On one level, there is a fairly unambiguous NO answer: if you bring two objects to rest relative to each other in an otherwise expanding universe, they will remain at rest relative to each other if no force is acting on them. This would argue in favor of the unit of length being constant.

On another level, as you noted in the question details, you are free to define a comoving coordinate system, in which the unit of length increases with the expansion. This is simply a consequence of general covariance, namely that the laws of general relativity are independent of the choice of coordinate system.

So ultimately, the choice of the unit of length, like dimensioned constants in general, has no fundamental significance.

What is important to recognize is that the expansion itself is not a force. Two things that are initially at rest relative to each other will remain at rest.

You may be wondering why, then, people talked about a possible "Big Rip" scenario. But there, the problem is the presence of phantom energy, whose energy density increases as the universe expands, and which responds to gravity as though gravity was repulsive. So it is the gravitational interaction between phantom energy and other lumps of matter, not the expansion itself, that causes the "rip".

V.T.,

I think that your argument holds true but that to an outside observer the two objects get bigger with respect to the outside observer or at least seem to change in size relative to the whole.

Dear George: I don't believe that an outside observer would see either the sizes or the distance between two objects get bigger. Suppose you float in space in an expanding universe. You place two ping-pong balls (or other light objects with no appreciable gravity) floating some distance from each other, at rest relative to each other and you. Even as the universe expands around you, the sizes of the ping-pong balls, or their relative distance will remain exactly the same as you continue measuring them (I am assuming that there are no gravitational sources or other forces acting on the balls, and that you yourself are far enough so that your own gravity doesn't affect the balls either.)

Relative to cosmological distances (e.g., relative to the distance from a far away galaxy cluster) all locally measured distances will get smaller of course. So if you choose a comoving length as your unit distance, then things that are not expanding will get smaller and smaller, in a universe of unchanging size (in these units). But if your unit of length is defined by a yardstick that you carry along, the sizes of the ping-pong balls and the distance between them will remain constant.

Dear Alfredo Oliveira, 

Length is invariant! you are right. But the problem exists in the Einstein interpretation to the Lorentz transformation depending on objectivity. That led to the concept of space-time continuum, and thus led to the constancy of the speed of light by proposing the concept of length contraction. According to the Einstein's interpretation of the Lorentz transformation, VSL is impossible, and because of that physicists do not understand or interpret if the speed of light is decreased by gravity or remains constant. And that led to the energy-momentum problem in GR still not solved till now, and the Pioneer anomaly! which I consider it as case not closed yet. I reinterpreted the Lorentz transformation according to the Copenhagen school. I found the Lorentz transformation according to my interpretation is vacuum energy dependent and the Lorentz factor is equivalent to the refractive index in optics. According to my transformation there is no space-time continuum, it is only time responsible for measuring a decrease in the speed of light or increase (faster than light) but that is not locally. Locally the speed of light is constant and equals to the speed of light in vacuum. According to my transformation space is invariant. And when I generalized my transformation in quantization of gravity, I found there is no graviton which mediates gravitation, it is photon. where the measured decrease in the speed of light (the red shift by gravity) depends on time only not on space, and there is no curved space-time, and that means frequency, and thus it depends on the gravitational potential not on the strength of the field. According to my transformation, the measured decrease is not only in the light speed in gravity , but also in the speed of the massive particles in gravity by the same factor, and this decrease is not not locally, and that leads to solve the energy momentum problem in gravity. where, my transformation leads to the Heisenberg uncertain principle and the wave-particle duality. To understand more about my paper read it in http://dx.doi.org/10.14299/ijser.2014.10.002

and then compare the results I reach in my paper with the recent experimental results

http://www.sciencedaily.com/releases/2014/12/141219085153.htm?utm_source=feedburner

http://physicsworld.com/cws/article/news/2014/jul/28/new-correction-to-speed-of-light-could-explain-sn1987-neutrino-burst

http://phys.org/news/2015-01-physicists-air.html

In my interpretation to the Lorentz transformation I refuse the reciprocity principle which was adopted by Einstein in the SRT and that led to the commutation relations in QM. Refusing the reciprocity principle in my theory leads to disappearing all the paradoxes in the SRT; the Twin paradox, Ehrenfest paradox, Ladder paradox and Bell's spaceship paradox. Furthermore, according to my interpretation I could reconcile and interpret the experimental results of quantum tunneling and entanglement (spooky action), —Casimir effect, Hartman effect.

