Latest revision 05-11-2020

Milne's Formula

$\def \hieruit {\quad \Longrightarrow \quad} \def \slechts {\quad \Longleftrightarrow \quad} \def \SP {\quad ; \quad} \def \MET {\quad \mbox{with} \quad}$ Quoting from What's the point of the Hoyle–Narlikar theory of gravity? (Physics Stack Exchange): Hoyle was really opposed to the big bang cosmological model, hence he really stuck with the quasi-static model all the way. [ .. ] Essentially, the advantage of the quasi-static model is that there is no "Why" left - you take the existence of the universe as a given. With a point of origin, like the Big Bang, you are then left asking why the Big Bang happened. [ .. ] in that model the universe did not 'happen': it has always existed. In short: The Big Bang Never Happened.

But how would this be possible? Haven't we defined just a minute ago that elementary particle rest mass is created at timestamp $T=A\,$? The only way out of this apparent paradox is to assume once again that orbital time and atomic time are different, so different that the universe has a beginning Alpha ($A$) in orbital time and it has always existed when measured in atomic time.
Two results from the previous, Time Dilation, are repeated for convenience: $$\frac{dT}{dt} = \sqrt{\frac{m}{m_0}} \slechts \frac{dt}{dT} = \sqrt{\frac{m_0}{m}}$$ Combining this with (our version of) Narlikar's Law, we obtain (trivially assuming that $T \ge A$): $$\frac{dt}{dT} = \sqrt{\frac{m_0}{m}} = \frac{T_0-A}{T-A}$$ Here $T=$ orbital time, $m=$ elementary particle rest mass, $m_0=$ reference mass (here and now), $T_0=$ gravitational reference time, which is the "nowadays" timestamp, $A=$ gravitational timestamp corresponding with a beginning (Alpha), at the time when rest mass is created out of nothing.
What we have now is an ordinary differential equation which can be solved easily: $$t(T) = C\ln\left(\frac{T-A}{T_0-A}\right)+D$$ Where $A$ still is the time of creation. $C$ and $D$ are constants to be determined by setting the atomic increment equal to the orbital increment at the reference timestamp $(0)$: $$\left.\frac{dt}{dT}\right|_{T=T_0} = 1 = \frac{C}{T_0-A} \quad \Longrightarrow \quad C = T_0-A \quad \Longrightarrow \\ t = (T_0-A)\ln\left(\frac{T-A}{T_0-A}\right) + D$$ Furthermore we must synchronize the clocks at the reference timestamp $(0)$ such that from $t=t_0$ it follows that $T=T_0$ and so $t_0=T_0$: $$t_0 = (T_0-A)\ln\left(\frac{T_0-A}{T_0-A}\right) + D = D \quad \Longrightarrow \quad D = t_0 \quad \Longrightarrow \\ \large \boxed{\;t-t_0 = (T_0-A)\ln\left(\frac{T-A}{T_0-A}\right)\;}$$ A picture says more than a thousand words. It is clearly seen that orbital time $T$ has a beginning $A$ - Alpha, the moment of creation - while atomic time $t$ has no beginning, it extends to minus infinite. In addition, both timespans $T$ and $t$ have no ending in our theory. At last, it should be noticed that

1. atomic clock ticks $\,dt\,$ are greater than orbital clock ticks $\,dT\,$ for $\,T < T_0\,$ and $\,t < t_0\,$ (the past)
2. atomic clock ticks $\,dt\,$ are smaller than orbital clock ticks $\,dT\,$ for $\,T > T_0\,$ and $\,t > t_0\,$ (a future)

So far so good. But this is what we found on the internet. From Euclid to Eddington: A Study of Conceptions of the External World, by Edmund Taylor Whittaker. From the chapter 19. THE BEGINNING AND END OF THE WORLD:

[ Professor Edward Arthur ] Milne distinguishes two kinds of time: ephemeral or dynamical time, $\tau$, which is the time recorded by a molar timekeeper such as the chronometer or the rotating earth, and absolute or kinematic kinematic time, $t$, which is the time recorded by a timekeeper based on atomic processes such as a radioactive clock. $\tau$ is the time of Newtonian dynamics, so that in terms of it, a particle moving under no forces has a constant velocity: it is such that the Newtonian constant of gravitation, which has the dimensions $[\mbox{Mass}]^{-1} [\mbox{Length}]^3 [\mbox{Time}]^{-2}$, remains permanently constant in time. On the other hand, the period of the radiations emitted by a radiating atom is constant only when it is measured in kinematic time, so $t$ is the time kept by an 'atomic' clock. The relation between $\tau$ and $t$ is: $$\tau = t_0\log\frac{t}{t_0}+t_0$$ where $t_0$ is the present value of $t$, i.e. the age of the universe in $t$-time, say $4\times 10^9$ years. With this disposition, we have at the present instant $t=\tau$ and $d\tau/dt=1$, so the two scales are momentarily indistinguishable. When the motion of a dynamical system such as the rotating earth is described in terms of $t$-time, the period of rotation shortens as we go backward in history, being in fact at any moment proportional to the value of $t$ at that moment, i.e to the kinematic time elapsed since the Creation. From this it can be shown that an infinite number of rotations would be required to take us back to the Creation: so that while Creation is separated from us by only a finite interval of $t$-time, it is infinitely remote in $\tau$-time: it is not accessible dynamically. The non-mathematician is apt to be puzzled by the statement that whether the Creation happened a finite or an infinite time ago, depends on how our clocks are graduated [ .. ]
Sober thinking reveals that professor Milne has discovered the same formula as we have, but with a little twist it seems: orbital time and atomic time are exactly the other way around. How can that happen? Well, the secret is in time kept. If clock ticks are smaller i.e. the clock runs faster, then the time measured between two events seems to last longer, because there are more of these shorter clock ticks between these two events. In general we can say that time measured is the inverse function of clock speed. Which could have explained the above "paradox".
Quite unfortunately, though, it doesn't. Digging further in A.E. Milne's theories reveals that his formula is indeed the other way around - when compared with ours - and it is penetrating all of his work on the subject:
 Milne UAC kinematic orbital dynamical atomic

I really don't understand Milne's way of deriving things quite well - and that's an understatement. What to think about the following utterings in the first reference at the opening page 324, where Milne claims that the laws of dynamics and the Newtonian approximation to the law of gravitation [ .. ] have been deduced rationally [ .. ] No appeal was made in the derivations to any empirical laws of dynamics or gravitation, or even to the principle of relativity or to the principle of the constancy of the velocity of light. No empirical laws?! Okay .. whatever; other titbits of prose from the same reference are quite worthwhile to consider. But, before doing so, we must complete the relationship between Milne's formulation and ours.
Last but not least, for the sake of being able to employ a Nondimensionalization (NDM) technique eventually, the Formula shall be slightly rewritten, as follows: $$t-t_0 = (T_0-A)\ln\left(\frac{T-A}{T_0-A}\right) \slechts \frac{t-T_0}{T_0-A} = \ln\left(1+\frac{T-T_0}{T_0-A}\right) \\ \slechts y = \ln(1+x) \MET x = \frac{T-T_0}{T_0-A} \SP y = \frac{t-T_0}{T_0-A}$$ Then the dimensionless relationship between $\,x\,$ and $\,y\,$ is like in the above picture, with $\,t_0=T_0=0\,$ and a dimensionless place of the Asymptote $\,=(A-T_0)/(T_0-A)=-1\,$ for $\,x=A\,$.

 Milne UAC NDM $\tau = t_0\log\frac{t}{t_0}+t_0$ $t-T_0=(T_0-A)\ln\left(\frac{T-A}{T_0-A}\right)$ $y = \ln(1+x)$

From which it can easily be derived that:

 Milne UAC NDM $t_0$ $T_0-A$ $t$ $T-A$ $t_0(1+x)$ $\tau$ $t-A$ $t_0(1+y)$ $\log$ $\ln$

Now we can quote, with our own notation and interpretation, starting at page 328.

We have that at the present epoch $\,t_0=T_0\,$ and $\,(dt/dT)_{T=T_0} = 1\,$, so that the orbital and atomic scales are indistinguishable in our present experience. But where we have experience extending over large tracts of time [ .. ] serious differences result. In particular, the epoch of " creation " $\,T-A=0\,$ on the orbital scale is measured by $\,t=-\infty\,$ on the atomic scale. [ .. ] The epoch $\,T=0\,$ is, in fact, atomically inaccessible in time, just as the absolute zero of temperature is thermodynamically inaccessible. This is of fundamentally evolutionary significance. [ .. ] This foreshadows a fundamental difference between atomic and nulear dynamics and macroscopic dynamics [ .. continuing on page 329 .. ]
It will be seen that we thus solve a question raised by de Sitter and we are led, in fact, rationally to his conjectural solution. He wrote: "We can easily relegate the catastrophe (of creation) to the time minus infinity by introducing another time variable, e.g., $\,cT=\kappa\log y\,$ which will make $\,y=0\,$ for $\,T=-\infty\,$. If for $\,\kappa\,$ we take the present value of $\,R_B\,$ [ ?? ] we will have at the present moment $\,dT/dt=1\,$ [ ?? ]. There is nothing in our experience of the physical world that would enable us to distinguish between the times $\,t\,$ and $\,T\,$. We do not know which of these times it is that we use as independent variable in the equations of celestial mechanics, or by which we measure the rate of progress of radioactive disintegration, or of the evolution of a star, or of any other physical process."
The quotation in red will be considered once again in our Galaxy project, where it is conjectured that de Sitter is wrong here.