Foreword - The Pythagorean Theorem and Physics

1
Foreword
1

 
2
General Comments and Facts
2

 
2.1
General Comments
2

 
2.2
Facts
2

 
2.3
Kepler's Laws (1609)
2

 
3
Overview
3

 
4
Structure of Gravity
7

 
 

Manuscript may be viewed in its entirety or by chapter and section as shown below.

Table of Contents
iii

 
Preface
ix

 
1
GIBBS ALGEBRA AND CLIFFORD ALGEBRA
1

 
2
PERIHELION ADVANCE OF MERCURY
7

2.1
Calculation of the Period of Revolution of Mercury
8

 
2.2
Relativistic Effect on the Period
10

 
2.3
Orbital Period in a Schwarzschild metric
11

 
2.4
Summary of the Sources of v2/c2 to Obtain a Total Value of 6v2/c2 Seconds of Arc per Century for the Advance of the Perihelion of Mercury
12

 
 
3
ASSOCIATIVE OR DIRECT PRODUCT. ROTATION OF VECTORS
13

3.1
Definition of Associative or Direct Product
13

 
3.2
Law of Cosines
14

 
3.3
Law of Sines
15

 
3.4
Clifford Algebra in 3-Dimensions
15

 
3.5
Rotation of Vectors
18

 
3.6
Rotation about an Arbitrary Axis n
21

 
 
