Big Beautiful Book

A book

Cover illustration

John Haverlack

© 2025 — CC BY-ND 4.0

Author: John Haverlack

Copyright: © 2025 John Haverlack

License: CC BY-ND 4.0

Version: 0.0.1

Date: 2025-11-05

Revolving Circles optical illusion

“Revolving Circles.” n.d. Accessed October 31, 2025.
Wikipedia source

The cover image presents a visual illusion of motion. When you focus on the central point and move the page toward or away from your eyes, the concentric circles appear to rotate.

This reflects a central theme of this work: perception shapes what we think of as “reality.” In this illusion, motion exists only in our minds. We perceive, we do not directly know; our brains construct experience.

Introduction

“If I have seen further it is by standing on the shoulders of Giants.”

– Isaac Newton (“Isaac Newton Letter to Robert Hooke, 1675,” n.d.)

Conventions

In this book we’ll use a few conventions.

New Concepts

As many of the topics discussed in this book are a mix of established math and physics, proposed dualistic interpretations of established ideas, and also speculative ideas that I don’t yet know how to address, I wanted a way to clearly distinguish these concepts. I’ve come up with the following convention to highlight these classes of concepts to indicate their level of mainstream acceptance.

In this book, established concept may be highlighted in green, and represent mainstream physics or math concepts.

Established Concept

Einsteins Relativistic Dynamics Equations E2 = (m0 ⋅ c2)2 + (p ⋅ c)2

New ideas proposed by the author which have not been peer reviewed, verified or tested, and should be looked at with scrutiny.

Proposed Concept

With the speed of light, c = 1: E2 = m02 + p2

Speculative Idea, that the author wonders about, but does not know how to demonstrate, or ideas that need further treatment to prove or disprove.

Speculative Concept

With the speed of light, c = 1: E2 = m02 + p2

Chapter 1

Blah blah blah

Example Content

In Rν the Planck Length is the universal unit for measurement of distance, and is defined approximately to be: $$\boxed{L_P=\sqrt{\hbar}=5.72928\times10^{-35}m=1 L}$$ Where 1 L, is 1 Planck Length of distance.

SI Conversion Factors

The following conversion factors can be used to convert observable quantities of measure from the SI system of units to Rν to ~6 significant digits.

Conversion Factor Symbol Value
meters to Planck Length χP $1.74542\times10^{34} \frac{L}{m}$
seconds to Planck Length τp $5.23264\times10^{42} \frac{L}{s}$
mass to Planck Length GP $1.62871\times10^8 \frac{L}{kg}$
energy to Planck Length EP $1.81219\times10^9 \frac{L}{J}$
momentum to Planck Length PP $5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}$
temperature to Planck Length kP $2.501998\times10^{-14} \frac{L}{K}$
charge to Planck Length CP $1.89007\times10^{18} \frac{L}{C}$

Physical Constants

Applying conversion factors from the table above, we can convert SI values to Reduced Natural Units. For example, performing this analysis on the the speed of light yields a unit-less number with a value of 1:

$c = 299792458 \frac{m}{s} = 299792458 \frac{m}{s} \cdot 1.74542\times10^{34} \frac{L}{m} \cdot \frac{1}{5.23264\times10^{42} \frac{L}{s}} = 1.00000$

Quantity Symbol SI ν
Speed of Light c $299792458 \frac{m}{s}$ 1
Reduced Gravitational Constant G0 $8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}$ 1
Boltzmann’s Constant k $k=1.380649\times10^-23 \frac{J}{K}$ 1
Permittivity of Free Space ϵo $8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}$ 1
Permeability of Free Space μo $\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}$ 1
Reduced Planck’s Constant ℏ $1.054571726\times10^-34 \frac{kg \cdot m^2}{s}$ 1L2
Mass of the Electron me 9.10938 × 10−31kg 1.48366 × 10−22L
Charge of the Electron e− −1.60218 × 10−19C −3.02822 × 10−1L
Unit Cycle Θ 2π = 6.28318... Radians 1τ = 6.28318... Radians

Fine Structure Constant

As a consistency check, we compute the Fine Structure Constant using Reduced Natural Units which is a unit less ratio that should be independent of our system of units.

$\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{2\tau}=0.00729735≈\frac{1}{137}}$

Dimensional Analysis

The reader should be familiar with high school physics and chemistry dimensional analysis.

