Dimensions check

Latest revision 03-08-2023

Dimensions check

At the website by Sahil Gupta, named The Big Circus Model , we find this remarkable formula.

Google up, if knowledge has become rusty: Gravitational constant $G$ , Speed of light $c$ , Planck's constant $h$ , Hubble constant $H$ , Proton mass $m_p$ , Electron mass $m_e$ .
The first thing I always do, when encountering a formula like this, is a check on physical dimensions.
Here goes: $[G]=length^3/mass/time^2$ , $[c]=length/time$ , $[h]=length^2\times mass/time$ , $[H]=time^{-1}$ , $[m_p]=[m_e]=mass$ .
Now treat the physical dimensions as if they were common variables. And simplify. $$ (time\times length/time) / ((length^2\times mass/time)/(mass\times length/time)) = \\ ((length^2\times mass/time)/(mass\times length/time)) / ((length^3/mass/time^2)/((length/time)^2)\times mass) \\ (time\times length/time) / ((length^2\times mass/time)/(mass\times length/time))) = 1 \\ ((length^2\times mass/time)/(mass\times length/time)) / ((length^3/mass/time^2)/((length/time)^2)\times mass) = 1 $$ Thus two dimensionless numbers to compare: 1 = 1. It cannot be better.
Upon substituting real world values Sahil proves that the left hand side divided by the right hand side is approximately unity.
Meaning that two physically "independent" large numbers have the same order of magnitude. Coincidence or not?
The advantage of a Computer Algebra System is that it can handle variables without a numerical value with the same ease.
Independent confirmation follows (with MAPLE).

c := 299792458 * length/time; # Speed of light
Mpc := 3.08567758*10^22; # Megaparsec
H := 70*1000/Mpc * 1/time; # Hubble tension
G := 6.67408*10^(-11) * length^3/mass/time^2; # Gravitational constant
m_e := 9.1093837015*10^(-31) * mass; # Electron mass
m_p := 1.67262192*10^(-27) * mass; # Proton mass
h := 6.62607015*10^(-34) * length^2*mass/time; # Planck constant
left := evalf((c/H)/(h/(m_e*c*2*Pi)));
right := evalf((h/(m_p*c*Pi/2))/(2*G/c^2*m_p));
quotient := evalf(left/right);
                                             39
                      left := 0.3422210705 10
                                              39
                      right := 0.3386412696 10

                       quotient := 1.010571071

It might raise more enthusiasm for the subject if we demonstrate how to pinpoint errors in a paper without understanding much of the math.

As an example, let's take Seeliger's Gravitational Paradox and the Infinite Universe.
On page 166 of this (published & accepted) paper the following formula is observed.

The first thing we have to know in order to establish the correctness of a formula is what the physical dimensions are.
Here goes again for the example at hand.
$dF=$ force. The unit of force is the Newton (Wikipedia) which is ($kg\,m/s^2$), giving $[dF] = mass\times length/time^2$ .
$2\pi$ is a pure math number; there is nothing physical with that: $[2\pi] = 1$ .
$G=$ gravitational constant. Google it up, then you will see that its unit ($m^3/kg/s^2$) , hence is physical dimension is: $[G] = length^3/mass/time^2$.
$m=$ mass, $\rho=$ density, which is mass divided by volume, so $[m] = mass$ , $[\rho] = mass/length^3$ . $[R,h,dR] = length$ , $[(h^2 + R^2)^{3/2}] = length^3$ .
Now we have all of the ingredients for simplifying the formula at hand into one with physical dimensions only.

$$ [dF] = \frac{[2\pi][G][m][\rho][h][R][dR]}{(h^2 + R^2)^{3/2}} \\ mass\times length/time^2 = (length^3/mass/time^2)\times mass\times (mass/length^3)\times length\times length\times length / length^3 = mass/time^2 $$ It follows that a dimension of length is missing at the right hand side, when simplified to length = 1 , which is not true.
Matching of physical dimensions, however, is a necessary condition. It is a Sine qua non.
Therefore, whatever theory is built on the formula as displayed cannot be alright.

