Latest revision 05-11-2020
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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
- atomic clock ticks $\,dt\,$ are greater than orbital clock ticks $\,dT\,$ for $\,T < T_0\,$ and $\,t < t_0\,$ (the past)
- 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:
- Kinematics, Dynamics, and the Scale of Time
- Kinematics, Dynamics, and the Scale of Time-II
- Kinematics, Dynamics, and the Scale of Time-III
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.