It is important to notice that provided light moves through a vacuum and provided electromagnetic radiation from the same atomic transition is used to measure both time and space intervals, then inevitably the speed of light is found to be unity, i.e. constant, such in concordance with INTRODUCTION 5 in the book Relativity Reexamined by Léon Brillouin: We witnessed the invention of atomic clocks of incredible accuracy, whose physical properties differ very much from the clocks Einstein imagined. This will be discussed in some detail in Chapter 3. Let us mention here a real difficulty resulting from internationally adopted definitions. The unit of length is based on the wavelength of a spectral line of krypton-86 under carefully specified conditions with accuracy 108 and the unit of time is based on the frequency of a spectral line of cesium with accuracy 1012. Hence, the same physical phenomenon, a spectral line, is used for two different definitions: length and time, and the velocity c of light remains undefined and looks arbitrary. It should be stated, once and for all, whether a spectral line should be used to define a frequency or a wavelength, but not both! Actually, the speed of light does not look "arbitrary", rather it is a constant ("unity") now by definition.
But the electronic charge $e$ always occurs in quantum mechanics in the fine-structure combination, $e^2/\hbar$, and the particle mass $m$ occurs either in the ratio $m/\hbar$ or as a ratio with respect to the mass of another particle, like $m_e/m_p$ for the electron and proton masses. [ ... ] all particle masses have the dimensionality of an inverse length - the Compton wavelength of a particle being just the reciprocal of its mass. [ ... ] Taken in a sensible way, both Planck's constant and the speed of light are unity, and they are so everywhere. Is that true? In UAC it is only true with atomic time, that is: as long as there is no (Newtonian) gravity involved.
Intervals of time and space can therefore be considered to be measured with respect to a unit determined by $m_e^{-1}$. Especially time. Hoyle doesn't seem to be aware of the fact that his hypothetical other (minus) side of the zero surface can never be observed, because the time ticks $\sim m_e^{-1}$ become infinite with $m_e \to 0$, while approaching the zero surface from the positive side. And what can never be observed, does that exist anyway? Moreover, the dimensionalities of all physical quantities can be expressed as some power of $m_e , m_e^*$ say. As examples, pressure and energy density have $n = 4$; current density and surface tension have $n = 3$; luminosity, force, and the electromagnetic field have $n = 2$; energy, mass, and frequency have $n = 1$; length has $n = -1$. With our own notation conventions, we subsequently have: $$ \frac{q}{q_0} = \left(\frac{m}{m_0}\right)^n \MET q = \mbox{quantity} \EN n = 4,3,2,1,-1 $$ Especially the energy density $u$ with power law $n=4$ is worthwhile to remember. It follows that (star) temperature $\Theta$ has power law $n=1$ because of the Stefan-Boltzmann law: $$ \frac{u}{u_0} = \left(\frac{m}{m_0}\right)^4 = \frac{\sigma\,\Theta^4}{\sigma\,\Theta_0^4} \hieruit \frac{\Theta}{\Theta_0} = \frac{m}{m_0} $$ Another way to derive the same is with the Ideal gas law; see below.
Every experiment consists, when its procedures are analyzed, in the counting of a dimensionless number, which is always made up as a product of physical quantities and their inverses in such a way that the sum of the dimensionalities add to zero. No physical quantity with $n \ne 0$ is ever measured, except as a ratio to another quantity of the same dimensionality. Hence it follows that, so long as $m_e(x)$ is only slowly variable with respect to the spacetime position $x$, as would be the case if $m_e$ were to vary only on a cosmological time scale, no local laboratory experiment can detect the variation. I wouldn't be so sure about this: see bottom lines of the Narlikar's Law page.
The failure to distinguish between the geometry (6) associated with $m_e$ and the geometry (7) associated with $m_e^*$ is total. No distinction is possible through any observation or through any experiment. We now take the view that to have attempted to distinguish between (6) and (7) was an irrelevant problem. What can never be observed does not exist.
Although the region over which the Einstein-de-Sitter model applies is only a small element of the whole universe, it nevertheless encompasses everything which the astronomer observes, even with the largest telescope. Hence no. Apart from redundancies such as the cosmological constant and other General Relativity stuff, the Einstein-de-Sitter model applies over the whole universe, as has been motivated in our Hubble Parameter section.
In this Minkowski conformal frame the electron mass $m_e^*$ is a function of $\tau$. Sufficiently near the zero surface, $m_e^*$ can be expanded in powers of $\tau$, $$ m_e^* = A \tau + B \tau^2 + \cdots \qquad (29) $$ The gravitational equations based on (4) turn out to require $A = 0$. Hence the leading term in the expression for $m_e^*$ is quadratic in $\tau$ [ ... ] The coefficient $B$ [ ... ] is positive. With other words, Narlikar's Law has been recovered, for Minkowski flat-space indeed.
