The possibility is considered that the red shift of the galaxies given by Hubble's law is not a consequence of the expansion of the Universe or of the Doppler effect but a direct consequence of what has been called the "Tempo rate law" in a non-expanding Universe. Considering that the exact form of the Tempo rate law remains undecided in the present version of the model, its compatibility with the observed Hubble red shifts can help to specify such Tempo law and therefore some other parameters of the aether.
Summarizing (from earlier sections of this work) the issues of interest for the present discussion:
It was postulated that there exist reference frames (called rectilinear) in which all the aetherinos move in straight lines at constant speeds. The observer associated with such reference frames is called the Ideal Observer IO. The Greek letter t is being used in this work for the time variable of the Ideal Observer.All material particles moving relative to the aether suffer the aether drag force. Due to such aetherinical drag force, the observer IO observes that the speed of any "free" material particle relative to the local aether (i.e. to the frame in which the aether can be considered at rest) decreases according to the same law:
[9-1] uI(t) = uI(0) f(t)where f(
t) is a yet undecided function of time.Note 1: Supposing that the aetherinos velocity’s distribution of the "undisturbed" aether does not change in time for IO (together with other simple suppositions of the model) it has been calculated (see for example [3-11] of Section 3) that the function f(
t) is of the form:[9-1b]
where m is a constant.
Considering that all the clocks used in our physical world are ultimately made of material particles, the rate of these clocks must also decrease for IO according to the same law [9-1]. But then an observer (called the Official Observer OO) using these real material clocks will observe no speed decrease of the "free" material particles. He will instead observe a continuous increase in the speed of any given aetherino. The letter t is being used in this work for the time variable of the Official Observer.
Note 2: In earlier sections of this work it has been brought into consideration that the speed increase of the aetherinos observed by the real official observer OO implies that the speed’s distribution of the aether observed by OO becomes increasingly "hot" with time. This would imply a time change of the fundamental physical constants. The experimental facts put a limit to this time evolution of the physical constants that the aether model might not be able to describe unless it is recognized that there is a mechanism that compensates for the speed increase of the aetherinos observed by OO. A compensating mechanism that seems straightforward within the paradigms of the model is to recognize that when the aetherinos collide with matter they emerge on the average with smaller speeds. This implies that the Ideal Observer would observe a "cooling" in the speed distribution of the aetherinos (i.e. for IO, the average speed of the aetherinos of the aether would decrease with time. The width or speed-spread itself of the distribution would decrease with time and that is why this assumption has been called "Shrinking Distribution" in earlier sections). But redoing the calculus of the slow down observed by IO in the material particles that move freely relative to a “shrinking” aether, a function different from [9-1b] is obtained and that is why, until the magnitude of the shrinking is finally elucidated, a general abstract function f(t) will be used in the discussion of this section.
"Tempo rate law" is the name given in this work to the function f(
t) that describes the IO speed decrease of the material particles moving through the aether (in absence of material forces and therefore subject only to the aether drag force). Considering that the Official Observer observes no speed decrease in such freely moving material particles, i.e. u(t) = constant, then synchronizing the IO and the OO clocks so that for example t = 0 at t = 0, which implies u(t) = uI(0), and considering that both observers IO and OO share the same space standards x, dx, etc,... then:[9-1c]
and therefore f(
t) also relates the rates of the IO and the OO clocks and that is why it has been called "Tempo rate law". It must also be remembered that this model interprets "the constancy of the speed of light" as meaning that it is relative to the material detector that the speed of light (in vacuum) has a specific value c (independent of the velocity of the detector across the aether). In a Galilean scenario this speed constancy can be implemented assuming that every light emitter "emits" radiation-modulated aetherinos at a plurality of speeds relative to the emitter but only those of speed c relative to the material detector are effective in reproducing the modulation (see Section 6 for a more precise description).The predictions made by the model of the frequency shifts in the atomic lines of distant galaxies at rest relative to the observer are not only conditioned by the exact form of f(
t) but also by the elucidation of what phenomena determine the value c for the "effective speed of light" and the problem here is that the study of what the model can say on this subject is at this stage incomplete (see Section 6). Different causes can a priori be made responsible that the effective speed of light takes a value c and they imply different predictions of whether c varies or not with the epoch (whose consequences are apparent in the case of light propagating across large distances). Therefore the following alternatives will be analyzed to interpret the speed of light:1) The speed of light is determined by the nature of the emission process.