Dear El Naschie

Your CANTORIAN SPACETIME is a formidable and fascinating work!

My line of research is peculiar and drives me along a different path. The question I put arises from it. It is a peculiar question because it can hardly be generated by human mind; the reason is that it is contrary to the knowledge automatically obtained by the routines of mind. Now, we have to be able to answer the question. You have seen my modest contribution. As far as I could understand, it is not incompatible with the fundamental ideas of your cantorian spacetime - maybe it can even provide a better scenario to support those ideas. I am very short of time now, but as soon as I can I will devote attention to the cantorian spacetime, I felt it inspiring and challenging. Thanks

The "standard" answer is that without a time-unchanging reference (which in theory does not exist if "the universe" is everything) it is impossible to tell whether distance or length units change, and generally physicists have labeled such questions "irrelevant."  It was on such a basis that Ether theories were finally discarded, i.e., not because they are incorrect but because they were on principle unobservable.

Now V Toth's answer that objects at rest with respect to each other remain at rest, i.e. the distance between them doesn't change, seems at first to refute the standard explanation above.  It took me a moment to think through it.  If A is at rest with respect to a distant object B, then A is in fact moving with respect to objects in its neighborhood, which I denote N(A).  Toward B.  Therefore to N(A) it appears that A may be reducing its distance to B.  According to N(A), either interpretation works. 

A number of years ago I realized that objects in the universe, over time, must self-sort by velocity.  This holds whether space is expanding or not, and in an initially random universe results, after some time, in the appearance of an expanding universe (this was published by Allen D. Allen in 1976 https://www.researchgate.net/publication/243171295_The_big_bang_is_not_needed ).

Therefore we can define a concept of "stationary"  An object A is stationary if it is in the neighborhood N(A).  Otherwise it is moving.  Stationary then becomes an absolute concept, like acceleration.  In the above example, A is non-stationary because it is moving relative to N(A).  In fact A belongs in the neighborhood N(B) and when the universe has expended so much that all the objects not in the neighborhood of B have departed from in between A and B, then A will be in N(B).

Now you can define the problem better, and the symmetry.  If the unit of length L is defined as a multiple of the molecular bonds in objects, A or B, we see that space expands.  The unit of length is unchanging by definition.

If we define a unit of cosmological length Lc as the distance between neighborhoods such as N(A) and N(B), then Lc is unchanging but we would find L decreasing.

So Lc is a unit of length based on space, and L based on molecular bonds.  Whichever one you pick, the other one is changing.  So the answer is truly a matter of measurement convention, but there is more to it than just an arbitrary choice of coordinates, because I have proposed a physically meaningful convention in either case.

We can further realize that L is the measurement convention natural to bound objects, and Lc is natural to unbound objects.  Molecules, galaxies, even clusters are bound by various forces. 

Article The big bang is not needed

Dear Robert Shuler,

I agree with you  the answer is truly a matter of measurement convention. Quantum theory built basis on observation and experimental measurement. The answer can be understood in a very simple way when we define the Lorentz transformation equations basis on the quantum theory and the Copenhagen school. That will lead the interpretation of the Lorentz transformation is depending on observation, and then refusing objectivity which was adopted in Einstein's interpretation of the Lorentz transformation in SRT. I answered this question completely in my paper theory of quantization of gravity and I succeeded http://dx.doi.org/10.14299/ijser.2014.10.002

There is no space-time continuum or curved space-time. It is only time! If you say the light speed is observed to be decreased globally by gravity which causing the red shift, while it is locally constant, This red shift caused by gravity as observed globally resulted not by the length expansion, it is resulted by the time dilation only. Length is invariant. And when I say length is invariant that means the length of the moving train is same for the observer stationary on the ground and for the rider of the moving train, and the length contraction exists in the passed distance of the moving train for the rider of the moving comparing to the measured passed distance of the moving train relative to the stationary observer on the ground at this time. It is impossible that the observer on the ground and the rider of the moving agree during the motion in constant speed that the moving train exists the same point in space at the same, and this is the core of the Heisenberg uncertainty principle. When the train becomes stationary, then they will agree which leads to vacuum fluctuations, and the concept of acceleration or deceleration according to my theory is vacuum fluctuations, because Lorentz transformation according to my theory in case of inertial frames is vacuum energy dependent not relative velocity dependent.

