Single-Particle Electrodynamics - Assassination Science

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correct. We must also multiply all expressions by the accelerative redshift factor λ(r), which for each of our two charges q 1 and q 2 is simply given by λ 1 = 1 + 1 ε(σ· ˙v), 2 λ 2 = 1 − 1 ε(σ· ˙v). (4.13) 2 For the computation of the power expression, we note that the velocities of the charges (in the dipole rest frame) are already of order ε; the factor of ε −1 in the values ±d/ε of the charges cancel this factor of ε; hence, there are therefore no “spare” factors of ε −1 left which could couple to the latter terms of (4.13). Hence, the power expression is given by the rest-frame value of (4.10) found above: P | v=0 = ( ˙ d ·E). (4.14) The rigorously relativistic computation of the force expression, however, is somewhat more subtle. Here, we find that the electric force on each charge may couple with the second factor of (4.13): ( F extra = ± d ) ( ± 1 ) ε 2 ε(σ· ˙v) E, where the upper (lower) signs apply to charge q 1 (q 2 ). We thus find an extra redshift force, F redshift = (d· ˙v)E. Thus, the total rest-frame force on the electric dipole is in fact given by F | v=0 = (d ·∇)E + (d· ˙v)E + ˙ d ×B. Now, in the interests of honesty, the author must confess that this extra “redshift force” was only discovered by him a few days before the final printing and binding of this thesis. It has ramifications for the radiation reaction calculations of Chapter 6; fortunately, the author was able to insert this additional force (and its corresponding “moment-arm” torque) into the program 125
adreact of Appendix G; the expressions found there and in Chapter 6, and the author’s comments on them in Chapter 6, have been updated to reflect the resulting changes to the radiation reaction equations of motion. In fact, this additional contribution simplified the final equations of motion somewhat; and it also removed one term that was clearly unphysical, which had been causing the author serious concern. Due to this increase in physical correctness of the radiation reaction calculations, the author most definitely considers the redshift force to be physically correct; he only wishes he had obtained it earlier. However, to the author’s knowledge, this redshift force has not been obtained in the literature to date; but then again neither has the accelerative redshift factor apparently been used at all, for the purposes of electrodynamical considerations. If the radiation reaction calculations yielded a bonanza with the redshift force, the published paper [65] listed in Appendix F did not fare quite so well. The derivation therein is now incomplete; and therefore the final “uncoupled” equations of motion, listed there, are incorrect. The reason for the author failing to obtain the redshift force from the manifestly covariant analysis presented there will be discussed in somewhat more detail in Section 4.3.4; basically, the author inappropriately used the covariant proper-time derivative, rather than the partial proper-time derivative, on one occasion: the difference between the two leads, somewhat remarkably, to the extra redshift force derived above! We now turn to the relativistically rigorous derivation of the torque equation of motion, in the rest frame. In this case, we note that the quantity z that is crossed into the force expression to obtain the “moment-arm” torque is in fact of order ε, so, as with the power equation of motion, there are no terms of order ε −1 remaining that could couple to the second terms in the redshift factors (4.13). Hence the rest-frame torque expression is still that given by (4.11) N| v=0 = d×E. (4.15) 126
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Single-Particle Electrodynamics Mr.
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Copyright Declaration I understand
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I apologise to Dr. Geoff Taylor for
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Contents 1 Overview of this Thesis
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3.2.2 Einsteinian rigidity . . . .
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6 Radiation Reaction 220 6.1 Introd
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A Notation and Conventions 314 A.1
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A.8.18 MCLF and CACS components . .
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E.2 Quantum field theory . . . . .
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G.7 test3int: Testing of 3-d integr
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lood pressure: such inane things ar
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1.2 What do I need to read? If the
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Section 2.7: Classical limit of qua
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Section 5.2: History of the retarde
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Section 6.7: Inverse-cube integrals
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Section A.7: Matrices Notation and
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Appendix C: Supplementary Proofs Se
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Section E.2: Quantum field theory T
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esults, not the establishment of ne
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e forgiven by their respective auth
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tron. The proton and neutron clearl
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most appropriate in the task of rem
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(see Section A.8): f ≡ d τ p. (2
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T 0i ≡ T i0 = E×B, T ij = 1 2 (E
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densities and the conservation laws
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2.4 Lagrangian mechanics An alterna
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canonical momentum: b a ≡ ∂ ˙q
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to be the appropriate degrees of fr
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task. Firstly, we need to select ap
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canonical energy, canonical momentu
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Let us return to the electric charg
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should also be carefully noted agai
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For an elementary particle, the mas
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2.6.7 Unit four-spin It is often ne
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the case. The reason, pointed out b
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period it appeared that the magnitu
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tionally reflect the difference bet
Page 75 and 76: has precessed by an amount ˙σ dt.
