NEWTON'S EMISSION THEORY OF LIGHT
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frequency f, speed c (relative to the source) and wavelength L. A
receiver on the ground receives light with frequency f', speed
c' (relative to the receiver) and wavelength L'. According to Newton's
emission theory of light:
f'=f(1+gh/c^2); c'=c(1+gh/c^2); L'=L
A rocket of length h accelerates with acceleration g. A light source
at the front end emits light with frequency f, speed c (relative to
the source) and wavelength L. A receiver at the back end receives
light with frequency f', speed c' (relative to the receiver) and
wavelength L'. At the moment of reception, the receiver has speed v
relative to the light source at the moment of emission. According to
Newton's emission theory of light:
f'=f(1+v/c); c'=c+v; L'=L
Einstein did not offer any reasonable alternative to the variation of
the speed of light in a gravitational field predicted by the emission
theory. Initially he was just using the equation c'=c(1+gh/c^2), then
quite stupidly (or dishonestly) replaced it with c'=c(1+2gh/c^2):
http://www.physlink.com/Education/AskExperts/ae13.cfm
"So, it is absolutely true that the speed of light is not constant in
a gravitational field [which, by the equivalence principle, applies as
well to accelerating (non-inertial) frames of reference]. If this were
not so, there would be no bending of light by the gravitational field
of stars....Indeed, this is exactly how Einstein did the calculation
in: 'On the Influence of Gravitation on the Propagation of Light,'
Annalen der Physik, 35, 1911. which predated the full formal
development of general relativity by about four years. This paper is
widely available in English. You can find a copy beginning on page 99
of the Dover book 'The Principle of Relativity.' You will find in
section 3 of that paper, Einstein's derivation of the (variable) speed
of light in a gravitational potential, eqn (3). The result is,
c' = c0 ( 1 + V / c^2 )
where V is the gravitational potential relative to the point where the
speed of light c0 is measured."
http://www.mathpages.com/rr/s6-01/6-01.htm
"In geometrical units we define c_0 = 1, so Einstein's 1911 formula
can be written simply as c=1+phi. However, this formula for the speed
of light (not to mention this whole approach to gravity) turned out to
be incorrect, as Einstein realized during the years leading up to 1915
and the completion of the general theory. In fact, the general theory
of relativity doesn't give any equation for the speed of light at a
particular location, because the effect of gravity cannot be
represented by a simple scalar field of c values. Instead, the "speed
of light" at a each point depends on the direction of the light ray
through that point, as well as on the choice of coordinate systems, so
we can't generally talk about the value of c at a given point in a non-
vanishing gravitational field. However, if we consider just radial
light rays near a spherically symmetrical (and non- rotating) mass,
and if we agree to use a specific set of coordinates, namely those in
which the metric coefficients are independent of t, then we can read a
formula analogous to Einstein's 1911 formula directly from the
Schwarzschild metric. (...) In the Newtonian limit the classical
gravitational potential at a distance r from mass m is phi=-m/r, so if
we let c_r = dr/dt denote the radial speed of light in Schwarzschild
coordinates, we have c_r =1+2phi, which corresponds to Einstein's 1911
equation, except that we have a factor of 2 instead of 1 on the
potential term."
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A light source emits light with frequency f, speed c (relative to the
source) and wavelength L. A receiver moving with speed v relative to
the source receives light with frequency f' (Doppler effect), speed
c' (relative to the receiver) and wavelength L'. According to Newton's
emission theory of light:
f'=f(1+v/c); c'=c+v; L'=L
Einstein's breathtaking alternative:
f'=f((1+v/c)/(1-v/c))^(1/2); c'=c; L'=L/((1+v/c)/(1-v/c))^(1/2)
The wavelength is determined by the light source so this dependence of
the wavelength on the speed of the receiver (if the receiver starts
accelerating the wavelength should start changing accordingly) is one
of the silliest corollaries of Einstein's 1905 false light postulate.
