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THE FUNDAMENTAL EQUATION OF SPECIAL RELATIVITY



 
 
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  #1  
Old December 5th 13, 12:24 AM posted to sci.astro
Pentcho Valev
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Default THE FUNDAMENTAL EQUATION OF SPECIAL RELATIVITY

http://www.youtube.com/watch?v=SC0Q6-xt-Xs
"Doppler effect - when an observer moves away from a stationary source. Pay attention to the velocity of the wave relative to the observer. When an observer moves away from a stationary source, the period of the wave emitted by a source is longer and the observed frequency is lower. Because the velocity of the wave relative to the observer is slower than that when it is still."

The observer starts moving away from the light source with speed v. The frequency he measures shifts from f=c/d to f'=(c-v)/d, where d is the distance between the pulses. (If v is small enough, the formula f'=(c-v)/d is virtually exact no matter whether the classical or relativistic Doppler effect is considered.)

The speed of the pulses relative to the moving observer is:

c' = d*f' = c - v = c

where c - v = c is the fundamental equation of special relativity:

http://img15.hostingpics.net/pics/86...uletableau.jpg

Pentcho Valev
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  #2  
Old December 5th 13, 07:49 AM posted to sci.astro
Pentcho Valev
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Posts: 7,546
Default THE FUNDAMENTAL EQUATION OF SPECIAL RELATIVITY

The Fundamental Equation of General Relativity

The top of a tower of height h emits light with frequency f, speed c and wavelength L (as measured by the emitter):

f = c/L

An observer on the ground measures the frequency to be f'=f(1+gh/c^2) (the Pound-Rebka experiment), the speed of the light to be c' and the wavelength to be L':

f' = c'/L'

The crucial questions a

c' = ? ; L' = ?

Einstein's general relativity gives a straightforward answer about c' but leaves open the question about L':

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

Given this prediction of general relativity, the only reasonable assumption about L' seems to be:

L' = L

So we have:

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

or:

1 = 2

This last equation, 1 = 2, is the fundamental equation of general relativity. Yet it is based on the assumption L' = L so if some Einsteinian wants to develop general relativity further, he/she may find it suitable to reject the fundamental equation and replace L' = L with an assumption about L' that is closer to the truth.

APPENDIX: References showing that, according to general relativity, the speed of light in a gravitational field varies in accordance with the equation c' = c(1+2gh/c^2):

http://arxiv.org/pdf/gr-qc/9909014v1.pdf
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."

http://www.speed-light.info/speed_of_light_variable.htm
"Einstein wrote this paper in 1911 in German. (...) ...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. (...) Namely the 1955 approximation shows a variation in km/sec twice as much as first predicted in 1911."

http://www.ita.uni-heidelberg.de/res...s/JeruLect.pdf
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 n1 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| "

http://www.mathpages.com/rr/s6-01/6-01.htm
"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."

http://poincare.matf.bg.ac.rs/~rvikt..._Cosmology.pdf
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
  #3  
Old December 5th 13, 04:33 PM posted to sci.astro
Pentcho Valev
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Posts: 7,546
Default THE FUNDAMENTAL EQUATION OF SPECIAL RELATIVITY

There is another fundamental equation in Divine Albert's world, 1 = 0, but it has been discovered by Einstein's followers and does not originate in Einstein's works. Einsteinians admit that, in a gravitational field, light falls like ordinary matter, as predicted by Newton's emission theory of light, and that this has been confirmed by the Pound-Rebka experiment:

http://sethi.lamar.edu/bahrim-cristi...t-lens_PPT.pdf
Dr. Cristian Bahrim: "If we accept the principle of equivalence, we must also accept that light falls in a gravitational field with the same acceleration as material bodies."

http://www.youtube.com/watch?v=FJ2SVPahBzg
"The light is perceived to be falling in a gravitational field just like a mechanical object would. (...) 07:56 : (c+dc)/c = 1+(g/c^2)dh [as predicted by Newton's emission theory of light]"

http://www.wfu.edu/~brehme/space.htm
Robert W. Brehme: "Light falls in a gravitational field just as do material objects."

http://courses.physics.illinois.edu/...ctures/l13.pdf
University of Illinois at Urbana-Champaign: "Consider a falling object. ITS SPEED INCREASES AS IT IS FALLING. Hence, if we were to associate a frequency with that object the frequency should increase accordingly as it falls to earth. Because of the equivalence between gravitational and inertial mass, WE SHOULD OBSERVE THE SAME EFFECT FOR LIGHT. So lets shine a light beam from the top of a very tall building. If we can measure the frequency shift as the light beam descends the building, we should be able to discern how gravity affects a falling light beam. This was done by Pound and Rebka in 1960. They shone a light from the top of the Jefferson tower at Harvard and measured the frequency shift. The frequency shift was tiny but in agreement with the theoretical prediction. Consider a light beam that is travelling away from a gravitational field. Its frequency should shift to lower values.. This is known as the gravitational red shift of light."

http://www.einstein-online.info/spot...t_white_dwarfs
Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."