How many kilometers are in 1 mile? $1\ mile = 1\ mile \times \frac{5280\ ft}{mile} \times \frac{12\ in}{ft} \times \frac{2.54\ cm}{in}\times \frac{1\ m}{100 cm} \times \frac{1\ km}{1000 m} = \frac{5280 \times 12 \times 2.54}{100 \times 1000}\ km = \frac{160934.40}{100000}\ km = 1.6\ km$ Note that each unit in the denominator cancels with one if the numerator until we are left with only km.

Newton’s Law of Gravity

The force of gravity (Fg) between 2 masses, m1 and m2 separated by distance r is given by Newton’s Law of Gravity:

$F_{g} = G \frac{m_{1} m_{2}}{r^{2}}$

Where G, is the Gravitational Constant.

$G = 6.67430 \times 10^{-11}\ N\frac{m^2}{kg^2}$

The strength of gravitational force follow the inverse square law distributing gravitational flux over the surface area of a sphere (4πr2).

Inverse Square Law

Any source of a signal strength (S0) that radiates isotropically in 3-dimensional space will distribute that signal strength (S0) over the surface area of a sphere (SA = 4πr) of radius (r). Such that the intensity (I) at distance (r) is:

$$I(r) = \frac{S_0}{4 \pi r^{2}}=\frac{S_0}{2 \tau r^{2}}$$ inverse square law #### Rν Reduced Gravitational Constant In this version of Newton’s Law of Gravity we introduce a new constant G0, the reduced gravitational constant to accommodate for the factor of 4π = 2τ which is has been integrated in the SI version of the gravitational constant.

$F_g =G \frac{m_{1} m_{2}}{r^{2}}= G_0 \frac{m_{1} m_{2}}{4 \pi r^{2}}=G_0 \frac{m_{1} m_{2}}{2 \tau r^{2}}$

Where:

$G = \frac{G_{0}}{2\tau} = 6.67384 \times 10^{-11} \frac{N \cdot m^2}{kg^2}$

Analyzing the units: $$\frac{N \cdot m^2}{kg^2} = \left( \frac{\left( kg \cdot \frac{m}{s^2} \right) \cdot m^2}{kg^2} \right)=\frac{m^3}{s^2 kg}$$ Converting seconds to meters with the SI speed of light as a conversion factor: $$\frac{m^3}{s^2 kg}\cdot\frac{1}{c^2}=\frac{m^3}{s^2 kg}\cdot\frac{s^2}{m^2}=\frac{m}{kg}$$

Thus where space and time are measured in units of meters, the reduced gravitational constant, is:

$$\boxed{G_0=\frac{2\tau G}{c^2}=\frac{2\tau \cdot 6.67384 \times 10^{-11}}{299792458^2} \frac{m}{kg} = 9.33135 \times 10^-27 \frac{m}{kg}}$$

Observation This implies that not only can space an time be measure in units of meters, but so can mass.

Relativistic Energy Momentum Relation

Einsteins Relativistic Energy Momentum relationship shows a Pythagorean relation between the total energy (E), rest mass (m0) and momentum (p) of a system.

E2 = (m0 ⋅ c2)2 + (p ⋅ c)2

Where space and time are both measure in units of meters, c=1.

E2 = (m0)2 + (p)2

From this we can see that Energy, Momentum and Mass have equivalent units.

While we do not really know what energy, mass and momentum are we know that they are fundamentally “made” out of the same stuff because they have the same units.

Objects of mass at rest

For an object at rest with no momentum (p = 0) we see Einstein’s famous equations:

E = m0 ⋅ c2

Or, with c = 1, this is much simpler to understand. Energy = Mass

E = m0 ##### Zero mass objects moving at the speed of light And for objects with no mass, like photos, (m0 = 0):

E = pc

Or, with c = 1, this is much simpler to understand. Energy = Momentum

E = p

Planck’s Constant

The Reduced Planck constant , ħ, represents a conversion factor for relating the frequency, ω (in 2π radians per second), of a photon to the energy of that photon. This can easily be seen from the simple but profound relationship:

E = ℏω

Where:

ℏ = 1.054571726 × 10−34J ⋅ s

and

$J \cdot s = {kg}\cdot\frac{m^2}{s}$

Reduced Planck’s Constant $\hbar = \frac{h}{2\pi} = \frac{h}{\tau}$

Simplifying our units by converting time and mass to units of meters: $$\boxed{\hbar=1.054571726 \times 10^{−34} {kg}\cdot\frac{m^2}{s}\cdot\frac{G_0}{c}=3.282462\times10^{-69}m^2}$$

Which suggest that the Plank constant can be interpreted as an areas for which the square root of is suspiciously close to the Plank length:

$$\boxed{\sqrt{\hbar}=\sqrt{3.282462\times10^{-69}m^2}=5.72928\times10^{-35}m}$$

Planck Area

The Planck Area is the square of the Planck Length.