Perhaps only length contraction, twin paradox etc. corollaries are
sillier, e.g. those according to which the long train is short (if
trapped inside a short tunnel), the 80m long pole is 40m long (if
trapped inside a 40m long barn and Einsteinians have forgotten to
reopen the doors of the barn "pretty quickly"), the bug is both dead
and alive:
http://www.youtube.com/watch?v=VSRIyDfo_mY&mode=related&search=
http://www.math.ucr.edu/home/baez/physics/Relativity/SR/barn_pole.html
"These are the props. You own a barn, 40m long, with automatic doors
at either end, that can be opened and closed simultaneously by a
switch. You also have a pole, 80m long, which of course won't fit in
the barn....So, as the pole passes through the barn, there is an
instant when it is completely within the barn. At that instant, you
close both doors simultaneously, with your switch. Of course, you open
them again pretty quickly, but at least momentarily you had the
contracted pole shut up in your barn."
http://hyperphysics.phy-astr.gsu.edu/Hbase/Relativ/bugrivet.html
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http://www.american.com/archive/2007/april-0407/the-man-who-made-our-world
The Man Who Made Our World
"In his 1905 papers, Einstein actually used two apparently
contradictory interpretations of the nature of light. In the paper on
light quanta, Einstein had argued forcefully for the return to
Newton's particulate conception of light. Yet in the same year, in his
development of special relativity, Einstein had rejected a particulate
emission theory of light and had used the traditional Maxwell wave
theory. This STRIKING ABILITY TO JUGGLE two viewpoints apparently at
odds was characteristic of Einstein's thought..."
http://www.astrofind.net/documents/the-composition-and-essence-of-radiation.php
The Development of Our Views on the Composition and Essence of
Radiation by Albert Einstein
Albert Einstein 1909: "A large body of facts shows undeniably that
light has certain fundamental properties that are better explained by
Newton's emission theory of light than by the oscillation theory. For
this reason, I believe that the next phase in the development of
theoretical physics will bring us a theory of light that can be
considered a fusion of the oscillation and emission theories. The
purpose of the following remarks is to justify this belief and to show
that a profound change in our views on the composition and essence of
light is imperative.....Then the electromagnetic fields that make up
light no longer appear as a state of a hypothetical medium, but rather
as independent entities that the light source gives off, just as in
Newton's emission theory of light......Relativity theory has changed
our views on light. Light is conceived not as a manifestation of the
state of some hypothetical medium, but rather as an independent entity
like matter. Moreover, this theory shares with the corpuscular theory
of light the unusual property that light carries inertial mass from
the emitting to the absorbing object."
http://books.google.com/books?id=JokgnS1JtmMC
"Relativity and Its Roots" By Banesh Hoffmann
p.92: "Moreover, if light consists of particles, as Einstein had
suggested in his paper submitted just thirteen weeks before this one,
the second principle seems absurd: A stone thrown from a speeding
train can do far more damage than one thrown from a train at rest; the
speed of the particle is not independent of the motion of the object
emitting it. And if we take light to consist of particles and assume
that these particles obey Newton's laws, they will conform to
Newtonian relativity and thus automatically account for the null
result of the Michelson-Morley experiment without recourse to
contracting lengths, local time, or Lorentz transformations. Yet, as
we have seen, Einstein resisted the temptation to account for the null
result in terms of particles of light and simple, familiar Newtonian
ideas, and introduced as his second postulate something that was more
or less obvious when thought of in terms of waves in an ether."
http://philsci-archive.pitt.edu/archive/00001743/02/Norton.pdf
John Norton: "Einstein regarded the Michelson-Morley experiment as
evidence for the principle of relativity, whereas later writers almost
universally use it as support for the light postulate of special
relativity......THE MICHELSON-MORLEY EXPERIMENT IS FULLY COMPATIBLE
WITH AN EMISSION THEORY OF LIGHT THAT CONTRADICTS THE LIGHT
POSTULATE."
http://query.nytimes.com/gst/abstract.html?res=9806EFDD113FEE3ABC4152DFB266838A639EDE
The New York Times, April 19, 1921
"Michelson showed that relative to the moving co-ordinate system K1,
the light traveled with the same velocity as relative to K, which is
contrary to the above observation. How could this be reconciled?
Professor Einstein asked."
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ENCYCLOPAEDIA BRITANNICA TEACHING LIES: "The fact that the speed of
light is the same for all observers is inexplicable in ordinary terms.