Re: Azzam "There is no space-time continuum or curved space-time. It is only time!"

While it is possible to compute the precession of Mercury on the basis of only time dilation, for the experimentally observed Shapiro Delay you get a longer delay than explainable by time dilation.  However, if radial length units are contracted by the reciprocal of the time dilation factor (or radial distances expanded by that factor) then everything works out.  So experimentally the spatial portion of spacetime changes has been observed in a direct way.

Dear Robert Shuler,

According to my theory I reach to the same result of the precession of Mercury same as in Einstein's GR in case of weak gravitational field. The deference between my model and the Einstein's model that Einstein's depended on the concept of space-time as in the Schwarzschild's  geometry, and that depended on the Einstein's interpretation of the Lorentz transformation as in SRT which led to the space-time continuum. But according to my model, The Lorentz transformation depended on time only. For the experimentally observed Shapiro Delay you get a longer delay than explainable by time dilation....Do you know why? because you depended on the Lorentz factor of time dilation derived by Schwarzschild geometry of the curved space-time. 

When we generalize my transformation in the case of noninertial frames in the case of gravity, we reach to the observed decrease in the speed of light in gravity or the observed decrease in the speed of massive objects in the gravitational field (red shift or blue shift) depend on the potential energy of the gravitational field not on the strength of the field as in Einstein equivalence principle. That led to solve also the Energy momentum problem in gravity, the Pioneer anomaly, the SN1987A. Franson  J D Franson 2014 New J. Phys. 16 065008

doi:10.1088/1367-2630/16/6/065008 calculated that, treating light as a quantum object, the change in a photon's velocity depends not on the strength of the gravitational field, but on the gravitational potential itself. However, this leads to a violation of Einstein's equivalence principle – that gravity and acceleration are indistinguishable – because, in a gravitational field, the gravitational potential is created along with mass, whereas in a frame of reference accelerating in free fall, it is not. Therefore, one could distinguish gravity from acceleration by whether a photon slows down or not when it undergoes particle–antiparticle creation. So review my paper in http://dx.doi.org/10.14299/ijser.2014.10.002

Also, review my equivalence principle and how my equivalence principle agreed exactly with the Pond-Rebka experiment. Proponents of the theory of General Relativity offer three different conflicting explanations of the results of the The Pound-Rebka experiment that are said to be equivalent to each other and therefore all equally correct. All make the claim that the results of the Pound-Rebka Experiment are “proof” of the Equivalence Principle even though nothing in these measurements suggests any need for the Equivalence Principle. 

Azzam, maybe you did not understand my post.  I said that you could get precession, but you cannot get Shapiro delay, by the method of time dilation only.  I made that mistake in 2011.  Pay attention, I'm trying to save you a lot of embarrassment.

Dear Robert Shuler,

I can get Shapiro delay exactly by the method of time dilation only according to my theory, not by the theory of GR. You are right for the experimentally observed Shapiro Delay you get a longer delay than explainable by time dilation according to GR depending on the Schwarzschild geometry.