$l_{P}= \sqrt{\frac{\hbar G}{c^3}}$

and $l_{P}^{2}= \frac{\hbar G}{c^3}$

In Rν units both c and Go are 1.

$l_{P} = \sqrt{\hbar}$

and lP2 = ℏ ## Bekenstein’s Bound After having recently read Three Roads to Quantum Gravity by Lee Smolin, I now suspect the meaning of this areas is related to the Bekensteins Law as applied to a surface areas surrounding a mass. Where the thermodynamic entropy, S, is proportional to the the enclosed surface area, A.

$S=\frac{1}{4}\cdot\frac{A}{G\hbar}$

$S=\frac{k c^{3} A}{4 G \hbar}$

$S \le \frac{2\pi k R E}{\hbar c} = \frac{\tau R k E}{\hbar c}$

From our new values for G0and ℏ we can likely rewrite this:

$S=\frac{\pi\cdot A}{\hbar G_0}$

With the limiting case being at the Plank scale.

$S=\frac{\pi\cdot \sqrt{\hbar}}{\hbar G_0}$

Planck Length

https://en.wikipedia.org/wiki/Planck_length

The concept of the Planck Length comes from exploring the limits of Quantum Mechanics and General Relativity. The limits of General Relativity can be seen a the event horizon of a black hole, described by the Schwarzschild Radius. And the limits of Quantum Mechanics can be found in the Compton Wavelength for a given quanta.

The Schwarzschild Radius is defined as the distance at which light cannot escape from the gravitational field of a mass (m):

Classic Derivation.

$r_S=\frac{2G m}{c^2}$

The reduced Compton Wavelength represents a lower limit on the wavelength for quanta that can interact with a quantum particle with mass (m):

$\lambda_C=\frac{h}{m c}$

$\bar{\lambda_C}=\frac{2\pi\hbar}{m c}=\frac{\tau\hbar}{m c}$

And set the Schwarzschild Radius equal to the Compton Wavelength: rS = λC

$\frac{2Gm}{c^{2}}=\frac{h}{m c}$

$m^{2}= \frac{hc}{2G}$

$m = \sqrt{\frac{hc}{2G}}$

$l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^2}$ $l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^{2}}= \sqrt{\frac{2Gh}{c^2}}$

With reduced Compton Wavelength $r_S=\bar{\lambda_C}$

$\frac{2Gm}{c^{2}}=\frac{\tau\hbar}{m c}$

$m^2=\frac{\tau\ \hbar\ c}{2G}$

$m = \sqrt{\frac{\tau\ \hbar\ c}{2G}}$

$l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\ c}{2G}}}{c^2}$

$l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\ c}{2G}}}{c^{2}}=\sqrt{\frac{4\ \tau\ G\ \hbar}{c^3}}$

If we reduce the units in these equation to those of mass and time measured in meters.

$l_P=\sqrt{4\tau\hbar G_{o}}$

and

$\lambda_C=\frac{\hbar}{m}$

$m=R_s=\lambda_C=\frac{\hbar}{m}$

This is known as the Planck Mass, MP. $M_P=m=\sqrt{\hbar}$

Solving the Compton Wavelength for distance we find the classic Plank Length:

$\lambda_C=\frac{\hbar}{\sqrt{\hbar}}=\frac{\sqrt{\hbar}}{\sqrt{\hbar}}\cdot\frac{\hbar}{\sqrt{\hbar}}=\sqrt{\hbar}=L_P$

Which is in precise agreement with the value we found in above. Thus the Plank Length is:

$L_P=\sqrt{\hbar}=5.72928\times10^{-35}m$

When we measure distance, time, and mass in units of distance, c=1, and the Plank Time, TP, is equal to Plank Length, LP, which is equal to the Plank Mass, MP:

$$\boxed{L_P=T_P=M_P}$$

Conversion Factor Symbol Value
meters to Planck Length χP $1.74542\times10^{34} \frac{L}{m}$
seconds to Planck Length τp $5.23264\times10^{42} \frac{L}{s}$
mass to Planck Length GP $1.62871\times10^8 \frac{L}{kg}$
energy to Planck Length EP $1.81219\times10^9 \frac{L}{J}$
momentum to Planck Length PP $5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}$
temperature to Planck Length kP $2.501998\times10^{-14} \frac{L}{K}$
charge to Planck Length CP $1.89007\times10^{18} \frac{L}{C}$

Applying conversion factors from the table above, we can convert SI values to Reduced Natural Units. $c=\frac{1}{\sqrt{\epsilon_o \mu_o}}$

Quantity Symbol SI ν
Speed of Light c $299792458 \frac{m}{s}$ 1
Gravitational Constant G0 $8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}$ 1
Boltzmann’s Constant k $k=1.380649\times10^-23 \frac{J}{K}$ 1
Permittivity of Free Space ϵo $8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}$ 1
Permeability of Free Space μo $\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}$ 1
Planck’s Constant ℏ $1.054571726\times10^-34 \frac{kg \cdot m^2}{s}$ 1L2
Mass of the Electron me 9.10938 × 10−31kg 1.48366 × 10−22L
Charge of the Electron e− −1.60218 × 10−19C −3.02822 × 10−1L

Fine Structure Constant

https://en.wikipedia.org/wiki/Fine-structure_constant As a consistency check, we compute the Fine Structure Constant using Reduced Natural Units which is a unit less ratio that should be independent of our system of units.

$\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{4\pi}=0.00729735≈\frac{1}{137}}$

This check confirms that our system of Reduced Natural Units has internally consistent values for c, ϵo, ℏ and e−. And also Go which was used to computer prior values is also consistent.

Sage Code

Unit Analysis computations have been performed with Sage Math.

# Define constance
one = 1.n(digits=6)
pi = pi.n(digits=6)
tau = 2 * pi
t = tau

# Define the units
meters = var('m')
m = one*meters

seconds = var('s')
s = seconds

kilograms = var('kg')
kg = kilograms

newtons = kg * m / s^2
N = newtons

joules = N * m
J = joules

print("pi    =", pi)
print("tau   =", t)

# Speed of light in meters/second
speed_of_light = 299792458 * meters/seconds
sol = speed_of_light
c = sol
print("si c  =", c)

rnu_c = c / c
print("R\u03BD c  =", rnu_c)

# Gravitational Constant
gravitational_constant = 6.67384e-11 * N*(m^2/kg^2)
G = gravitational_constant
print("si G  =", G)

rnu_G = 4*pi*G/c^2
Go = rnu_G
print("R\u03BD Go =", Go)


# Planck's Constant
reduced_plancks_constant = 1.054571726e-34 * J*s
h_bar = reduced_plancks_constant
print("si \u210F  =", h_bar)

rnu_h_bar = h_bar * Go / c
print("R\u03BD "u"\u210F  =", rnu_h_bar)

# Planck Length
rnu_h_bar_str = str(rnu_h_bar) 
numerical_part_str = rnu_h_bar_str.split('*')[0]  
numerical_part_str = numerical_part_str.strip('()')
numerical_part = float(numerical_part_str)
rnu_sqrt_h_bar = numerical_part^(1/2)
# ^ Sage cannot process sqrt on units... Lame.
lP = rnu_sqrt_h_bar * m
print("R\u03BD \u221A\u210F =", lP)

Output

pi    = 3.14159
tau   = 6.28319
si c  = 299792458*m/s
Rν c  = 1
si G  = (6.67384e-11)*m^3/(kg*s^2)
Rν Go = (9.33135e-27)*m/kg
si ℏ  = (1.05457e-34)*kg*m^2/s
Rν ℏ  = (3.28246e-69)*m^2
Rν √ℏ = (5.72928e-35)*m
si lP = (1.61620e-35)*sqrt(m^2)
Rν lP = (2.77455e-47)*sqrt(m^3/kg)

Terminology

Citations

“Isaac Newton Letter to Robert Hooke, 1675.” n.d.