index $ \def \hieruit {\quad \Longrightarrow \quad} \def \met {\quad \mbox{with} \quad} \def \EN {\quad \mbox{and} \quad} \def \SP {\quad ; \quad} \def \half {\frac{1}{2}} $

Time Dilated Past

There exists a critical essay, written by Léon Brillouin at the end of his life. The title of the booklet is Relativity Reexamined. Quoting from page 33: Einstein's clocks were supposed to emit extremely short signals and to accurately measure time intervals between signals emitted and received. In a word, an Einstein clock was a radar system, and its requirements were thus very different from those of a frequency standard. Your "light clock" is such a radar system. But okay, that's not necessarily bad.

Question here is consistency. Let's see, departing from the formulas we do agree upon. $$ -t = \frac{1}{H}\log(1+z) \quad ; \quad \tau = \frac{1}{H}\left(e^{Ht}-1\right) $$ And so $$ \tau = \frac{1}{H}\left(e^{H.-1/H\log(1+z)}-1\right) = \frac{1}{H}\left(\frac{1}{1+z}-1\right) = -\frac{1}{H}\frac{z}{1+z} $$ Hence - inevitably - the associated (Minkowski) distance is, indeed $$ d = -c\tau = \frac{c}{H}\frac{z}{1+z} \quad \mbox{and} \quad 0 \le d \le \frac{c}{H} $$ There is an independent Cosmic distance ladder though. Are you sure that these distances are limited to the Hubble length $c/H \approx$ 14.4 billion light-years ?

Update

• The (HdB version of) the Variable Mass Theory corresponds with a BB flat Universe with Matter alone
• The Time Dilated Past hypothesis by @mikehelland corresponds with a BB flat Universe with Radiation alone
• Eric Lerner's Static Euclidean Universe corresponds with a BB flat Universe with Λ alone (and $d_P=c/H.z$ )

With the variable mass theory it is exactly the other way around. The speed of light steadily increases with distance, in some sense. Mathematically. With $t=$ atomic time, $T=$ orbital time and $H=$ (intrinsic) Hubble parameter we have $$ \frac{1}{2}H.t = \ln\left(1+\frac{1}{2}H.T\right) \quad \Longrightarrow \quad \frac{dt}{dT} = \frac{1}{1+H/2.T} $$ On the other hand, with $m=$ variable mass and $_0=$ here and now, according to Narlikar's law: $$ \frac{m}{m_0} = \left(1+\frac{1}{2}H.T\right)^2 \quad \Longrightarrow \quad \frac{dt}{dT} = \sqrt{\frac{m_0}{m}} = \sqrt{1+z} $$ The speed of light $c$ (orbital) or $c_0$ (atomic) can be expressed with the two time scales. Let $s=$ distance, then $$ c_0 = \frac{ds}{dt} \quad \mbox{and} \quad c = \frac{ds}{dT} = \frac{ds}{dt}\frac{dt}{dT} = c_0\sqrt{\frac{m_0}{m}} = c $$ According to the variable mass theory, mass is increasing with time. Therefore $c$ in the past has been larger than it is now, because of $m_0 \gt m$ , in some sense (: orbital frame). It follows that (mass)energy, as calculated within the orbital time frame, is conserved. $$ \large \boxed{mc^2=m_0c_0^2} $$ (supposing we are near the universe centre).
You are saying this repeatedly. But, Matt, we are always in the centre of our own observable part of the universe.
So we could have redshift linked to time dilation and mass variation.
Yes, atomic clocks with lighter electrons run slower and send redshifted light.

Note.

Meanwhile, the evil has been done ! Spectral lines are used for both length and time. And thus defined these are associated with atomic time, not with orbital time. As a consequence, the lightspeed $c_0$ in the atomic timeframe is a constant by definition, not only here and now, but a universal constant. We have done it ourselves; it's not a "law of nature".

Looking for feedback on metric for the universe, with time dilation but no expansion

@mikehelland :
I do not understand why you are stoking another little campfire that seems to replace your Time Dilated Past hypothesis.
In a Static Euclidean infinite and eternal Universe there are many timescales that show a Time Dilated Past: $$ t=\frac{1}{H}\ln\left(\frac{1}{1+z}\right)=\frac{p}{H}\ln\left(\left[\frac{1}{1+z}\right]^{1/p}\right)=\frac{p}{H}\ln\left(H/p.\tau+1\right) $$ $$ H/p.\tau+1=(1+z)^{-1/p} \quad\Longrightarrow\quad d\tau=\frac{p}{H}(-1/p)\frac{dz}{(1+z)^{1+1/p}} $$ $$ dt=-\frac{1}{H}\frac{dz}{1+z} \quad\wedge\quad d\tau=-\frac{1}{H}\frac{dz}{(1+z)(1+z)^{1/p}} \quad\Longrightarrow\quad \large \boxed{dt=(1+z)^{1/p}.d\tau} $$ The special cases $p=1$ and $p=2$ have been covered in that previous post.

Very much agreed !! This is more or less similar to my findings in a previous posting. To my surprise, there are infinitely many time scales that lead to the well-known distance formula $r=c/H\ln(1+z)$. Therefore it's impossible to decide - based upon that formula - which Time Dilated Past would be the physical relevant one. Mike's favorite is $dt=d\tau(1+z)$ ; my favorite is $dt=dT\sqrt{1+z}$ ; other favorites are thinkable.

Update

An application

# Age of the earth
age := 4.543*10^9;
# Seconds in a year
year := 31556926;
# Seconds in a day
day := 60*60*24;
# Rotation of the earth
omega := 2*Pi/day;
# Retardation
slower := evalf(omega/(age*year));