Remember Shapiro wrote according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path, these time delays should thereby be increased by almost 2x10-4 sec when the radar pulses pass near the sun. Such a change, equivalent to 60 km in distance, could now be measured over the required path length to within about 5 to 10% with presently obtainable equipment. But as I informed you Franson calculated that, treating light as a quantum object, the change in a photon's velocity depends not on the strength of the gravitational field, but on the gravitational potential itself. However, this leads to a violation of Einstein's equivalence principle – that gravity and acceleration are indistinguishable – because, in a gravitational field, the gravitational potential is created along with mass, whereas in a frame of reference accelerating in free fall, it is not. Therefore, one could distinguish gravity from acceleration by whether a photon slows down or not when it undergoes particle–antiparticle creation. That is exactly what I predicted in my quantization of gravity. According to my theory also I got also the exact solution of the Pioneer anomaly see my paper http://vixra.org/abs/1109.0058 which means  according to my theory solving of the energy momentum problem in gravity. You do not make me embarrassed, because most of the Proponents of the theory of General Relativity tried to prove the inconsistency of my theory, but now they understood that my theory in quantization of gravity is completely right, specially the team that proposed the Thermal Origin of the Pioneer Anomaly. review my paper http://vixra.org/abs/1205.0006 about Comments on the Thermal Origin of the Pioneer Anomaly. 

Dear Robert Shuler,

I give you also some recent experiments that agree completely with what I predicted in my paper http://dx.doi.org/10.14299/ijser.2014.10.002

http://www.sciencedaily.com/releases/2014/12/141219085153.htm?utm_source=feedburner

http://phys.org/news/2015-01-physicists-air.html

http://physicsworld.com/cws/article/news/2014/jul/28/new-correction-to-speed-of-light-could-explain-sn1987-neutrino-burst

Also The Pond rebka experiment, The Pioneer anomaly (the redshift and blue shift in gravity), and there are tens of other experiments.

Re: "the speed of a light wave depends on the strength of the gravitational potential along its path"

To get Shapiro delay correct then speed (if one means coordinate velocity relative to a distant observer) must depend on the square of potential radially, and on the unsquared potential transverse to radius.

In order to have local clocks ticking at the same rate regardless of orientation, this implies there must be more radial units of length than expected from R=2π/C.

Dear Robert Shuler,

The problem here is the observed decrease in the light speed depends on the strength of the field as predicted by Einstein in his GR and the equivalence principle. Or it depends on the gravitational potential itself, where that violates the Einstein's equivalence principle. According to the experimental results, the observed decrease in the speed of light by gravity depends on the gravitational potential. That means there is no curved space-time, it is only time as I told you. And since the GR produced from the Einstein interpretation  of the Lorentz transformation depending on objectivity, which resulted the concept of space-time continuum. Then reinterpretation of the Lorentz transformation by quitting the space-time continuum will lead the Lorentz transformation is vacuum energy depending and the Lorentz factor equivalent to the refractive index in optics. That required to the remove the reciprocity principle in the Lorentz transformation and thus it is resulted the commutation relations in QM. In this case the Lorentz transformation will lead to the wave-particle duality and the Heisenberg uncertainty principle. When we generalize the new model of the Lorentz transformation, and the wave-particle duality, we get as the light speed decrease by gravity depending on the gravitational potential, then massive particle speed of free fall must be decreased by the same factor, which resulted solving the energy momentum problem in gravity.  classical theories, special and general relativity, are built on the notion of exact Lorentz invariance (at least in infinitesimal neighborhoods, i.e., "local" Lorentz invariance). Similarly, quantum field theory, being a relativistic theory, is manifestly Lorentz invariant. But why at least in infinitesimal neighborhoods, i.e., "local" Lorentz invariance? because it depends on objectivity, and that resulted the concept of space-time continuum, and from here it is resulted the difficulty in the unification between quantum and relativity.

By quitting the space-time continuum by considering space is invariant, then the observed decrease in the light speed depends on the gravitational potential at any point in space. At any point in space the Lorentz transformation is linear, and then you can define local" Lorentz invariance at any point in space. That interprets the experimental result of the Pond-Rebka experiment, that the gravitational time dilation produced as the free fall particle moving in a speed of the escape velocity. Proponents of the theory of General Relativity offer three different conflicting explanations of the results of the The Pound-Rebka experiment that are said to be equivalent to each other and therefore all equally correct. All make the claim that the results of the Pound-Rebka Experiment are “proof” of the Equivalence Principle even though nothing in these measurements suggests any need for the Equivalence Principle. In strong gravitational field there is a huge amount of the rest mass change to photons which required to define the escape velocity as a relativistic, not classical as in Einstein GR. GR is not completely relativistic. 