4
SPACE-INVARIANT. SPACE-TIME INTERVAL INVARIANT
25

4.1
Space-time Formalism to Special Relativity. Space-time Interval Invariant
25
 
4.2
Space-Time Interval. Space-Time "Rotation"
28
 
4.3
Acceleration and Force in Special Relativity
32
 
4.4
4-Velocity, 4-Acceleration
34
 
4.5
Orthogonality of A and V
36
 
4.6
Evaluation of A and A2 when the Path is Curved
37
 
4.7
Equation of Motion
40
 
4.8
Velocity as a Function of t
41
 
4.9
Total Work Done on Charge by Electric Field or Gravitational Field
43
 
4.10
Equation of Motion of a Non-Rotating Mass Pulled Into a Black Hole (Flat Space)
44
 
4.11
Transformation of Electromagnetic Bivector under Lorentz Transformation
46
 
4.12
Summary
48
 
 
5
MAXWELL'S EQUATIONS
49

5.1
Derivation of Maxwell's Equations using Space-time Algebra Format
49
 
5.1.1
Summary
56
 
5.2
Comments on the Source Current J
59
 
5.3
Derivation of Maxwell's Equations in a Schwarzschild Metric
60
 
5.4
Schwarzschild Metric
60
 
5.5
Addendum: Laplacian Gradient in Schwarzschild Metric
69
 
5.6
Standard 3-D Spherical Coordinates
70
 
5.7
Evaluation of  
71
 
 
6
REVERSION
81

6.1
Evaluation of     Assuming Complex Coefficients for the Multivectors in  
83
 
6.2
Evaluation of     Complex Coefficients
84
 
6.3
Vector Terms
86
 
6.4
Vector Terms. Real.
88
 
6.5
Bivector Terms
88
 
6.6
Bivector Terms. Real.
90
 
6.7
Trivector Terms. Trivector terms are imaginary
91
 
6.8
Quadvector Terms. Imaginary.
92
 
6.9
Quadvector Term. Pure Imaginary
92
 
6.10
Scalar Terms. Scalar terms are imaginary
92
 
6.11
Summary
93
 
 
7
MOTION IN A SCHWARZSCHILD METRIC
95

7.1
General Motion in a Schwarzschild Metric
95
 
7.2
Planar Planetary Motion in a Schwarzschild Metric
98
 
7.3
Conclusion
102
 
 
8
MULTIVECTOR FORM OF THE BIOT-SAVART LAW AND FARADAY'S LAW
103

8.1
Biot-Savart Law
104
 
8.2
Faraday's Law
105
 
8.3
Multivector Forces
107
 
8.4
Indirect Force
109
 
8.5
Definition of Contraction
109
 
8.6
Vector Identities in Clifford and Gibbs Algebra. Evaluation of (BsVt - VtBs)/2
112
 
8.7
Coulomb's Law and Faraday's Law
114
 
8.8
Gravity
114
 
8.9
Evaluation of Term II in Eq. (8.16)
116
 
8.10
Evaluation of Term I in Eq. (8.16)
122
 
8.11
Contraction of the Force P's on P't
123
 
8.12
Summary
127
 
8.13
Deflection of Light in the Earth's Gravitational Field
131
 
8.14
Gravitational Redshift
132
 
8.15
Tables of Multivector "Charges"
132
 
8.15.1
Generalization, Property Charges, Higgs Boson
134
 
8.15.2
Multivector Tables
135
 
8.16
Tables of Property Algebras
137
 
 
9
EVALUATION OF LINEAR AND ANGULAR MOMENTUM INTEGRALS
157

9.1
Linear Momentum Imparted to the Field by qsvs and qtvt
158
 
9.2
Evaluation of Integrals to Obtain Equations (9.4) and (9.5)
161
 
9.3
Part III. Gravity
175
 
9.4
Summary. Direct Torques
176
 
9.5
Indirect Torques
176
 
9.6
Details Describing Conversion to Dimensionless Variables
177
 
 
10
CALCULATION OF DIRECT AND INDIRECT FORCES AND TORQUES
179

10.1
Action and Reaction Between Two Moving Charges. Gravitational-Orbital Motion
179
 
10.2
Verification that Action and Reaction are Not Equal and Opposite in the Faraday Law
180
 
10.3
1. Magnetic Force
180
 
10.4
2. Coulomb Force
181
 
10.5
3. Indirect Force (Electromagnetic Force)
182
 
10.6
Possible Alternate Grouping, Not Adopted
189
 
10.7
Angular Momentum and Torques
191
 
10.8
Evaluation of     and    
194
 
 
11
EVALUATION OF THE TERM  

197

11.1
Reactive Force on a Moving Mass
197
 
11.2
Value of G Resulting from the Circular Motion of a Single Charge
201
 
 
12
FIELD PRODUCED BY A CONSTANT VELOCITY CHARGE

203

 
13
ALTERNATE DERIVATION OF THE ANGULAR MOMENTUM EQUATION

209

13.1
Alternate Derivation of the Angular Momentum Equation for Interacting Particles
209
 
13.2
Proca Equations as Modified Maxwell Equations
214
 
 
14
VERIFICATION THAT THE NON-COMMUTATIVE FIELD FORCE IS ZERO
219

14.1
Vector Forces Associated with the Transport of the Multivectors of Space-time Algebra
219
 
14.2
Indirect Force
220
 
14.3
Direct Force
223
 
14.4
Values for Vt on Vs to Obtain the Indirect Force
224
 
14.5
Detailed Evaluation of the Vector Direct Force
227
 
14.6
Evaluation of the Indirect Force (cont.)
229
 
14.7
Summary
232
 
 
15
MULTIVECTOR FORCES. TRIVECTOR, NEGATIVE GRAVITY FORCE.
237

15.1
Rotation Curve
237
 
15.2
Multivector Interactions
239
 
15.2.1
Direct Force
239
 
15.2.2
Indirect Force
242
 
15.3
General Formulation of the Forces Generated by qsvs and qtvt
244
 
15.3.1
Contraction of Terms
247
 
15.3.2
Vector Force
248
 
15.4
Contraction (a repeat)
250
 
15.4.1
Indirect Force
250
 
15.5
Summary
253
 
15.6
Mixed Multivector Interactions
258
 
 
16
SCHWARZCHILD METRIC
259

16.1
Derivation of the Schwarzschild Metric
259
 
16.2
Space-time Schwarzschild Metric
263
 
16.3
Schwarzschild Metric in Curved Space. 3 Dimensional Hypersphere
264
 
16.4
Comments on the de Sitter Metric
266
 
 
17
LIGHT IN A SCHWARZCHILD METRIC. PLANCK VALUES.
269

17.1
Velocity of Light in Various Directions in a Schwarzschild metric
269
 
17.2
Planck Values for M, L, T and Other Physical Quantities
270
 
 
18
FACTOR OF i IN TABLES.
275

18.1
On the Use of the Tables in Chapter 8
275
 
 
19
PARITY. DIRECT PRODUCT OF TWO TRIVECTORS
277

19.1
Electro-Weak "Force": 3D Formulation of the Biot-Savart Law
277
 
19.2
Associative Product of Trivector T and Trivector T'
286
 
19.3
Parity
287
 
19.4
Remarks on Inversion
289
 
 
20
MIXED MULTIVECTORS
293

20.1
Mixed Multivector Interactions: Preliminary
293
 
 
21
NEGATIVE GRAVITATION: SUPPLEMENTARY REMARKS
297

21.1
Direct Product of 2 Space-time Multivectors
297
 
21.2
Vector, Trivector and Contraction. Alternate Discussion
298
 
 
22
APPENDIX: ELECTROMAGNETISM. SOME BASICS
301

22.1
Review of a Few Introductory Topics in Electromagnetism.
301
 
22.2
Magnetic Field at Distance a from a Long Wire Carrying a Current, Is.
306
 
22.3
Kepler's Three Laws of Planetary Motion
309
 
 
23
References
311