But it is important to remember that such proper matching is only a necessary condition. It certainly is not sufficient (as e.g. Bud Rapanault will be happy to confirm).
For the sake of completeness, here is the correct formula. An infinitesimal length $dh$ has been missing. $$ dF = 2\pi G m \rho h\,{\bf dh}\,\frac{R}{(R^2+h^2)^{3/2}}dR $$ From the Seeliger's Paradox section at my website.

Core reference is a paper by Patrick G. Tonin entitled The Quantum Bang Hypothesis: An Alternative to Dark Matter and Dark Energy.
We start at section 2 of the paper. There the author says: We hypothesize that the quantum realm and the cosmos are linked by a scale relation. It is the first thing, actually the only thing we are going to investigate here. It has been argued in my ACG e-mail posts that the paper is at least mathematically correct. So attention can be restricted to physics only.
The scale relation, according to the author, is supposed to be defined as follows. $\alpha_G$ is the gravitational coupling constant defined using a pair of protons ($\alpha_G = (m_{pr}/m_p)^2$).
That definition didn't make sense to me at first sight. Later on the author explained in an e-mail what the meaning is of the symbols. The main cause of the confusion is that our own notation is a bit different: $m_{pr}\to m_p =$ proton mass, $m_p \to m_P=$ Planck mass $=\sqrt{\hbar c/G}$. Is that a pair of protons? I think not. Whatever. After having resolved my misunderstanding the algebra is simple: $$ \alpha_G = \left(\frac{m_p}{m_P}\right)^2 = \frac{m_p^2}{(\sqrt{\hbar c/G})^2} = \frac{G\,m_p^2}{\hbar c} $$ But meanwhile I had started a reverse engineering procedure in order to find a more satisfactory definition.
And this is what I found, for a real pair of protons placed at a distance $\,r\,$ from each other.

$\alpha_G=$ (Newton's law) / (Coulomb's law) $\times$ (Fine-structure constant) $$ \alpha_G = \Large \frac{\frac{G\,m_p\,m_p}{r^2}}{\frac{e\,e}{4\pi\varepsilon_0\,r^2}} \normalsize \times \alpha \quad \mbox{with} \quad \alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c} \\ \mbox{Simplified:} \quad \alpha_G = \frac{G\,m_p^2}{\hbar c} $$ Here: $G=$ gravitational constant, $m_p=$ proton mass, $\hbar=$ reduced Planck constant, $c=$ lightspeed.
The result ${\bf \alpha_G}$ is a dimensionless constant, namely (force / force) x $\alpha$ : it is a pure number. Then the paper says: We propose the following three novel equations from which our cosmological model is deduced. Three equations. Let's check if there is numerical agreement between Patrick's findings and ours. A computer algebra system (MAPLE) is employed, thus avoiding errors.

c := 299792458; # Speed of light
G := 6.67408*10^(-11); # Gravitational constant
m_e := 9.1093837015*10^(-31); # Electron mass
m_p := 1.67262192*10^(-27); # Proton mass
h := 6.62607015*10^(-34); # Planck constant
h_bar := h/(2*Pi); # Reduced Planck constant
alpha_G := evalf(G*m_p^2/(h_bar*c)); # The coupling constant
lambda_e := h/(m_e*c); # Electron Compton length
t_e := h/(m_e*c^2); # Electron Compton time
t_0 := evalf(t_e/(Pi*alpha_G)); # First formula
year := 31556926; evalf(t_0/year/10^9);
R_h := c*t_0; # Second formula
mu := m_p/m_e;
M_u := mu^2*m_e/alpha_G^2; # Third formula
M_u := evalf(R_h*c^2/(2*G)); # The same
Mpc := 3.08567758*10^22; # Megaparsec
H_0 := 1/t_0 * Mpc/1000;
.......................................
                                            -38
                  alpha_G := 0.5905954709 10
.......................................