Page 77 and 78: is in terms of the quantities of Sp
Page 79 and 80: 2.6.10 The FitzGerald three-spin In
Page 81 and 82: such systems. On the other hand, fo
Page 83 and 84: equations, one then finds, for eith
Page 85 and 86: i.e., the torque. Clearly, the corr
Page 87 and 88: The component ∆L 0 is our warning
Page 89 and 90: vanishes, for v = 0: P | v=0 = 0; (
Page 91 and 92: wish of them—not least of which b
Page 93 and 94: The author will, in the remaining c
Page 95 and 96: (see equation (A.58) of Section A.8
Page 97 and 98: 6(c 1·c 3 ) + 4c 2 2 = 0, 8(c 1·c
Page 99 and 100: and reverting t(τ) from (2.91), on
Page 101 and 102: Chapter 3 Relativistically Rigid Bo
Page 103 and 104: than considering the rest frame of
Page 105 and 106: We also note Pearle’s proof of hi
Page 107 and 108: properties of the constituent r are
Page 109 and 110: where we set u(τ) = r (3.7) becaus
Page 111 and 112: + 1 { [1 ] .... + (r· ˙v) v + 2 [
Page 113 and 114: applied than the former.) We shall
Page 115 and 116: which varies with time. We deem tha
Page 117 and 118: desirable properties—but unfortun
Page 119 and 120: the magnitude d of d describes the
Page 121 and 122: shall compute the net force on the
Page 123 and 124: we shall make the analysis relativi
Page 125: Now, let us consider such a spin-ha
Page 129 and 130: 4.2.2 The magnetic-charge dipole Ma
Page 131 and 132: ut its employment of moving constit
Page 133 and 134: point of the loop at angle θ to th
Page 135 and 136: Now, the power into each positive c
Page 137 and 138: or, on reärranging the terms, −(
Page 139 and 140: Again noting the relations (4.23) a
Page 141 and 142: its discovery. Mathematically, the
Page 143 and 144: direction: E ≡ jE y . Now, let us
Page 145 and 146: (i.e., 2πε/v orb with v = 1, the
Page 147 and 148: a sufficient amount of inertia, the
Page 149 and 150: The fact that the Penfield-Haus eff
Page 151 and 152: tum of ±µ×E. Firstly, one can sh
Page 153 and 154: concepts precisely and correctly, a
Page 155 and 156: on the structure of spacetime, one
Page 157 and 158: 4.3.4 Relativistic Lagrangian deriv
Page 159 and 160: we know that the operator equations
Page 161 and 162: invariance requirements. They sough
Page 163 and 164: That (4.73) is an amazing result is
Page 165 and 166: is an important example. Clearly, t
Page 167 and 168: let us therefore add a Pauli term (
Page 169 and 170: On the other hand, for a Pauli mome
Page 171 and 172: moment actually cancels with the Di
Page 173 and 174: Chapter 5 The Retarded Fields I hav
Page 175 and 176: to particles carrying dipole moment
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workers. In 1969, Cohn [53] unfortu
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manifestly-covariant field expressi
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as a mathematical aid—it not bein
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integrate over the proper time τ,
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Equation (5.28) is, in the above no
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this is Jackson’s equation (14.13
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moments of a particle. There are a
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It can be noted that equation (5.51
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the overall Lorentz-gauge four-pote
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Applying (5.61) to the delta functi
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The results above are for a particl
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(One should carefully note that the
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under differentiation, the latter e
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Because the field expressions diver
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expression (5.81) in the direction
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We now note that the non-zero elect
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clearly, for any ε, this parametri
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For a point electric dipole moment,
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where the convenient constant η 1
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the former generates solely an elec
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The position of the mechanical cent
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when the particle is in arbitrary m
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Chapter 6 Radiation Reaction Accord
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perhaps, fortunately,—such a “f
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Lorentz performed this calculation
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6.2 Previous analyses To the author
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other than pedagogy, in the past fi
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immediate practicality, since then
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6.3 Infinitesimal rigid bodies We n
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enefit that, by Gauss’s law, ther
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particular times depend on the acce
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ground our understanding of the qua
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whence the reverse transformation i
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We define η 0 ≡ 1 ( ) 4 −2 ∫
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and ending values of each loop are
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= 1 8( 4 3 π(2ε)3 ) 2 , (6.23) wh
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= 4 3 π(2ε)5 { 1 5 − 3 4 α +
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where by the notation ∫ d 2 n d w
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6.4.6 Angular outer integrals We no
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6.4.7 Is there an easier way? It mi
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6.5.3 Final expression for the reta
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Using the fact that t ret ≡ −R,
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6.5.9 Modified retarded normal vect
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6.6 Divergence of the point particl
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“godlike” parameter, i.e., one
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Using the chain rule on (6.74), we
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Our first use of (6.84) is to compu
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The significance of this result is
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lead to delta-function contribution
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(6.99) then gives { (a·b) − 5(n
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6.7.1 The bare inverse-cube integra
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6.7.2 Inverse-cube integral with tw
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we immediately find that the r d -i
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6.8.4 Duality symmetry consideratio
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in order that Maxwell’s equations
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This asserted result of the author