If a passenger in a train moving at 100 km per hour shoots an arrow in
the train’s direction of motion at 200 km per hour, a trackside
observer would measure the speed of the arrow as the sum of the two
speeds, or 300 km per hour (see figure). In analogy, if the train
moves at the speed of light and a passenger shines a laser in the same
direction, then common sense indicates that a trackside observer
should see the light moving at the sum of the two speeds, or twice the
speed of light (6 × 108 metres per second). While such a law of
addition of velocities is valid in classical mechanics, THE MICHELSON-
MORLEY EXPERIMENT SHOWED THAT LIGHT DOES NOT OBEY THIS LAW."
Einsteinians who are somewhat less dishonest than those who have
written Encyclopaedia Britannica:
http://philsci-archive.pitt.edu/archive/00001743/02/Norton.pdf
John Norton: "Einstein regarded the Michelson-Morley experiment as
evidence for the principle of relativity, whereas later writers almost
universally use it as support for the light postulate of special
relativity......THE MICHELSON-MORLEY EXPERIMENT IS FULLY COMPATIBLE
WITH AN EMISSION THEORY OF LIGHT THAT CONTRADICTS THE LIGHT
POSTULATE."
http://books.google.com/books?id=JokgnS1JtmMC
"Relativity and Its Roots" By Banesh Hoffmann
p.92: "Moreover, if light consists of particles, as Einstein had
suggested in his paper submitted just thirteen weeks before this one,
the second principle seems absurd: A stone thrown from a speeding
train can do far more damage than one thrown from a train at rest; the
speed of the particle is not independent of the motion of the object
emitting it. And if we take light to consist of particles and assume
that these particles obey Newton's laws, they will conform to
Newtonian relativity and thus automatically account for the null
result of the Michelson-Morley experiment without recourse to
contracting lengths, local time, or Lorentz transformations. Yet, as
we have seen, Einstein resisted the temptation to account for the null
result in terms of particles of light and simple, familiar Newtonian
ideas, and introduced as his second postulate something that was more
or less obvious when thought of in terms of waves in an ether."
James H. Smith "Introduction à la relativité" EDISCIENCE 1969 pp.
39-41: "Si la lumière était un flot de particules mécaniques obéissant
aux lois de la mécanique, il n'y aurait aucune difficulté à comprendre
les résultats de l'expérience de Michelson-Morley.... Supposons, par
exemple, qu'une fusée se déplace avec une vitesse (1/2)c par rapport à
un observateur et qu'un rayon de lumière parte de son nez. Si la
vitesse de la lumière signifiait vitesse des "particules" de la
lumière par rapport à leur source, alors ces "particules" de lumière
se déplaceraient à la vitesse c/2+c=(3/2)c par rapport à
l'observateur. Mais ce comportement ne ressemble pas du tout à celui
d'une onde, car les ondes se propagent à une certaine vitesse par
rapport au milieu dans lequel elles se développent et non pas à une
certaine vitesse par rapport à leur source..... Il nous faut insister
sur le fait suivant: QUAND EINSTEIN PROPOSA QUE LA VITESSE DE LA
LUMIERE SOIT INDEPENDANTE DE CELLE DE LA SOURCE, IL N'EN EXISTAIT
AUCUNE PREUVE EXPERIMENTALE. IL LE POSTULA PAR PURE NECESSITE
LOGIQUE."
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The Albert Einstein of our generation clearly accepts Newton's
emission theory of light but then finds it suitable to confuse other
Einsteinians:
http://www.amazon.com/Brief-History-Time-Stephen-Hawking/dp/0553380168
Stephen Hawking, "A Brief History of Time", Chapter 6:
"Under the theory that light is made up of waves, it was not clear how
it would respond to gravity. But if light is composed of particles,
one might expect them to be affected by gravity in the same way that
cannonballs, rockets, and planets are.....In fact, it is not really
consistent to treat light like cannonballs in Newton’s theory of
gravity because the speed of light is fixed. (A cannonball fired
upward from the earth will be slowed down by gravity and will
eventually stop and fall back; a photon, however, must continue upward
at a constant speed...)"