Dear V. T.

A single fact establishes the superiority of Newton’s model over Ptolemy model: the fact that it has no unknown entity. So, the answer to the old question “rotates the universe or Earth?” is “Earth moves” because no unknown entity is required to explain data in this case.

You described the present understanding and used a “phantom energy”, which has basically the same role of the celestial spheres of Ptolemy: to explain the motion of distant celestial bodies. However, on considering a comoving length unit, we can obtain a model with no unknown entities. As I see things, this defines the answer to the above question.

You have a simple two-page summary of my analysis in Appendix I of http://vixra.org/abs/1412.0170, so in 5 minutes you can compare the two cases – invariant matter or invariant space. If the answer is that decreases the length unit, we have been working in a kind of Ptolemy model and to continue to do so will not be very fruitful.

http://vixra.org/abs/1412.0170

Dear Azzam

Einstein used the argument of symmetry because he had to find the value of a parameter and he did not knew what to do -  in one of its texts on SR he mentioned it and asks something like "what else could I have done?" You are right, the value of the parameter is not "1" as considered by Einstein because he knew not what else to do and the reciprocity, which is the cause of  SR paradoxes, does not hold. However, we have to start from the beginning in order to establish the correct physical models of the universe; and the first step, as I see it, is the answer to the question I present here. And not only by physical reasons, but also by psychological ones - if we conclude that the Big Bang model is just a kind of Ptolemy model and that space does not expands but it is the matter that decreases, then we can begin reanalyzing other theories. Until then, they are as sacred as the fundaments of any religion (am I wrong?)

We can see a two-page summary of my analysis in the appendix I of http://vixra.org/abs/1412.0170

Dear Azzam Almosaliami, Robert Shuler, and Alfredo Oliveira

I thank Robert Shuler for remembering my 1976 paper from Foundations of Physics. That paper attracted a lot of attention at the time, including from the late Carl Sagan. But back then we didn’t know that the expansion of the universe was accelerating. Indeed, cosmologists still do not really believe this. They always give the age of the universe as t = 1/H, where H is the observed value for the Hubble constant. This is equivalent to having a universe containing two objects that are initially superimposed and fly apart at a uniform speed v because we obtain H = 1/t when the distance between the objects is D = vt. If the two objects fly apart at a uniform rate of acceleration a, then the universe is twice as old t = 2/H when we solve for H with D = ½at2. See A.D. Allen, The age of the Universe is twice the standard estimate. Physics Essays 2009:22;344-347.

I’ve another paper that should interest Azzam even more. Consider a rigid rod with a finite proper length oriented parallel to the x-axis. Assume also a point-particle that is moving along the x-axis, relative to the rod, with a uniform relativistic speed. Finally, assume that the point-particle has a differential extent along the x-axis and cannot be Lorentz contracted. Then the time it takes the rod and the particle to fly by each other is shorter for the particle due to Lorentz contraction of the rod. For a formal relativistic proof see A.D. Allen, When moving clocks run fast. Physics Essays 2013:26;58-60.

Dear Allen D Allen,

In fact your paper is very fascinating and I'm really interested in it. It is in full agreement with my paper relative to quantization of gravity http://dx.doi.org/10.14299/ijser.2014.10.002 

