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Pentcho Valev

Posts: 6,212
Registered: 12/13/04
Posted: Jan 17, 2014 3:12 AM
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An observer emits light from the top of a tower of height h downwards. The initial frequency, as measured by this observer, is f=c/L, where L is the wavelength. As the light reaches an observer/receiver on the ground, its frequency, as measured by this observer and confirmed in numerous experiments, is:

f' = f(1+gh/c^2) = c(1+gh/c^2)/L = c'/L

where c'=c(1+gh/c^2) can obviously be nothing else than the speed of the light relative to the observer/receiver on the ground, at the moment of reception.

On the other hand, there are predictions about the speed of light independent of any frequency measurement. As the light reaches the observer on the ground, its speed relative to this observer is:

A) c' = c(1+gh/c^2) (Newton's emission theory)

B) c' = c(1+2gh/c^2) (Einstein's general relativity)

C) c' = c (Richard Epp, Stephen Hawking, Brian Cox)

Clearly the frequency shift f'=f(1+gh/c^2) is consistent with A and incompatible with both B and C.

References showing that, according to Einstein's general relativity, the speed of light varies in a gravitational field in accordance with the equation c'=c(1+2gh/c^2):
Steve Carlip: "It is well known that the deflection of light is twice that predicted by Newtonian theory; in this sense, at least, light falls with twice the acceleration of ordinary "slow" matter."
"Einstein wrote this paper in 1911 in German. (...) 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. (...) Namely the 1955 approximation shows a variation in km/sec twice as much as first predicted in 1911."
LECTURES ON GRAVITATIONAL LENSING, RAMESH NARAYAN AND MATTHIAS BARTELMANN, p. 3: " The effect of spacetime curvature on the light paths can then be expressed in terms of an effective index of refraction n, which is given by (e.g. Schneider et al. 1992):
n = 1-(2/c^2)phi = 1+(2/c^2)|phi|
Note that the Newtonian potential is negative if it is defined such that it approaches zero at infinity. As in normal geometrical optics, a refractive index n>1 implies that light travels slower than in free vacuum. Thus, the effective speed of a ray of light in a gravitational field is:
v = c/n ~ c-(2/c)|phi| "
"Specifically, Einstein wrote in 1911 that the speed of light at a place with the gravitational potential phi would be c(1+phi/c^2), where c is the nominal speed of light in the absence of gravity. In geometrical units we define c=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. (...) ...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."
Relativity, Gravitation, and Cosmology, T. Cheng

p.49: This implies that the speed of light as measured by the remote observer is reduced by gravity as

c(r) = (1 + phi(r)/c^2)c (3.39)

Namely, the speed of light will be seen by an observer (with his coordinate clock) to vary from position to position as the gravitational potential varies from position to position.

p.93: Namely, the retardation of a light signal is twice as large as that given in (3.39)

c(r) = (1 + 2phi(r)/c^2)c (6.28)
[end of quotation]

Pentcho Valev

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