The original Einstein also believed in and even used Newton's emission
theory of light but confused Einsteinians in a different way:
Pentcho Valev
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Pentcho Valev
> A light source emits light with frequency f, speed c (relative to the
> source) and wavelength L. A receiver moving with speed v relative to
> the source receives light with frequency f' (Doppler effect), speed
> c' (relative to the receiver) and wavelength L'. According to Newton's
> emission theory of light:
>
> f'=f(1+v/c); c'=c+v; L'=L
>
> Einstein's breathtaking alternative:
>
> f'=f((1+v/c)/(1-v/c))^(1/2); c'=c; L'=L/((1+v/c)/(1-v/c))^(1/2)
>
> The wavelength is determined by the light source so this dependence of
> the wavelength on the speed of the receiver (if the receiver starts
> accelerating the wavelength should start changing accordingly) is one
> of the silliest corollaries of Einstein's 1905 false light postulate.
The above objection against the dependence of the wavelength on the
speed of the receiver (which amounts to REDUCTIO AD ABSURDUM) refutes
both Einsteiniana's interpretation of the Doppler effect and the
"expanding universe" hypothesis:
http://www.amazon.com/Brief-History-Time-Stephen-Hawking/dp/0553380168
Stephen Hawking, "A Brief History of Time", Chapter 3:
"In the 1920s, when astronomers began to look at the spectra of stars
in other galaxies, they found something most peculiar: there were the
same characteristic sets of missing colors as for stars in our own
galaxy, but they were all shifted by the same relative amount toward
the red end of the spectrum. To understand the implications of this,
we must first understand the Doppler effect. As we have seen, visible
light consists of fluctuations, or waves, in the electromagnetic
field. The wavelength (or distance from one wave crest to the next) of
light is extremely small, ranging from four to seven ten-millionths of
a meter. The different wavelengths of light are what the human eye
sees as different colors, with the longest wavelengths appearing at
the red end of the spectrum and the shortest wavelengths at the blue
end. Now imagine a source of light at a constant distance from us,
such as a star, emitting waves of light at a constant wavelength.
Obviously the wavelength of the waves we receive will be the same as
the wavelength at which they are emitted (the gravitational field of
the galaxy will not be large enough to have a significant effect).
Suppose now that the source starts moving toward us. When the source
emits the next wave crest it will be nearer to us, so the distance
between wave crests will be smaller than when the star was stationary.
This means that the wavelength of the waves we receive is shorter than
when the star was stationary. Correspondingly, if the source is moving
away from us, the wavelength of the waves we receive will be longer.
In the case of light, therefore, means that stars moving away from us
will have their spectra shifted toward the red end of the spectrum
(red-shifted) and those moving toward us will have their spectra blue-
shifted."
Einsteinians now know that Einstein's 1905 false light postulate has
produced a wrong cosmology but.....it would be so painful to destroy
beautiful Potemkin's villages:
http://www.sciscoop.com/story/2008/10/30/41323/484
"Does the apparently constant speed of light change over the vast
stretches of the universe? Would our understanding of black holes,
ancient supernovae, dark matter, dark energy, the origins of the
universe and its ultimate fate be different if the speed of light were
not constant?.....Couldn't it be that the supposed vacuum of space is
acting as an interstellar medium to lower the speed of light like some
cosmic swimming pool? If so, wouldn't a stick plunged into the pool
appear bent as the light is refracted and won't that affect all our
observations about the universe. I asked theoretical physicist Leonard
Susskind, author of The Black Hole War, recently reviewed in Science
Books to explain this apparent anomaly....."You are entirely right,"
he told me, "there are all sorts of effects on the propagation of
light that astronomers and astrophysicists must account for. The point
of course is that they (not me) do take these effects into account and
correct for them." "In a way this work is very heroic but unheralded,"
adds Susskind, "An immense amount of extremely brilliant analysis has
gone into the detailed corrections that are needed to eliminate these
'spurious' effects so that people like me can just say 'light travels
with the speed of light.' So, there you have it. My concern about
cosmic swimming pools and bent sticks does indeed apply, but
physicists have taken the deviations into account so that other
physicists, such as Susskind, who once proved Stephen Hawking wrong,
can battle their way to a better understanding of the universe."
http://www.renewamerica.us/columns/hutchison/080616
"Grigori Potemkin (1739–1791) was a general-field marshal, Russian
statesman, and favorite of Empress Catherine the Great. He is alleged
to have built facades of non-existent villages along desolate
stretches of the Dnieper River to impress Catherine as she sailed to
the Crimea in 1787. Actors posing as happy peasants stood in front of
these pretty stage sets and waved to the pleased Empress....What has
all this to do with Einstein? The science establishment has a powerful
romantic desire to believe in Einstein. Therefore, they are not only
fooled by Einstein's tricks, they are prepared to defend his Potemkin
villages. A Potemkin village is a pretty picture to fool the gullible
romantic. Einstein was romantically infatuated with pretty pictures.