Lorentz symmetry causes all the paradoxes in SRT. Proponents of relativity offer conflicting solutions for the paradoxes by proposing acceleration, and they forget that SRT is for inertial frames. Lorentz symmetry is only to keep on objectivity not for the Lorentz invariance. Because of objectivity it is impossible measuring a faster than light or interpret the experimental results of quantum tunneling and entanglement (spooky action), —Casimir effect, Hartman effect by Einstein interpretation of the Lorentz transformation in SRT. Because of that all the attempts are failed to unify between quantum and relativity. I succeeded in the quantization of gravity and I found there is no graviton, it photon mediates gravitation. All the experimental results are agreed completely with my theory, and I could solve the most problems regarded to quantum and relativity. I found there is no the concept of acceleration as Einstein adopted in his GR, and in his equivalence principle, In fact it is vacuum fluctuations because Lorentz transformation must be vacuum energy dependent  which is quantized after removing symmetry, and then removing objectivity, which leading to disappearing all the paradoxes in SRT and then reconciling and interpreting the experimental results of quantum tunneling and entanglement (spooky action), —Casimir effect, Hartman effect according to our new transformation. That is can be appeared in case of strong gravitational field where in case of strong gravitational field the escape velocity must be defined as relativistic not classical as in Einstein GR. Recently two published papers in European Physical Journal D challenge established wisdom about the nature of vacuum. In one paper, Marcel Urban from the University of Paris-Sud, located in Orsay, France and his colleagues identified a quantum level mechanism for interpreting vacuum as being filled with pairs of virtual particles with fluctuating energy values. As a result, the inherent characteristics of vacuum, like the speed of light, may not be a constant after all, but fluctuate. Meanwhile, in another study, Gerd Leuchs and Luis L. Sánchez-Soto, from the Max Planck Institute for the Physics of Light in Erlangen, Germany, suggest that physical constants, such as the speed of light and the so-called impedance of free space, are indications of the total number of elementary particles in nature. Also, two separate research groups, one of which is from MIT, have presented evidence that wormholes — tunnels that may allow us to travel through time and space — are “powered” by quantum entanglement. Furthermore, one of the research groups also postulates the reverse — that quantum entangled particles are connected by miniature wormholes. These ideas are agreed and predicted in my new transformation equations.

Dear Allen D Allen, 

Do you know why you concluded t = 2/H... You are right but do know why??

When you measured the red shift, the measured red shift is depended on a relativistic quantized motion, that means according to my theory of quantization of gravity this red shift must be equivalent to a relativistic quantize force Fq. Now in case of weak gravitational field if you try to compute the acceleration equivalent to Fq, it will give you aq half the classical acceleration (Newton's acceleration) , where aq=1/2 ac. where ac is classical acceleration. Because of that if you would like to compute D then D=1/2 ac t^2=1/2 2aq t^2.  Review my paper about the the relativistic quantized force http://dx.doi.org/10.14299/ijser.2014.10.002 and there are more and more!

Forget Einstein equivalence principle, it is wrong! read this paper http://physicsworld.com/cws/article/news/2014/jul/28/new-correction-to-speed-of-light-could-explain-sn1987-neutrino-burst

Dear Azzam:

The reason is much, much simpler. You can easily derive H = 2/t by substituting the quantity ½at2 for distance D in the definition of the Hubble constant, H = v/D. The quantity ½at2, in turn, can be derived as the indefinite integral (anti-derivative) of dD/dt = at = v, which is the definition of (uniform) acceleration a. The latter takes knowledge of the differential calculus. But the former only requires algebra.

I read your paper but haven’t had enough time to digest it. Since you know your work best, let me ask what you make of Shuler’s advice that you are repeating a mistake he made. He’s pretty good at making mistakes, finding them, and then disclosing it to the community. He’s like a mine sweeper going before us.   

Alfredo,

Questions:

"One may [think] that if the length unit were varying, physical laws would not hold. However, that is not true! One can define a comoving system of units where space is invariant and physical laws hold, and all constants keep invariant."

Can you explain what you are saying here? Are you saying that "space is invariant and physical laws hold, and all constants keep invariant" even if the length unit were varying? It seems if the former is true the length scale would not change. What do you mean by "space is invariant"? Can you explain clearly how the length unit would vary? Is the speed of light changing in this case?  Thanks.

