of Gravitational Potential; a Minimum Energy Path Calculation

This calculation is based entirely on a conservation of energy concept considering the gradient of the gravitational ﬁeld of the sun acting directly on the rapidly moving ionized material particles of the thin plasma atmosphere of the sun. The calculation considers a minimum energy path for rays of light. The resulting calculation was found to be totally independent of frequency. The rapidly moving ionized particles of the solar plasma is assumed to be bounded by the gravitational potential of the sun given by

φ(rr==∞ R ) =

r=∞ r=R

GM r2

dr

=

GM R

.

(A1)

It may be assumed that the plasma particles of the ionized solar rim move with random velocities such

that

their

kinetic

energies

are

as

dictated

by

1 2

mv2

=

3 2

kT

+ φm,

where

m

is

the

mean

mass

of

the

plasma

particles of temperature T (K◦) and v is the velocity of the plasma particle bounded by the gravitational

potential φ. The random velocity v of the moving ions may be assumed to have an upper bound velocity

of v =

2GM R

,

the

escape

velocity

of

the

solar

gravity

at

the

surface

of

the

sun.

The

solar

plasma

particles

bounded by gravity in the solar rim may be considered as a dynamic lens under the intense gravitational

gradient ﬁeld of the sun. It is theoretically shown here, and in detail in Dowdye2 (2007), that a minimum

energy path for light rays propagating in the solar plasma rim, subjected to the gradient of the gravitational

ﬁeld

of

the

sun,

yields

the

mathematical

results

of

4GM Rc2

.

It is shown that the moving ions acting as secondary sources within the plasma rim, move with random

velocities not to accede the escape velocity. The frequency and wavelength of a lensed light ray exposed to

the moving plasma are redshifted as:

ν

=

ν0(1

−

v2 c2

)

=

ν0(1

−

2GM Rc2

)

(A2)

λ

=

λ0(1

−

v2 c2

)−1

=

λ0(1

−

2GM Rc2

)−1

(A3)

λ

≈

λ0(1

+

2GM Rc2

).

(A4)

From this, the number of wavelengths per unit length along the minimum energy path for any given light ray propagating within the plasma rim may be given as

n=

1 λ

=

1

λ0(1

−

2GM Rc2

)−1

=

1 (1 λ0

−

2GM Rc2

).

(A5)

Thus, the photon energy density , joules per unit length of a light ray along the minimum energy path is

=

0(1 −

2GM rc2

).

Consequently,

the

number

of

re-emitted

waves

per

unit

length

along

the

photon

path

and

thus the energy per unit length increases as r increases. This translates to a downward, re-emitted path

of the bent light ray, along a minimum energy path for the approaching segment of the light ray. Theory

and

details are

published

in [13].

If

d dr

=+

2GM 0 r2c2

or δ

=

+

0

2GM r2c2

δR,

then

the

re-emission

of

the

light

ray

in the atmosphere of ions will occur such that the total energy along the minimum energy (conservation of

energy) path for a given light ray would not change. If is the energy per unit length along the light ray and

δ is the change in energy in the direction of the gradient potential φ(r), then the angle of change during

the approach segment of the light ray is

δθapp = δ app = +

r=R r=∞

2GM r2c2

dr

=

−

2GM Rc2

(A6)

9

and the path change for the receding segment of the light ray is

δθrec = δ rec = +

r=∞ r=R

2GM r2c2

dr

=

+

2GM Rc2

.

The net change in the path of the light ray is

δθ

=

δθrec

−

δθapp

=

4GM Rc2

.

(A7) (A8)

B. Appendix: The Einstein Ring Equation; the General Case (DL = DSL)
The general case for the Einstein ring equation involves all values for the distances, whereby DL is the distance between the observer and the lens and DSL is the distance between the lens and the source. These are cases where DL is not necessarily equal to DSL. The general case for the radius of the Einstein ring in units of radians is

δθ(rad) =

DSL 4GM DL + DSL DLc2

(B1)

The radius of the Einstein ring at the image location the distance of (DL + DSL) expressed in meters is

R(meters) = (DL + DSL)δθ(rad)

(B2)

where DL and DSL are also expressed in meters. The impact parameter (IP) corresponding to the Einstein ring is the nearest point of approach of the light rays to the point-like lensing mass, when observed at a distance of DL meters away from the observer, for the rays of light coming from the light source to the observer. Since this is a 3 dimensional problem, the impact parameter of the light rays that would produce an Einstein ring is also a ring itself. It is a virtual ring for purpose of the analysis of the problem. The impact parameter in meters is

RIP (meters) = (DL)δθ(rad)

(B3)

where RIP is the nearest point of approach of the gravitationally lensed light rays to the lensing star. It is that distance the lensed light rays will pass over the plasma rim of the lensing star, moving through the empty vacuum space well above the plasma rim of the lensing stars, moving along astronomical distances from the source to the observer. The radius of the predicted Einstein ring, according Equation (B1) and the light bending rule of General Relativity, will be nearly 15 times the radius of a sun-like lensing star, the same mass and radius as the sun, when both are observed at the same distance DSL = 4 light years away. Adjusting the parameter DSL would cause the radius of the Einstein ring to change. Increasing DSL would cause the image of the Einstein ring to increase in radius (an increase in magniﬁcation), assuming the validity of General Relativity. Setting DL = DSL, Equation (B1) becomes Equation (5), the special case.

10