He deliberately sought theories that were aesthetically beautiful in
their harmony, symmetry, and simplicity. He romantically believed
something akin to Keats' famous poetic summation: "Beauty is truth and
truth, beauty." This is pure romanticism. I thought science was hard-
headed stuff, not romantic fluff. Silly me. Einstein used blackboard
mathematics to develop the pretty pictures that became the new physics
of a romantic age. That is why he is beloved by modernists who thrill
to the romance of his new cosmos. As we shall see, no such cosmos
exists."
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of light:
http://www.mfo.de/programme/schedule/2006/08c/OWR_2006_10.pdf
Jean Eisenstaedt: "At the end of the 18th century, a natural extension
of Newton's dynamics to light was developed but immediately forgotten.
A body of works completed the Principia with a relativistic optics of
moving bodies, the discovery of the Doppler-Fizeau effect some sixty
years before Doppler, and many other effects and ideas which represent
a fascinating preamble to Einstein relativities. It was simply
supposed that 'a body-light', as Newton named it, was subject to the
whole dynamics of the Principia in much the same way as were material
particles; thus it was subject to the Galilean relativity and its
velocity was supposed to be variable. Of course it was subject to the
short range 'refringent' force of the corpuscular theory of light --
which is part of the Principia-- but also to the long range force of
gravitation which induces Newton's theory of gravitation. The fact
that the 'mass' of a corpuscle of light was not known did not
constitute a problem since it does not appear in the Newtonian (or
Einsteinian) equations of motion. It was precisely what John Michell
(1724-1793), Robert Blair (1748-1828), Johann G. von Soldner
(1776-1833) and Fran¸cois Arago (1786-1853) were to do at the end of
the 18th century and the beginning the 19th century in the context of
Newton's dynamics. Actually this 'completed' Newtonian theory of light
and material corpuscle seems to have been implicitly accepted at the
time. In such a Newtonian context, not only Soldner's calculation of
the deviation of light in a gravitational field was understood, but
also dark bodies (cousins of black holes). A natural (Galilean and
thus relativistic) optics of moving bodies was also developed which
easily explained aberration and implied as well the essence of what we
call today the Doppler effect. Moreover, at the same time the
structure of -- but also the questions raised by-- the Michelson
experiment was understood. Most of this corpus has long been
forgotten. The Michell-Blair-Arago effect, prior to Doppler's effect,
is entirely unknown to physicists and historians. As to the influence
of gravitation on light, the story was very superficially known but
had never been studied in any detail. Moreover, the existence of a
theory dealing with light, relativity and gravitation, embedded in
Newton's Principia was completely ignored by physicists and by
historians as well. But it was a simple and natural way to deal with
the question of light, relativity (and gravitation) in a Newtonian
context. Einstein himself did not know of this Newtonian theory of
light and he did not rely on it in his own research. But he was not
very far from these ideas: he wrote to Freundlich in 1913 that 'it was
rather natural that the idea of a bending of light appeared at the
time of the theory of emission'."
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> frequency f, speed c (relative to the source) and wavelength L. A
> c' (relative to the receiver) and wavelength L'. According to Newton's
> emission theory of light:
>
> f'=f(1+gh/c^2); c'=c(1+gh/c^2); L'=L
>
> A rocket of length h accelerates with acceleration g. A light source
> at the front end emits light with frequency f, speed c (relative to
> the source) and wavelength L. A receiver at the back end receives
> light with frequency f', speed c' (relative to the receiver) and
> wavelength L'. At the moment of reception, the receiver has speed v
> relative to the light source at the moment of emission. According to
>
>
> the speed of light in a gravitational field predicted by the emission
> theory. Initially he was just using the equation c'=c(1+gh/c^2), then
> quite stupidly (or dishonestly) replaced it with c'=c(1+2gh/c^2):
>
In 1915 Einstein knew how to "outdo" Newton (by replacing the equation
c'=c(1+gh/c^2) given by Newton's emission theory of light with the
silly equation c'=c(1+2gh/c^2)) but in 1911 he was still desperately
trying to camouflage his involuntary allegiance to the emission
theory. So in 1911 Einstein devised "gravitational time dilation": an
absurd concept implying that, in a gravitational field, the frequency
and the wavelength change but the speed of light remains constant.