[ ... ] gravitation behaves peculiarly in the Minkowski frame, not only because the particle masses change with time, but
because the gravitational "constant" $G$ also changes. In the Einstein frame $G$ is indeed constant; but being of dimensionality $n=-2$,
$G$ varies like $m_e^{-2}$. Such in concordance with the following.
The Planck mass is defined by:
$$
m = \sqrt{\frac{\hbar\,c}{G}} \approx 21.7651\,\mu g
$$
Herefrom it follows that:
$$
G = \frac{\hbar\,c}{m^2}
$$
Let's assume that Planck particles have
some physical significance. My own interpretation is as follows. Tiny black holes do not exist,
for the simple reason that singularities never really exist in physics. But it may be that they are
sort of upper bound to the existence of real elementary particles, meaning for example that
particles heavier than Planck particles cannot come into existence. I do not believe in any fundamental
meaning of the Planck units.
At best, they provide upper and lower bounds for certain quantities such as length, mass, time and temperature.
Whatever. With subscript $0$ for nowadays values and assuming a variable Planck mass, we can derive that the
Gravitational Constant may be variable as well:
$$
G_0 = \frac{\hbar\,c}{m_0^2} \quad \Longrightarrow \quad
\boxed{\frac{G}{G_0} = \left(\frac{m_0}{m}\right)^2}
$$
Conclusion: the Gravitational Constant may be inversely proportional to the square of (varying rest) mass of elementary particles.
Only when measured in atomic time, however. Remember that orbital clocks - as opposed to atomic clocks -
are defined with the explicit assumption that the gravitational constant $\,G\,$ is indeed a constant!
But there is an easier way to derive the atomic case. Newton's laws for acceleration and gravitation are considered once again,
for non-empty space and non-orbital (ipse est atomic) time.
$$
F = m.a = G\,\frac{M.m}{r^2} \slechts \frac{m}{m_0}\frac{m_0/m}{(m_0/m)^2} = \frac{G}{G_0}\frac{(m/m_0)^2}{(m_0/m)^2}
\slechts \left(\frac{m}{m_0}\right)^2 = \frac{G}{G_0}\left(\frac{m}{m_0}\right)^4 \\ \slechts
\frac{G}{G_0} = \left(\frac{m_0}{m}\right)^2 \EN \boxed{\frac{F}{F_0} = \left(\frac{m}{m_0}\right)^2}
$$
And once again for orbital time, only the force, leading to the same - obviously reliable - result:
$$
\vec{F} = m\vec{a} \;\approx\; m\frac{\vec{r}(T+\Delta T)-2\vec{r}(T)+\vec{r}(T-\Delta T)}{(\Delta T)^2} \hieruit \\
\vec{F} \sim m\frac{1}{(\Delta T)^2} \slechts \frac{F}{F_0} = \frac{m/m_0}{\left(\sqrt{m_0/m}\right)^2} = \left(\frac{m}{m_0}\right)^2
$$
Especially the latter is important, because it gives us another argument for deriving the elementary rest mass dependence of temperature,
namely according to the Ideal gas law. Here $p=$ pressure $=F/A$ with $F=$ force and $A=$ area, $V=$ volume, $n=$ amount of substance
in moles, $R=$ ideal gas constant, $\Theta=$ temperature.
$$
p.V = n.R.\Theta \slechts \frac{(m/m_0)^2}{(m_0/m)^2}\left(\frac{m_0}{m}\right)^3 = \frac{\Theta}{\Theta_0}
\slechts \boxed{\frac{\Theta}{\Theta_0} = \frac{m}{m_0}}
$$
According to the Kinetic theory of gases,
with $N=$ number of molecules, $m=$ mass of molecules, $v=$ speed of molecule:
$$
p.V = \frac{2}{3}N.\half m\overline{v^2} \slechts \frac{m}{m_0} = \frac{m}{m_0} \slechts
\overline{v^2}/\overline{v_0^2} = 1
$$
All measured with atomic time. As we shall see in our Galaxy project, mixtures of atomic time
as observed and orbital time as calculated will give rise to quite another perspective.
For the sake of completeness, let's pay attention to the other Planck units as well. And check for consistency with the questionable $\,G=(m_0/m)^2\,$, featuring in atomic time anyway:
| Name | Expression | Varying with |
|---|---|---|
| Planck length | $l_P=\sqrt{\hbar G/c^3}$ | $m_0/m$ |
| Planck mass | $m_P=\sqrt{\hbar c/G}$ | $m/m_0$ |
| Planck time | $t_P=\sqrt{\hbar G/c^5}$ | $m_0/m$ |
| Planck temperature | $T_P=\sqrt{\hbar c^5/(Gk_B^2)}$ | $m/m_0$ |
The questionable $\,G=(m_0/m)^2\,$. Let us assume that Hoyle is right and that there are two basic frames to be distinguished.
