Special Algebra for Special Relativity: Second Edition: Proposed Theory of Non-Finite Numbers

328 Pages·2·4.938 MB·English

by  Paul C Daiber

SPECIAL
ALGEBRA
FOR
SPECIAL
RELATIVITY

SECOND EDITION
2
SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

The other book by Paul C Daiber:
ALIEN INVASION MATH STORY
SPECIAL
ALGEBRA
FOR
SPECIAL
RELATIVITY

Proposed Theory of Non-Finite Numbers

Paul C Daiber

SECOND EDITION
4
SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Copyright © 2020 by Paul C Daiber
All rights reserved. This book or any portion thereof may not be reproduced or
used in any manner whatsoever without the express written permission of the
publisher except for the use of brief quotations in a book review, scholarly
journal or other critical document.
Daiber, Paul C, 1960 –
Special Algebra for Special Relativity
p. cm.
Includes index.
Paperback ISBN (see book cover)
1. Special Relativity, Electricity, Waves, Algebra, Mathematics, Infinity, Math,
Abstraction I. Title.

A special thank you to amazon for making it so simple to publish a book
For My Wife Sue

Table of Contents

Finite Precision for Numbers ........................................................................ix
Chapter 1 – Numbers .................................................................................. 1
1.1 Process from Descartes ......................................................................... 1
1.2 Geometric-Vectors ................................................................................ 2
1.3 Quaternions ......................................................................................... 4
1.4 Translation Back to Geometry.............................................................. 19
1.5 Singular-Label-Numbers ...................................................................... 20
1.6 Exercises ............................................................................................ 21
Chapter 2 – Particles ................................................................................. 33
2.1 Hypercomplex-Plane ........................................................................... 33
2.2 Inertial Reference Frames ................................................................... 36
2.3 The Unspecified-Speed-Parameter ....................................................... 38
2.4 Compound-Label-Numbers and Components ........................................ 39
2.5 Adding Hyperbolic-Angles .................................................................... 42
2.6 Energy, Time Dilation, Length Contraction ........................................... 46
2.7 Space-Like and Time-Like Invariants .................................................... 48
2.8 Electric Current Density ....................................................................... 52
2.9 Motion Faster than Light ..................................................................... 55
2.10 Anti-Matter ........................................................................................ 64
2.11 Distributed Material Theory ............................................................... 72
2.12 Exercises .......................................................................................... 83
Chapter 3 – Fields ..................................................................................... 89
3.1 Geometric-Vector Notation .................................................................. 89
3.2 All-Number Notation ........................................................................... 94
3.3 Gauges and Super-Potentials ............................................................. 105
3.4 Lorentz Transformation ..................................................................... 108
3.5 Biot-Savart Law ................................................................................ 116
3.6 Electric Energy-Momentum of an Electron .......................................... 119
3.7 Maxwell’s Wave Equation .................................................................. 126
3.8 Forces Using Geometric-Vector Notation............................................. 132
3.9 Force Density Invariant ..................................................................... 133
3.10 Area and Volume Differential Operators ............................................ 143
3.11 Exercises ........................................................................................ 151
Chapter 4 – Waves .................................................................................. 159
4.1 Differential Operator ......................................................................... 159
4.2 Development of the Dirac Equation .................................................... 162
4.3 Solutions to the Dirac Equation .......................................................... 166
4.4 Particle Properties ............................................................................. 169
4.5 Two Alternative Arrangements ........................................................... 173
4.6 Lorentz Transformation of a Dirac Spinor ........................................... 175
4.7 Exercises .......................................................................................... 181
Chapter 5 – Proposed Theory ................................................................... 187
5.1 Local-Real Numbers .......................................................................... 187
5.2 Cantor’s Theory of Infinite Sets.......................................................... 196
5.3 Algebra Field for Local-Real Numbers ................................................. 204
5.4 Lorentz Transformation with Non-Finite Numbers ............................... 209
5.5 Dirac Equation Development.............................................................. 229
5.6 Force Density Using the Complex-Conjugate ...................................... 237
5.7 Spin of a Photon ............................................................................... 245
5.8 Exercises .......................................................................................... 248
Appendix A – Octonions and Sedonions .................................................... 257
Appendix B – Spooky Action at a Distance ................................................ 277
Appendix C – Discovering an Abstraction .................................................. 289
The Storybook ........................................................................................ 295
Glossary ................................................................................................. 296
Index ..................................................................................................... 307
Back Cover ............................................................................................. 310
Preface