Spin-offs from this particular camouflage of Einstein's:
http://www.amazon.com/Brief-History-Time-Stephen-Hawking/dp/0553380168
Stephen Hawking, "A Brief History of Time", Chapter 2:
"Another prediction of general relativity is that time should appear
to slower near a massive body like the earth. This is because there is
a relation between the energy of light and its frequency (that is, the
number of waves of light per second): the greater the energy, the
higher frequency. As light travels upward in the earth’s gravitational
field, it loses energy, and so its frequency goes down. (This means
that the length of time between one wave crest and the next goes up.)
To someone high up, it would appear that everything down below was
making longer to happen. This prediction was tested in 1962, using a
pair of very accurate clocks mounted at the top and bottom of a water
tower. The clock at the bottom, which was nearer the earth, was found
to run slower, in exact agreement with general relativity. The
difference in the speed of clocks at different heights above the earth
is now of considerable practical importance, with the advent of very
accurate navigation systems based on signals from satellites. If one
ignored the predictions of general relativity, the position that one
calculated would be wrong by several miles! (...) Consider a pair of
twins. Suppose that one twin goes to live on the top of a mountain
while the other stays at sea level. The first twin would age faster
than the second. Thus, if they met again, one would be older than the
other."
http://www.astronomynotes.com/relativity/s4.htm
"Prediction: light escaping from a large mass should lose energy---the
wavelength must increase since the speed of light is constant.
Stronger surface gravity produces a greater increase in the
wavelength. This is a consequence of time dilation. Suppose person A
on the massive object decides to send light of a specific frequency f
to person B all of the time. So every second, f wave crests leave
person A. The same wave crests are received by person B in an interval
of time interval of (1+z) seconds. He receives the waves at a
frequency of f/(1+z). Remember that the speed of light c = (the
frequency f) (the wavelength L). If the frequency is reduced by (1+z)
times, the wavelength must INcrease by (1+z) times: L_atB = (1+z)
L_atA. In the doppler effect, this lengthening of the wavelength is
called a redshift. For gravity, the effect is called a gravitational
redshift."
http://helios.gsfc.nasa.gov/qa_sp_gr.html
"Is light affected by gravity? If so, how can the speed of light be
constant? Wouldn't the light coming off of the Sun be slower than the
light we make here? If not, why doesn't light escape a black hole?
Yes, light is affected by gravity, but not in its speed. General
Relativity (our best guess as to how the Universe works) gives two
effects of gravity on light. It can bend light (which includes effects
such as gravitational lensing), and it can change the energy of light.
But it changes the energy by shifting the frequency of the light
(gravitational redshift) not by changing light speed. Gravity bends
light by warping space so that what the light beam sees as "straight"
is not straight to an outside observer. The speed of light is still
constant." Dr. Eric Christian
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http://www.speed-light.info/speed_of_light_variable.htm
"Einstein wrote this paper in 1911 in German (download from:
http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1911_35_898-908.pdf).
It predated the full formal development of general relativity by about
four years. You can find an English translation of this paper in the
Dover book 'The Principle of Relativity' beginning on page 99; you
will find in section 3 of that paper Einstein's derivation of the
variable speed of light in a gravitational potential, eqn (3). The
result is: c'=c0(1+phi/c^2) where phi is the gravitational potential
relative to the point where the speed of light co is measured......You
can find a more sophisticated derivation later by Einstein (1955) from
the full theory of general relativity in the weak field
approximation....For the 1955 results but not in coordinates see page
93, eqn (6.28): c(r)=[1+2phi(r)/c^2]c. Namely the 1955 approximation
shows a variation in km/sec twice as much as first predicted in 1911."
Sooner or later Einsteinians will have to answer the simple question:
The gravitational redshift factor 1+phi/c^2 experimentally confirmed
by Pound and Rebka is consistent with:
(A) Einstein's 1911 equation c'=c(1+phi/c^2) given by Newton's
emission theory of light?
(B) Einstein's 1955 equation c'=c(1+2phi/c^2)?
Then Einsteinians will have to explain why they have been so silent
about this simple problem for so long.
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