This means that the speed of light depends only on the velocity distribution of the aetherinos surrounding the emitter at the epoch of emission, or perhaps simply on some specific speed characterizing such distribution (e.g. on the speed V
M at which the distribution reaches its maximum or on some specific value related with VM ). In Section 6 it is shown how can this alternative be implemented and made compatible with the fact that the speed of light is independent of the velocity of the emitter relative to the detector. What is specific to this alternative is that the observed speed of light is determined by the nature of the emission process and therefore such speed can depend on the observed light source, not because of the velocity of the source relative to the detector but because the emissions might occur in very different epochs.What is specific to this alternative is that c is determined by the nature of the emission process and therefore the value c measured in the reference frame of the detector can depend of the observed light source, not because of the velocity of the source relative to the detector but because the emissions might occur in very different epochs.
2) The speed of light is determined only by the nature of the reception.
This means that the capability of the incoming aetherinos in manifesting themselves as light depends only on their speed relative to the detector at the epoch of detection. (The effective relative speed c would on its turn be conditioned by the velocity distribution of the aetherinos surrounding the detector at the epoch of reception. It could for example happen that the only flows of aetherinos that are able to activate the detector are those of a specific speed c that depends on the epoch of reception because such particular speed flow of aetherinos is the only one that can interfere in a constructive way with the secondary waves produced at the matter of the detector. The other radiation flows not having the appropriate speed are suspected either to suffer a destructive interference or to cross the detector and follow their way but in either case leaving no energy at the detector of the type called electromagnetic radiation).
It could thus be said that in option 1 the speed of light depends on the laboratory speed of light at the epoch and position of the emission. In option 2 instead, the speed of light depends on the laboratory speed of light at the epoch and position of the detection. The reference to "the laboratory" is to remark that such speed of light is measured using an emitter that is close to the detector and therefore the aetherinos travel a short distance along which their speed can be considered constant not only for IO but also for the Official Observer. Notice also that in the case that the emitter is far away from the detector it can no longer be assumed that the aetherinos travel at constant speed for the Official Observer but with increasing speed, (since it has been postulated in Section 1 that the aetherinos always travel at a constant velocity relative to the Ideal Observer).
Hubble's law.
It will be assumed that the official observer OO measures at all epochs the same laboratory frequency in any given atomic spectral line. This supposition is reasonable if the time standard of the official observer is directly based on atomic transitions. (The frequency of a given atomic line is supposed to decrease in time for the Ideal Observer IO but for this observer the OO clocks slow down at the same rate).
As said in Section 5, not being yet evident from experimental or theoretical facts what rate of shrinking must be supposed to occur in the aetherinos speed distribution, the exact quantitative expression of the aether drag force is yet undecided. Therefore the slow down seen by IO in a free material body moving in the aether at speed u
I will be written in the generic form:[9-1] u
I(t) = uI(0) f(t)and hence the "Tempo rate law" relating the readings of the IO clocks with those of OO will be written as:
[9-2]
As a direct consequence of [9-2] (see also [5-23, 24]), a IO speed u
I is related to its OO value uF by:[9-2b] u
I = uF . f(t)In mainstream Physics, the red shift z of a radiative atomic transition is defined as:
[9-3]
where nR is the shifted frequency measured under the pertinent circumstances and n0 is the frequency of the same transition measured in the proximity of the source (in the lab) by a detector at rest relative to the emitter.
According to the experimental facts the Hubble law is:
[9-3b]
where H is the Hubble constant whose experimental value is approximately H @ 75 Km sec-1 Mpc-1
But since it has been assumed that the lab frequency of a given atomic transition does not change in time for the Official Observer OO (i.e. at whatever epoch, the OO clocks next to the emitter measure the same period DtE = constant = 1/n0 for that specific transition) hence:
[9-4] 
Consider now a light detector R that receives a light pulse emitted by a distant stable emitter E. Suppose that E and R are at relative rest and let D be the separation between them. Let tE be the IO epoch of emission of the light pulse that is detected at R at the IO epoch tR.