Dear Allen D Allen,

My paper is talking about quantization of gravity. In my paper I adopted the principle of Heisenberg when he said in his book "it is impossible to reach to the unified theory without defining the force in terms of frequency". So acceleration is changing of velocity in a unit of time. And when the velocity of a particle changed, that means its kinetic energy changed, and that means it gained energy or lost energy. Energy is quantized in classical motion (low velocities) or in relativistic velocity (high velocity), and energy is defined according to quantum theory by hf, where h is Planck's constant, and f is frequency, and frequency is the number of cycle per unit of time. Now in classical motion or in relativistic motion when the velocity of a particle or a train changed from 0 to v1  that means the particle  or the train gains kinetic energy equals to hf1, and when the particle gain another photon f2, then it will move in velocity v equivalent to kinetic energy h(f1+f2). So how can you define the concept of classical acceleration according to the concept of quantized energy, and then how you define the concept of Newton's second law according to the concept of quantized energy. Also there is no uniform acceleration, because according to relativity, mass changes, when velocity changes. So Newton's second law must be define as relativistic and quantized.

Classical physics and Newton's law built basis on continuity. while energy is discrete!!!! According to continuity in classical physics the quantity ½at2, in turn, can be derived as the indefinite integral (anti-derivative) of dD/dt = at = v, which is the definition of (uniform) acceleration a!!!

I interested when you informed about your paper when deriving  H = 2/t. this factor of 2 is related also to the factor of 2 in the Schwarzschild solution of Einstein field equation (2GM/c^2r) which coincidentally equals to the square  of the classical escape velocity/c^2, while GM/r is the potential energy, and the recent experimental results illustrated the decrease in the speed of light as quantized depended on the gravitational potential GM/r not on the strength of the field (classical acceleration) as in Einstein's equivalence principle. where according to Franson's paper depends on GM/c^2r not on 2GM/c^2r as in Einstein's GR.

Also review the Pond-Rebka experiment, and compare it with the equivalence principle of Einstein. Also this factor of 2 which is resulted by the classical motion not relativistic or quantized illustrated to you why according to GR which depended on the concept of classical acceleration it must graviton of spin 2 mediates gravitation not photon of spin 1. Also why according to GR the Schwarzschild radius must be 2GM/c^2 while in quantum GM/c^2. So, read my paper and you will understand all of that. 

Relative to Shuler’s advice; if you read my paper about the exact solution of the pioneer anomaly http://vixra.org/abs/1109.0058 which is the same factor of the decreased light speed by gravity, which is leading also to solve the energy momentum problem in gravity, you will understand also Shapiro Delay, the mercury precession you get by my theory is the same experimentally observed.

Alfredo: your question "Does the expansion of space increase or decreases the unit length?"  May be answered considering "replication" of the unit length, the Plank´s length, so that there is no increase or decrease of the unit length: it is not afected by the expansion of the universe. It just replicates. Today it has replicated about 10^61 times, that is how expanded is the universe around us now.

Dear Alfredo,

When the light is passing through the gravitational field, it is same to pass through a medium of refractive index n. Now suppose a light beam passes through a tube of length L and  refractive index n, then for an observer on the lab. the Length of the tube locally is L same if the tube is empty because it is define according the vacuum of the lab. and when the observer of the lab. measures the time required for light beam to pass the length of the tube of refractive index n, he will get the light beam will take longer time to pass the length of the tube than when the tube is empty (vacuum) by a factor of n (refractive index). Now according to my theory, the length of the tube measured locally for an observer located in the lab by a meter stick put outside the tube according to the vacuum of the lab. which gives the observer the length of the tube to be same if the tube empty (vacuum). Also for an observer lives inside the tube, the length of the tube locally for him is same if the tube is empty according to his meter stick inside the tube. So locally for the observer inside the tube the length of the tube is same if the tube is empty, and also locally for the observer of the lab the length of the tube is same if the tube is empty. Because of that the measured length of the tube locally is same in x, y and z for the observer inside the tube, and for the observer of the lab, Because of that  we say length is invariant! But according to the clock of the observer of the lab, when the light beam passed the length of the tube of refractive index n, it takes longer time separation than when the tube is empty. Because of that we say in optics the light speed is decreased, because the length of tube is measured locally in the lab according to the length of the vacuum of the lab. Now the light speed for the observer inside the tube must be same if the tube is empty, where he must measure according to his local length and local time (clock) the speed of light to be c the speed of light in vacuum. So   measuring the decrease in the speed of light inside the tube for the observer of the lab comparing to the observer inside the tube depends on time only not on space. Space here is invariant. Where the clock of the lab is moving in a faster rate than the clock of observer inside the tube (time dilation).  Now if we consider the light speed is locally constant for the observer inside the tube and the observer of the lab. and equals c the speed of light in vacuum. Thus in this case we must conclude the decrease in the speed of light measured by the observer of the lab inside the tube is resulted as if the light beam passes longer length inside the tube, and because of that it takes longer time to pass the tube. That means within the medium of the  length must expand, and then the light must take longer time to pass the length of the tube. Here Lorentz transformation is concerned only within the range of the boundaries of space and time inside the tube or the moving train. in this case Lorentz transformation is linear (inertial frame), and in this boundaries you keep at the Lorentz invariance. But the space and time out of the boundaries of the tube or the moving train, for example when you use the local length of tube according to the lab, in this case the vacuum is fluctuated and Lorentz invariance broken, and the frame in this case is non-inertail. which is equivalent as an accelerated frame. Because of that you measure a decrease in the speed of light, or an increase in the speed of light (faster than light).

