Exploring Physics with Geometric Algebra, Book II., , C December 2016 COPYRIGHT
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peeter joot [email protected] EXPLORINGPHYSICSWITHGEOMETRICALGEBRA,BOOKII. EXPLORINGPHYSICSWITHGEOMETRICALGEBRA,BOOKII. peeter joot [email protected] December 2016 – version v.1.3 Peeter Joot [email protected]: Exploring physics with Geometric Algebra, Book II., , c December 2016 COPYRIGHT Copyright c 2016 Peeter Joot All Rights Reserved This book may be reproduced and distributed in whole or in part, without fee, subject to the following conditions: • The copyright notice above and this permission notice must be preserved complete on all complete or partial copies. • Any translation or derived work must be approved by the author in writing before distri- bution. • If you distribute this work in part, instructions for obtaining the complete version of this document must be included, and a means for obtaining a complete version provided. • Small portions may be reproduced as illustrations for reviews or quotes in other works without this permission notice if proper citation is given. Exceptions to these rules may be granted for academic purposes: Write to the author and ask. Disclaimer: I confess to violating somebody’s copyright when I copied this copyright state- ment. v DOCUMENTVERSION Version 0.6465 Sources for this notes compilation can be found in the github repository https://github.com/peeterjoot/physicsplay The last commit (Dec/5/2016), associated with this pdf was 595cc0ba1748328b765c9dea0767b85311a26b3d vii Dedicated to: Aurora and Lance, my awesome kids, and Soﬁa, who not only tolerates and encourages my studies, but is also awesome enough to think that math is sexy. PREFACE This is an exploratory collection of notes containing worked examples of more advanced appli- cations of Geometric Algebra (GA), also known as Cliﬀord Algebra. These notes are (dis)organized into the following chapters • Relativity. This is a fairly incoherent chapter, including an attempt to develop the Lorentz transformation by requiring wave equation invariance, Lorentz transformation of the four- vector (STA) gradient, and a look at the relativistic doppler equation. • Electrodynamics. The GA formulation of Maxwell’s equation (singular in GA) is devel- oped here. Various basic topics of electrodynamics are examined using the GA toolbox, including the Biot-Savart law, the covariant form for Maxwell’s equation (Space Time Algebra, or STA), four vectors and potentials, gauge invariance, TEM waves, and some Lienard-Wiechert problems. • Lorentz Force. Here the GA form of the Lorentz force equation and its relation to the usual vectorial representation is explored. This includes some application of boosts to the force equation to examine how it transforms under observe dependent conditions. • Electrodynamic stress energy. This chapter explores concepts of electrodynamic energy and momentum density and the GA representation of the Poynting vector and the stress- energy tensors. • Quantum Mechanics. This chapter includes a look at the Dirac Lagrangian, and how this can be cast into GA form. Properties of the Pauli and Dirac bases are explored, and how various matrix operations map onto their GA equivalents. A bivector form for the angular momentum operator is examined. A multivector form for the ﬁrst few spherical harmonic eigenfunctions is developed. A multivector factorization of the three and four dimensional Laplacian and the angular momentum operators are derived. • Fourier treatments. Solutions to various PDE equations are attempted using Fourier series and transforms. Much of this chapter was exploring Fourier solutions to the GA form of Maxwell’s equation, but a few other non-geometric algebra Fourier problems were also tackled. I can not promise that I have explained things in a way that is good for anybody else. My audience was essentially myself as I existed at the time of writing (i.e. undergraduate engineer- ing background), but the prerequisites, for both the mathematics and the physics, have evolved continually. xi Peeter Joot [email protected] xii CONTENTS Preface xi i relativity 1 1 wave equation based lorentz transformation derivation 3 1.1 Intro3 1.2 The wave equation for Electrodynamic ﬁelds (light)4 1.3 Verifying Lorentz invariance4 1.4 Derive Lorentz Transformation requiring invariance of the wave equation5 1.4.1 Galilean transformation7 1.4.2 Determine the transformation of coordinates that retains the form of the equations of light7 1.5 Light sphere, and relativistic metric 10 1.6 Derive relativistic Doppler shift 10 2 equations of motion given mass variation with spacetime position 11 2.1 11 2.1.1 Summarizing 16 2.2 Examine spatial components for comparison with Newtonian limit 16 3 understanding four velocity transform from rest frame 19 3.1 19 3.1.1 Invariance of relative velocity? 