Some alternative possibilities will now be studied.
OPTION 1
The aetherinos associated with the manifestation of light at the detector R are those that depart the emitter E at the laboratory speed of light at the epoch of emission.
This Option 1 is considered less plausible than the Option 2 (see below) and its analysis has now been removed from this document but its main features can still be consulted here.
-------------------------------
OPTION 2
Suppose that a flow of aetherinos manifests itself as light only if it has a specific speed c relative to the detector.
This speed c can in principle change with the epoch. For example, it seems plausible that this speed c changes at the same rate that do the characteristic speeds (e.g. the so called VM) of the aetherino’s speed distribution. In a first analysis, it will be supposed that c = k VM where k is a constant and VM is the value taken at the epoch of reception by some speed characteristic of the distribution. Two special cases can be singled out:
2-a) The aetherinos speed distribution does not change with the epoch for the Ideal observer IO. In this case the speed of the aetherinos implementing the effective light does not change in time for IO but it increases in time for the official observer OO.
2-b) The aetherinos speed distribution does not change with the epoch for the Official Observer and it therefore "shrinks" for the Ideal Observer. Then the IO speed of the aetherinos that manifest the light must change with the epoch since it must have a specific value relative to OO (and therefore relative to the characteristic speeds of the distribution) when it arrives at the detector.
In general according to the main supposition of Option 2, a light signal emitted at E at the IO epoch tE reaches the detector R at the epoch:
[9-30]
Option 2-a: suppose that the aetherino's canonical distribution suffers no shrink.
Hence the IO speed of light is independent of the epoch (because that speed is expected to depend only on the aetherino's distribution at the laboratory close to the detector and, in this option 2-a, that distribution does not change in time for IO). Therefore it can be written:
[9-31]
which leads to the same prediction of Option 1-a and is therefore unable to explain the experimental facts.
Option 2-b: suppose that there is shrinking for IO of the canonical distribution.
Let fS(t) be the rate at which the distribution shrinks. That means for example that the IO speed VM at which the canonical aetherino's distribution reaches its maximum decreases in time according to:
[9-31b] VM(t) = VM(0) fS(t)
and therefore the assumptions of option 2 imply that the IO speed of light appearing in Eq[9-30] varies as:
[9-32] cI(tR) = cI(0) fS(tR)
which means that although the speed of the effective aetherinos is constant for IO during all the journey E-R, this speed depends on the epoch of reception in the same way that the aetherino's distribution near the detector.
Therefore an event that arrives at the detector at the IO epoch tR must have departed the emitter at an IO epoch tE such that:
[9-33]
Differentiating [9-33]:
[9-34]
An intuitive interpretation of [9-34] is that an event at the emitter of duration dtE takes a time interval dtR to cross a light detector placed at a distance D (due to the fact that the aetherinos transporting the information of the beginning of dtE have a different speed from those transporting the information of the end of dtE). Both durations dtE and dtR correspond to what would observe an Ideal Observer (IO). But for an Official observer, using real clocks (that decrease their rate relative to that of the Ideal Clocks according to the Tempo Rate Law dt/dt = f(t) ), the duration of those events would be respectively: dtE = dtE f(tE) and dtR = dtR f(tR)
and therefore, using the z shift definition [9-4]:
[9-35] 
From the context of its deduction, [9-35] is a prediction of the redshift observed by OO in the oscillations of an emitter of a given IO period dtE. Equation [9-35] predicts therefore, for example, the OO redshifts when observing the same given oscillatory event at different distances D. But that is a different thing from Hubble’s Law that corresponds to the observation at the same epoch of different events that have taken place at different epochs since they come from different distances. But it is reasonable to assume that the lab oscillation periods of the atomic transitions are the same in all epochs when measured (in the lab) with Official clocks but not with Ideal clocks. In other words, it should not be assumed (as has been implicitly done) that the observed events have a constant period for IO and therefore [9-35] does not describe the Hubble Law scenario. If it is instead reasonably assumed that the IO period dtE of the observed oscillations is not always the same but corresponds to the period of a given atomic transition that therefore increases with the epoch (at the same rate that IO sees the OO time unit to increase with time), then the observed period dtE of [9-35] must be replaced by dtE f(tE) and therefore the predicted redshift of an atomic transition takes the form:
[9-35b] 
The evaluation of the z given in [9-35b] is simplified by taking the epoch of reception as tR = 0. And since the epoch zero is the one at which the IO and OO clocks are synchronized, it is f(0) = 1 and fS(0) = 1. Today, the epoch in which the earth's laboratory speed is c, is the epoch tR = 0 of reception of the studied light and therefore cI(0) = c. The Eq[9-35b] can then be written in the simpler form:
[9-36b] 
where according to [9-30] (with tR = 0), the IO epoch of emission of the pertinent light can now be substituted by tE = -D/c.