Dear Alfredo,

What do you mean by unit of length? If you choose for instance the Compton wavelength of the electron, this is always the same in GR. From the analysis of atomic spectra in distant objects (quasars) we know that this is true to a very high accuracy.

Dear Robert Shuler

Well, we agree in a number of things... I have reached a different result but there are common understandings... I think you would like to take a quick look to my paper http://vixra.org/abs/1107.0016. You can also see a two-pages summary in Appendix I of  http://vixra.org/abs/1412.0170 - I am sure that at least you will find inspiring reasoning.

Paul K. Sherard

A system with invariant description (the same physical laws, invariant constants) is not necessarily an invariant system - this is a characteristic of any system having a self-similar transformation when described using units intrinsic to the system. For instance, when inflating a ballon, the shape of this one is invariant if the length unit is a segment marked in the surface of the balloon.

What happens in the universe is that matter is suffering a self-similar transformation, an evanescence, and we use units that are intrinsic to matter - units where the concept of reference body and clock hold. So, the description of bounded systems of bodies remains invariant. However, when we use these units to mesure "space", i.e., the distance between bodies not bounded, this distance increases - because our length unit is decreasing in relation to space.

And no, light speed does not changes - it is not only the standard length unit that changes, also the standard units of time, mass and charge change; they are all intrinsic to matter and they all change with matter at the same rate. Because of that, velocity unit is invariant; and also light speed.

Of course that trying to explain all the process in few words is not feasible; but you can see a two-page summary in Appendix I of  http://vixra.org/abs/1412.0170. The full work is available at http://vixra.org/abs/1107.0016.

Dear Antonio Alfonso-Faus

I have answered the question "expands the space or decreases the length unit" without considering any unknown phenomena or entity (you can see how in the two-pages summary presented in Appendix I of http://vixra.org/abs/1412.0170); so, no need to consider imaginative processes like "replication". 

Dear Azzam

I think you are making an analogy that is not valid; a medium is not the same that a gravitational field in spite of both delaying light, so I have difficulty in following your reasoning.

 Dear Norbert

I mean the standard length unit - the SI or any other accepted length unit because they are all proportional to the SI unit - the reason is that they all have to verify the concept of reference body so physical laws may hold valid. They are all "intrinsic" to matter, i.e., such that matter properties are invariant when measured with them.

I have found that we can define a different system of units, a comoving system, with a comoving length unit and also different mass, charge and time units, where physical laws hold and gravitational and dielectric constants and light speed hold invariant. Using this comoving system, current physical laws fit both cosmic and local data without the need to use any unknown entity or imaginative phenomenon.

You can see a two-page summary in appendix I of http://vixra.org/abs/1412.0170.

Dear Alfredo Oliveira,

Re: a medium is not the same that a gravitational field.

first have you read my paper http://dx.doi.org/10.14299/ijser.2014.10.002 

I hope to introduce to me a physical criticism of my model to understand your point of view. Also to understand if I'm wrong.