Translated into our own parlance it would mean that there is an Atomic frame (= Minkowski frame) and
an Orbital frame (= Einstein frame).
In the Orbital frame there is no varying mass dependency of the gravitational constant, for the reason that
there exists NO gravitational shielding.
This means that in Newton's law for gravitational attraction $\,F = G.Mm/r^2\,$ the distance $\,r\,$ entirely behaves as if it is
in empty space. But empty space doesn't suffer from Length Contraction. So on the right hand side we only have
a proportionality $\,\sim (m/m_0)^2\,$, which is in complete agreement with the same varying mass dependency of the force $\,F\,$ on
the left hand side.
In the Atomic frame there is a varying mass dependency of the gravitational constant. All space acts as if it is filled with
(virtual) matter. There is Length Contraction everywhere. However, on the right hand side a proportionality
$\,\sim (m/m_0)^2\,$ is obtained again, which is in complete agreement with the same varying mass dependency of the force $\,F\,$ on
the left hand side.
Let's summarize our findings in a table. It is Force that enables us to tie the two frames together.
| Atomic frame | Orbital frame |
| $L/L_0 = m_0/m$ | $L/L_0 = 1$ |
| $\Delta t/\Delta t_0 = m_0/m$ | $\Delta T/\Delta T_0 = \sqrt{m_0/m}$ |
| $G/G_0 = (m_0/m)^2$ | $G/G_0 = 1$ |
| $F/F_0 = (m/m_0)^2$ | $F/F_0 = (m/m_0)^2$ |
At this point, I had the intention to proceed with an explanation of the
Tully-Fisher relation.
But how is that possible if Wikipedia fails to make a clear distinction between e.g. Luminosity (with $\,n=2\,$ according to Hoyle) and mass (with $\,n=1\,$ according to Hoyle). Sober thinking reveals the following.
The luminosity $\,\Lambda\,$ should be defined according to the Stefan-Boltzmann law, as a decent approximation for stellar radiation. And the outcome should be multiplied by the (real) area
of the radiating body, as measured and subsequently calculated by the observer. Thus we get:
$$
\frac{\Lambda}{\Lambda_0} =\frac{\Theta^4}{\Theta_0^4}\frac{A}{A_0} \slechts
\frac{\Lambda}{\Lambda_0} = \left(\frac{m}{m_0}\right)^4\left(\frac{m_0}{m}\right)^2 = \left(\frac{m}{m_0}\right)^2
$$
The latter outcome is indeed in concordance with the Hoyle's $\,n=2\,$ findings. As we shall see in our Galaxy project, mixtures of atomic time as observed and orbital time as calculated will give rise to another perspective for
certain quantities, such as the orbital velocity $\,\vec{u}\,$ in atomic time. We repeat:
$$
\vec{u} = \frac{d\vec{s}}{dt} = \frac{d\vec{s}}{dT}\frac{dT}{dt} = \vec{v} \frac{\sqrt{m_0/m}}{m_0/m} = \vec{v} \sqrt{m/m_0}
$$
Now imagine that we as an observer are in the position $\,\vec{s}\,$, then the velocity $\,\vec{v}\,$ must appear as a here-and-now
velocity $\,\vec{u_0}\,$ for us, in a Newtonian frame of reference. With absolute values (i.e. speed $\,u\,$) we thus have:
$$
\frac{u}{u_0} = \sqrt{\frac{m}{m_0}} \slechts \left(\frac{u}{u_0}\right)^4 = \left(\frac{m}{m_0}\right)^2 = \frac{\Lambda}{\Lambda_0} \\
\boxed{\frac{\Lambda}{\Lambda_0} = \left(\frac{u}{u_0}\right)^4} \qquad \boxed{\frac{u}{u_0} = \sqrt[\Large 4]{\frac{\Lambda}{\Lambda_0}}\,}
$$
The rightmost formula is the one I was triggered by, and has motivated present search. It is found at page 76 of the book by
[Fahr 2016].
It has been established in our Galaxy project that, in some sense,
MOdified Newtonian Dynamics is equivalent to our Unified Alternative Cosmology.
Among the Observational evidence for MOND we find indeed Tully-Fisher.
The Tully-Fisher relation is special in that it has no other physical ground than our peculiar way of measuring time with
two different clocks. Quite in general, the following may be said.