Supposing that both the rate of shrinking of the distribution and the rate of the Tempo Rate Law (according to which the "free material bodies slow down for IO) are given by the same function, say:
[9-37b]
the following inadequate prediction of z is obtained:
[9-38b]
Option 2-c:
The possibility has also be considered that the rate of shrinking of the distribution and the Tempo Rate Law are not quantitatively equal. Suppose for example that
[9-37c]
where an additional constant k has been introduced in the exponent defining the rate of shrinking fS(t) (observed by IO in the speed distribution).
Suppose again for simplicity that "today" (epoch of detection of the light that allows us to experiment with the Hubble Law) is assigned the epoch tR = tR = 0. It happens again that f(0) = 1 and fS(0) = 1 and therefore Eq[9-36b] is still valid. But now, when assuming the expressions [9-37c] the derivative fS'(0) takes a different value and the following expression is obtained:
[9-38c]
Some plots have been done of the atomic line’s redshift predicted by [9-38c] to try several values of the constant k. It has been found that "interesting" predictions are obtained if k is approximately equal to 2.
Taking Hubble's constant H = 75 Km s-1 Mpc-1 the Hubble's red shift of a galaxy whose distance to the Earth "is" 10 Mpc (or rather "was" at the epoch when it emitted the light that is today being received) is:
z = H D/c = 75*10/300000 = 0.0025
An estimate of the constant m can be obtained requiring that the experimental point {D = 10, z = 0.0025} belongs to the function [9-38c] with k=2. The distance of 10 Mpc has been chosen for calibration because it is considered that the experimental evidence supporting Hubble's law is more reliable for such small distances which can be measured with smaller error. Taking c = 1 light-year year -1, D = 10 Mpc =10*3.26*106 light-year, it can be seen that if
m = 1.06 * 10-9 year -1 the experimental point {D = 10, z = 0.0025} belongs to the function [9-38c] with k=2.
The following graphic illustrates the evolution of [9-38c] for a wider range of distances:
Fig [9-38c]. z predicted by option 2-c
NOTE 9-2 The z predicted by option 2-c has been called "interesting" because of the following:
1st
- It is considered controversial that the experimental Hubble law z=H D/c is relating the observed red shifts to the correct distances of the galaxies at the epoch of emission of their light. The review papers consulted give the impression that the distances of the far away galaxies are deduced relating their apparent luminosity to their absolute luminosity applying only the inverse square distance law. But it is believed that the effect that the z shift itself has on the apparent luminosity must also be accounted for. From wave theory, the intensity of an electromagnetic wave is proportional to the square of its frequency. Then
Let I0 be the intensity received by a detector placed at a unit distance from of a light source whose rest frequency is n0 . If this source is observed from a larger unknown distance with an intensity I the astronomers are assigning to the source a distance d such that d2 = I0/I. But if it is known that its light has a red shift z , it should also be taken into account that its observed intensity must have suffered another attenuation given by z = n0/n -1 = Sqrt [I0/I] -1 that is to say that where an intensity I0 should be observed (in absence of red shift) an intensity I = I0 / (z+1)2 will be observed (in presence of red shift). Then, if as well the source is at the real distance D, due to the inverse square distance law, its received intensity will be I = I0 / [(z+1)2 D2]. Comparing this "correct" distance D with the "incorrect" one given by d2 = I0/I it follows D = d /(z+1). Therefore the "corrected" Hubble experimental law should be:
[9-40] 
whose representation with (H=75, c=3*105) is:
Fig [9-40]. Suggested revision of Hubble's experimental law.