Therefore, if the top of a tower of height h emits light with frequency f=c/L (as measured by the emitter; L is the wavelength), an observer on the ground will measure the frequency to be f'=f(1+gh/c^2), as the Pound-Rebka experiment showed. Accordingly, the speed of the light as measured by the observer on the ground is:

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

where 1 = 0, the fundamental equation, is obviously used.

Einsteinians who successfully apply the fundamental equation 1 = 0:

http://www.oapt.ca/newsletter/2004-0...Searchable.pdf
Richard Epp: "One may imagine the photon losing energy as it climbs against the Earth's gravitational field much like a rock thrown upward loses kinetic energy as it slows down, the main difference being that the photon does not slow down; it always moves at the speed of light."

http://www.amazon.com/Brief-History-.../dp/0553380168
Stephen Hawking, A Brief History of Time, Chapter 6: "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..."

http://www.amazon.com/Why-Does-mc2-S.../dp/0306817586
Brian Cox, Jeff Forshaw, p. 236: "If the light falls in strict accord with the principle of equivalence, then, as it falls, its energy should increase by exactly the same fraction that it increases for any other thing we could imagine dropping. We need to know what happens to the light as it gains energy. In other words, what can Pound and Rebka expect to see at the bottom of their laboratory when the dropped light arrives? There is only one way for the light to increase its energy. We know that it cannot speed up, because it is already traveling at the universal speed limit, but it can increase its frequency."

Pentcho Valev
  #4  
Old December 6th 13, 07:36 AM posted to sci.astro
Pentcho Valev
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Posts: 7,546
Default THE FUNDAMENTAL EQUATION OF SPECIAL RELATIVITY

Applications of c - v = c, the fundamental equation of special relativity:

http://www.cmmp.ucl.ac.uk/~ahh/teach...24n/lect19.pdf
Tony Harker, University College London: "The Doppler Effect: Moving sources and receivers. The phenomena which occur when a source of sound is in motion are well known. The example which is usually cited is the change in pitch of the engine of a moving vehicle as it approaches. In our treatment we shall not specify the type of wave motion involved, and our results will be applicable to sound or to light. (...) Now suppose that the observer is moving with a velocity Vo away from the source. (....) If the observer moves with a speed Vo away from the source (...), then in a time t the number of waves which reach the observer are those in a distance (c-Vo)t, so the number of waves observed is (c-Vo)t/lambda, giving an observed frequency f'=f(1-Vo/c) when the observer is moving away from the source at a speed Vo."

If in a time t the number of waves which reach the observer are those in a distance (c-Vo)t, then the speed of the waves relative to the observer is:

c' = (c - Vo)t/t = c - Vo

In a world different from Divine Albert's world the validity of this result would be obvious: As the observer starts moving away from the light source, the wavecrests (or pulses of light separated by some distance) start hitting him les frequently which can only be due to the fact that their speed relative to the observer has decreased:

http://www.youtube.com/watch?v=SC0Q6-xt-Xs

In Divine Albert's world the above "validity" is invalid because the fundamental equation c - v = c is valid:

c' = (c - Vo)t/t = c - Vo = c

So in Divine Albert's world it is absolute truth that the speed of light relative to the observer is independent of the speed of the observer, in perfect accordance with the Revelations of Divine Albert's Divine Theory:

http://www.extremetech.com/wp-conten...ed-640x326.jpg

Pentcho Valev
  #5  
Old December 6th 13, 07:15 PM posted to sci.astro
Pentcho Valev
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Posts: 7,546
Default THE FUNDAMENTAL EQUATION OF SPECIAL RELATIVITY

The fundamental equation of special relativity, c-v = c, can be applied in general relativity as well:

http://physics.ucsd.edu/students/cou...ecture5-11.pdf
"In 1960 Pound and Rebka and later, 1965, with an improved version Pound and Snider measured the gravitational redshift of light using the Harvard tower, h=22.6m. From the equivalence principle, at the instant the light is emitted from the transmitter, only a freely falling observer will measure the same value of f that was emitted by the transmitter. But the stationary receiver is not free falling. During the time it takes light to travel to the top of the tower, t=h/c, the receiver is traveling at a velocity, v=gt, away from a free falling receiver. Hence the measured frequency is: f'=f(1-v/c)=f(1-gh/c^2)."

The frequency measured at the bottom of the tower is f=c/L, where L is the wavelength. The frequency measured by a stationary observer at the top of the tower is:

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

Accordingly, the speed of light relative to the observer at the top of the tower is:

c' = c-v = c

From the equivalence principle, c' = c-v = c is also the speed of light relative to an observer moving, in gravitation-free space, away from the emitter with speed v. Clearly the fundamental equation c-v = c justifies Steve Carlip's declaration that the speed of light is "constant by definition":

http://math.ucr.edu/home/baez/physic..._of_light.html
Steve Carlip: "Is c, the speed of light in vacuum, constant? At the 1983 Conference Generale des Poids et Mesures, the following SI (Systeme International) definition of the metre was adopted: The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. This defines the speed of light in vacuum to be exactly 299,792,458 m/s. This provides a very short answer to the question "Is c constant": Yes, c is constant by definition!"

Pentcho Valev
 




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