                      H_0 := 70.73997502 # km/s/Mpc

The universe as proposed by the author has the critical Schwarzschild radius: $$ R_h = \frac{2GM_u}{c^2} $$ All of our own results are in agreement with those by Patrick. And because of $\,R_h := c t_0\,$ and $\,M_u = R_h c^2/(2 G)\,$ we only have to check the first one: $t_0 = 4.361999816\times 10^{17}\;s$ , which is OK.
Worth noticing is Tonin's value for the Hubble constant $\,H_0\,$. This value falls between the Planck Collaboration 2018 estimate [4] of $H_0 = 67.37 \pm 0.54\; km\,s^{-1}Mpc^{-1}$ and the distance-ladder estimate from the SH0ES project 2019 [5] of $H_0 = 74.03 \pm 1.42 \; km\,s^{-1}Mpc^{-1}$. Another consequence is a well known formula involving the mean mass density $\,\rho_u\,$ of the universe (whatever that is supposed to mean): $$ R_h = \frac{2GM_u}{c^2} = \frac{2G}{c^2} \rho_u \frac{4}{3}\pi R_h^3 \\ 1 = \frac{2G}{c^2}\rho_u \frac{4}{3}\pi\left(\frac{c}{H_0}\right)^2 \quad ; \quad H_0^2 = \frac{8}{3}\pi\,G\,\rho_u \\ \boxed{ H_0 = \sqrt{\chi\,G\,\rho_u} \quad \mbox{with} \quad \chi = \frac{8}{3}\pi \quad (: \Lambda\mbox{CDM}) } $$ The advantage of the doubt is on hold at section 5, where the author writes: In order for Equation (1) to be valid as the Universe ages, either the gravitational coupling constant $\alpha_G$ or the electron Compton time $t_e$, or both, have to vary. Indeed Equation (1) and later on Equation (4) represents a problem, provided that the quantities involved are all true constants of nature. If $t_0$ is the age of the universe given by Equation (1), then that age cannot become larger now as time proceeds. Which is indeed absurd. However, a remedy by making quantities varying with time, as proposed by Patrick, is that going to work? According to the paper we have: $$ \alpha_G(t) \sim t^{-2} \quad (9)\; ; \quad m_e(t) \sim t^{-2} \quad (16)\; ; \quad G(t) \sim t^{-1} \quad (18)\; ; \quad h(t) \sim t^{-3} \quad (19) $$ Prediction 4: The mass and size of particles decrease with cosmic time. So likewise: $m_p(t) \sim t^{-2}$. It follows that (9) is OK: $$ \alpha_G = \frac{G\,m_p^2}{\hbar c} \sim \frac{t^{-1}t^{-2}t^{-2}}{t^{-3}} = t^{-2} $$ But how about the Fine-structure constant. $$ \alpha(t) = \frac{e^2}{4\pi\varepsilon_0 \hbar c} \sim t^{-3} $$ This seems to be contradictory to experience; empirical evidence has revealed that the fine structure constant is the same everywhere in the universe and does not vary. But I'm not sure, because nothing has been said about the time dependence of electrical and thermodynamical quantities.

It's interesting to make a comparison between $\alpha_G$ and the right hand side of the formula by Sahil Gupta.

The inverse of the latter can can be simplified to: $$ \frac{2G}{c^2}.m_p \times \frac{m_p.c.\pi/2}{h} = \frac{1}{2}\frac{G\,m_p^2}{\hbar\,c} = \alpha_G/2 $$ Numerical confirmation:

right := evalf((h/(m_p*c*Pi/2))/(2*G/c^2*m_p));
alpha_G := 2/right;
                                            -38
                  alpha_G := 0.5905954709 10

How about Sahil's left hand side? Referring to the paper by Patrick G. Tonin we find with equation (4) for the inverse of that number: $$ H_0 = \frac{c}{R_h} = \frac{\pi \alpha_G}{t_e} \quad \mbox{with} \quad t_e = \frac{h}{m_e c^2} \\ \frac{h}{m_e.c.2\pi} \times \frac{H}{c} = \frac{h}{m_e c^2}\frac{1}{2\pi} \times \frac{\pi \alpha_G}{t_e} = \alpha_G/2 $$ Conclusion: From The Quantum Bang Hypothesis by Patrick G. Tonin it follows that, in the formula by Sahil Gupta, left hand side and right hand side are exactly equal:

$= 2/\alpha_G$ $$ \boxed{\frac{1}{\mbox{left}} = \frac{1}{\mbox{right}} \; \mbox{(Gupta)} = \frac{\alpha_G}{2} \; \mbox{(Tonin)}} $$