which is qualitatively similar to the predicted z of the option 2-c (Eq[9-38c]) of this aether model. But from the quantitative point of view Eqs [9-38c] and [9-40] are very different.
2nd
- The qualitative form of Eq[9-38c] suggests a new interpretation of some experimental observations (quasars, X-ray and Gamma bursts, ...) which mainstream Physics has difficulty to understand.
It must first be noticed that, with the standard definition of z = DtR/DtE -1, blue shifts such that z<-1 imply that DtR/DtE < 0. This only seems possible if DtR and DtE are of different sign. Considering that DtE is positive by definition it must then happen that DtR is negative. In other words it seems that the possible meaning of shifts such that z < -1 is to admit that, for large distances between the emitter and the detector (e.g. for D>4000 Mpc in fig [9-40]), if a wave front 2 is emitted soon enough after the emission of wave front 1 it will happen (due to the above hypothesis about their speeds) that wave front 2 reaches the detector "before" wave front 1. If that is the case it must also be noticed that as long as | DtR | > |DtE | the experimenter will measure a wave frequency "smaller" than the laboratory standard no matter if z £ -2. He would call that a red shift.
Notice finally that in the blue shift zone in which now | DtR | < | DtE | (i.e. -2 < z < -1) and specially in the asymptotic right part of the figures in which z tends to -1 it will happen that a big time interval in the life of the observed galaxy will be received in a small time interval at the detector. This predicts that the observation of very far galaxies will appear as very bright, short living, high frequency sources that can perhaps explain the observed X and Gamma ray bursts.
NOTE 9-3 A suggested experiment.
CMBR = Cosmic Microwave Background Radiation
MR = Microwave Radiation.
This model of the aether predicts therefore in a natural way that in a non-expanding universe we must observe frequency shifts (related with the distance) in the spectral lines from distant sources. Although some quantitative discrepancies have yet to be solved, the model suggests that there is no need to assume an expanding universe and a Big Bang to explain the Hubble redshifts. But assuming a non-expanding universe as the most simple hypothesis the problem is now to explain the CMBR (Cosmic Microwave Background Radiation) without a Big Bang.
An intuition is that the CMBR is just the "noise of the aether" that by its very nature is able to "activate" detectors of radiation. The random collisions of the aetherinos of an undisturbed aether with a detector would behave as a blackbody type distribution of frequencies. If that were so, then the CMBR is not coming from the depths of the universe but should be detectable everywhere where the aether is present. For example a detector oriented to some direction where we have placed a screen that shields any hypothetical microwave radiation coming from the outer space, should also detect microwave radiation of the CMBR type (blackbody of 2.7º K).
It is suspected that no experiment has yet oriented its detectors to a screen able to shield the MR (Microwave Radiation) from outer space, because material screens are in general much hotter than 2.7º K and emit so much blackbody radiation that it would be a hard task to deduce if the screen is shielding or not the CMBR. But perhaps a specific experiment can be designed to solve that problem; perhaps enclosing a detector of MR in a container made of a material that is known to shield the MR from the outside, and cooling both the container and the detector below 2.7º K.
Another feature that should be experimented inside that cold shielding container is the anisotropy of the observed radiation (using a detector with good angular resolution and able to observe in all directions of space). The prediction is that the MR detected inside the container should have an anisotropy coincident in direction and "speed" to the one that has been observed in the CMBR of the sky. That is because the aether model interprets such anisotropy as due to the velocity of the detector relative to the aether and this effect should not be affected by enclosing the detector in a container since the great majority of the aetherinos (like do the neutrinos) penetrate big amounts of matter without colliding with it (and therefore without changing their velocities).