The first four chapters of Special Algebra for Special Relativity present an
all-number mathematical structure for Special Relativity. The fifth
chapter restricts a measurable quantity to finite precision by limiting
place-value digits to a maximum before and after the decimal point.
For example, a physically real square of unit length on a side has
only finite precision for each unit length and, likewise, the “2” diagonal
also has only finite precision. The ideal of unit length by use of the
integer one is not geometrically possible in the physical world, and
neither is the ideal of a perfectly precise “2” irrational number.
The finite imprecision larger than the measurable rational number
is the division reciprocal of the finite imprecision smaller than the rational
number. In Special Relativity the larger and smaller imprecision are
added to the time-space hyperbolic angle “” (that relates to speed by
“v = c*tanh”) using a Lorentz Transformation. The small magnitude
imprecision is trivial. The large magnitude imprecision models
electromagnetism because Maxwell’s Equations are derived from the
Dirac Equation. Precision improves with time.
The mathematics for predicting electric current density from the
Dirac Spinor results in the electromagnetic field force density invariant,
and it includes the empirically derived energy density and Poynting
Vector. The union of those empirically derived models is new and
suggests quantities in our geometric world actually do have finite
precision, and that finite precision numbers apply to the more modern
theories of physics.
ix
SYNOPSIS – FINITE PRECISION FOR NUMBERS

Finite Precision for Numbers

The proposed Theory of Non-Finite Numbers assigns finite precision to
numbers. The count of place-value digits in a number, both before and
after the decimal point, is limited to Aristotle’s potential infinity, a finite
number. The remaining finite imprecise portion conforms to probability
theory and is the non-finite number that applies to existing mathematical
models of physics.
The example application is the hundred-year-old Dirac Equation.
A Lorentz Transformation that adds a finite imprecision term both larger
and smaller than the finite rational number time-space hyperbolic angle
results in Maxwell’s Equations and a uniting of the measured Poynting
Vector and electromagnetic energy density into the measured force
density invariant.
That unity of different measured electromagnetic phenomena into
one theoretical model is something new, and it suggests numbers as
quantities in our physical world actually do have finite precision. The
same technique of assigning finite precision can next be applied to more
modern mathematical models of physics.

   

The algebra field for rational numbers requires

• Any sum or product of two rational numbers be a rational number,
as well as negatives and reciprocals (except division by zero)
• Zero and one be included as identity elements
• Commutative and associative properties for addition and
multiplication as well as the distributive property of multiplication
over addition

An irrational number, for example “2” or “log 3”, can be included
2
in the rational numbers’ algebra field because none of the above criteria
are violated.
To prove “2” is irrational, set “2 = p/q” from which “p2 = 2*q2”,
from which “p” must be even and “q” must be even, and that observation
is inconsistent with the ability of either “p” or “q” to be odd because both
x
SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
can be divided by “2” until one is odd. The inconsistent observations
mean no ratio “p/q” of natural numbers “{1, 2, 3, …}” possibly equals
“2”. To prove “log 3 = p/q” is irrational, derive “2^(log 3) = 2^(p/q)” and
2 2
then “2p = 3q” and observe no natural numbers “{1, 2, 3, …}” for “p” and
“q” apply because an even number “2p” cannot equal an odd number “3q”.
The irrational numbers with the rational numbers form the set of
real numbers, and, therefore, the algebra field as defined for rational
numbers also applies to real numbers.
Cantor defined real numbers in the late 1800’s by stating real
numbers had a quantity “2^N ” over any finite or infinite (N ) interval.
0 0
“N ” (called “aleph null”) was forced to be a positive actual infinity
0
through his Continuum Hypothesis: No set has a quantity between “N ”
0
and “2^N ”.
0
Attempt to prove “2” and “log 3” are included in Cantor’s set of
2
real numbers. If “2” equals the ratio of two infinity numbers, both are
“2^N ” and so, maybe, both are even and cannot be odd, and therefore
0
it appears “2” is excluded. If “log 3” equals a ratio of two infinity
2
numbers, then “2^(2^N )” equates to “3^(2^N )”, but there is no algebra
0 0
for “3^(2^N )” because the later developed algebra field theory applies
0
only to finite numbers. Rather than conclude “2” and “log 3” are not
2
included in Cantor’s set of real numbers, we wonder if we are mistakenly
forcing “N ” to be finite in the proofs.
0
Cantor’s general approach of organizing numbers into sets was
structured into Axiomatic Set Theory. The Axiom of Infinity addressed
only Aristotle’s potential infinity, a finite number, and not a positive actual
infinity. To insert an actual infinity into Axiomatic Set Theory, Cantor’s
Continuum Hypothesis was added as another axiom, effectively.
Also, Axiomatic Set Theory had no axiom that addressed the
reciprocal of zero. Propose a reciprocal-of-zero axiom to specify what
calculations are not possible: No operation that includes a reciprocal of
the integer zero can result in a finite number. Per the proposed axiom,
“0/0” and “1/0 - 1/0” are not permitted operations because they cannot
result in a finite number. The positive or negative feature of a zero is
not specified, and therefore “1/0 + 1/0” has no result and does not
necessarily equal “2*(1/0)”. The operations “1/0 + 7 = 1/0” and “(1/0)*7 =
1/0” are accepted. Also “2^(1/0) = 3^(1/0) = 0 or 1/0”. “1/0” is the absolute
maximum magnitude of numbers and is both or either positive and
negative.