21 3.1.2 General invariance? 22 3.2 Appendix. Omitted details from above 23 3.2.1 exponential of a vector 23 3.2.2 v anticommutes with γ0 24 4 four vector dot product invariance and lorentz rotors 25 4.1 25 4.1.1 Invariance shown with hyperbolic trig functions 26 4.2 Geometric product formulation of Lorentz transform 26 4.2.1 Spatial rotation 27 4.2.2 Validity of the double sided spatial rotor formula 27 4.2.3 Back to time space rotation 29 4.2.4 FIXME 31 4.2.5 Apply to dot product invariance 31 5 lorentz transformation of spacetime gradient 33 xiii xiv contents 5.1 Motivation 33 5.1.1 Lets do it 33 5.1.2 transform the spacetime bivector 35 6 gravitoelectromagnetism 37 6.1 Some rough notes on reading of GravitoElectroMagnetism review 37 6.2 Deﬁnitions 37 6.3 STA form 38 6.4 Lagrangians 39 6.4.1 Field Lagrangian 39 6.4.2 Interaction Lagrangian 40 6.5 Conclusion 41 7 relativistic doppler formula 43 7.1 Transform of angular velocity four vector 43 7.2 Application of one dimensional boost 45 8 poincare transformations 47 8.1 Motivation 47 8.2 Incremental transformation in GA form 47 8.3 Consider a speciﬁc inﬁnitesimal spatial rotation 50 8.4 Consider a speciﬁc inﬁnitesimal boost 53 8.5 In tensor form 54 ii electrodynamics 55 9 maxwell’s equations expressed with geometric algebra 57 9.1 On diﬀerent ways of expressing Maxwell’s equations 57 9.1.1 Introduction of complex vector electromagnetic ﬁeld 57 9.1.2 Converting the curls in the pair of Maxwell’s equations for the electro- dynamic ﬁeld to wedge and geometric products 59 9.1.3 Components of the geometric product Maxwell equation 59 10 back to maxwell’s equations 61 10.1 61 10.1.1 Regrouping terms for dimensional consistency 62 10.1.2 Refactoring the equations with the geometric product 63 10.1.3 Grouping by charge and current density 64 10.1.4 Wave equation for light 64 10.1.5 Charge and current density conservation 65 10.1.6 Electric and Magnetic ﬁeld dependence on charge and current den- sity 66 10.1.7 Spacetime basis 66 10.1.8 Examining the GA form Maxwell equation in more detail 70 contents xv 10.1.9 Minkowski metric 71 11 macroscopic maxwell’s equation 73 11.1 Motivation 73 11.2 Consolidation attempt 73 11.2.1 Continuity equation 75 12 expressing wave equation exponential solutions using four vectors 77 12.1 Mechanical Wave equation Solutions 77 13 gaussian surface invariance for radial field 81 13.1 Flux independence of surface 81 13.1.1 Suggests dual form of Gauss’s law can be natural 83 14 electrodynamic wave equation solutions 85 14.1 Motivation 85 14.2 Electromagnetic wave equation solutions 85 14.2.1 Separation of variables solution of potential equations 85 14.2.2 Faraday bivector and tensor from the potential solutions 86 14.2.3 Examine the Lorentz gauge constraint 87 14.2.4 Summarizing so far 88 14.3 Looking for more general solutions 89 14.3.1 Using mechanical wave solutions as a guide 89 14.3.2 Back to the electrodynamic case 90 14.4 Take II. A bogus attempt at a less general plane wave like solution 91 14.4.1 curl term 92 14.4.2 divergence term 92 14.5 Summary 93 15 magnetic field between two parallel wires 95 15.1 Student’s guide to Maxwell’s’ equations. problem 4.1 95 15.1.1 95 15.1.2 Original problem 96 16 field due to line charge in arc 99 16.1 Motivation 99 16.2 Just the stated problem 99 16.3 Field at other points 100 16.3.1 On the z-axis 101 17 charge line element 103 17.1 Motivation 103 17.2 Calculation of electric ﬁeld for non-inﬁnite length line element 103 18 biot savart derivation 107 18.1 Motivation 107 xvi contents 18.2 Do it 107 18.2.1 Setup. Ampere-Maxwell equation for steady state 107 18.2.2 Three vector potential solution 108 18.2.3 Gauge freedom 109 18.2.4 Solution to the vector Poisson equation 109 19 vector forms of maxwell’s equations as projection and rejection oper- ations 111 19.1 Vector form of Maxwell’s equations 111 19.1.1 Gauss’s law for electrostatics 111 19.1.2 Gauss’s law for magnetostatics 112 19.1.3 Faraday’s Law 113 19.1.4 Ampere Maxwell law 114 19.2 Summary of traditional Maxwell’s equations as projective operations on Maxwell Equation 115 20 application of stokes theorem to the maxwell equation 117 20.1 Spacetime domain 117 20.2 Spatial domain 123 21 maxwell equation boundary conditions 127 21.1 Free space 127 21.2 Maxwell equation boundary conditions in media 130 21.3 Problems 133 22 tensor relations from bivector field equation 137 22.1 Motivation 137 22.2 Electrodynamic tensor 137 22.2.1 Tensor components 138 22.2.2 Electromagnetic tensor components 139 22.2.3 reciprocal tensor (name?) 140 22.2.4 Lagrangian density 140 22.2.5 Field equations in tensor form 145 22.2.6 Lagrangian density in terms of potential 147 23 four vector potential 149 23.1 149 23.1.1 150 23.1.2 Dimensions 150 23.1.3 Expansion in terms of components.