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obtain a non-vanishing integral ove
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− 1 2 ˜µ2 η 1 ( ˙v· ¨σ)σ
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If one examines the author’s calc
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performed the electric charge radia
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and the spin renormalisation term f
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motion of the particle. This has, i
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chrotron” radiation (i.e., the ra
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6.10.2 The Ternov-Bagrov-Khapaev ef
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where the characteristic polarisati
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detail, for arbitrary values of g,
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Substituting (6.151) into (6.152),
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importantly, the hyperbolic tangent
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twice, then it clearly will not do
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Appendix A Notation and Conventions
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Filename bibliog.tex claspcle.tex c
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Filename algebra.h algebra1.c algeb
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costella, british or american must
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fonts of other resolutions, or font
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typographically, the mathematical n
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quantity which has been designated
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noted that the inline fraction symb
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The d’Alembertian operator is den
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It should be noted that quantum mec
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A.3.15 Pointlike objects If an obje
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as a mathematically G-covariant qua
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The adjective enumerated will be us
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Each of the i m take values from th
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distributed factor; for example, A
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A.7.2 Square matrices Matrices with
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A.8.2 Lorentz tensors Rank-zero, ra
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AB denotes the four-tensor that is
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and G are four-tensors, then ε(A,
Page 353 and 354:
The use of this signature of metric
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ackets; e.g., C α | MCLF ≡ [C α
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The first way is to simply measure
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is referred to as the parity operat
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three-vector conversion. In any cas
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A.9.9 Magnitude of a three-vector T
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A.9.13 Cross-products An alternativ
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B.2.2 The field strength tensor The
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B.2.9 Explicit electromagnetic comp
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Also, (a×b)·(c×d) ≡ (a·c)(b·
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{ (σ·∇)rd m r s (n s·A)n d = r
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For two electric charges q 1 and q
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where z is the position of the char
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Appendix D Retarded Fields Verifica
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(−2) ≡ − e { 3 a U mρ 2 2 ρ
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F µν + 1 2ȧUM [να U α R µ]
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Comparing (D.11) and (D.12), we see
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efore neutral particles were discov
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Appendix E The Interaction Lagrangi
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G µ ≡ G 1 k/k µ + G 2 k/B µ +
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u 2 σ µν u 1 −→ ˜Σ µν ,
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powers of ∂ 2 acting on the field
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International Journal of Modern Phy
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we shall not claim any predictive c
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least, if one wants to derive the e
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and electric charge interaction ter
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way that the “Lagrangian-based”
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Θ ≡ ζ 2 (E·B) , ∂ ′ ≡
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prospective solution of the combine
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Appendix G Computer Algebra Mrs. Du
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) dependence to extract the three-g
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mately chosen, both for simplicity,
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appendical reference. In following
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Program Filename Description kinema
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G.4 kinemats: Kinematical quantitie
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The reason for us naming it after F
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[ ˙ Σ 0 ] = γ 2 (v· ˙σ) + γ
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− 25γ 10 (γ + 1) −2 ˙v 2 (v
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+ 124γ 11 (v· ˙v) 2 (v·¨v), (
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+ 2γ 5 (γ + 1) −2 ( ˙v·σ)¨v
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( Σ ... ) 0∣ ∣ ∣v=0 = 0, (
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+ 4γ 8 (v· ˙v) 2 (v·σ ′ ), (
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fields. These products are labelled
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+ 3γ 4 (γ + 1) −1 (v· ˙σ)¨v
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G.5.2 Covariant field expressions S
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B ′d 1 = κ 2 ¨σ ′ ×n + κ 3
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Expanding out these expressions exp
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G.6 radreact: Radiation reaction G.
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Firstly, iterating the differential
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we define the redshift factor λ
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G.6.8 Sum and difference variables
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− 1 16 r d(r d· ˙v) 3 − 1 8 r
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− 1 120 r4 d (n d·.... v)n d + 1
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+ 1 4 r2 d (n d· ˙v) 2 ˙v − 1
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+ 1 8 r2 d r s ˙v 2 (n s· ˙v)n d
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− 1 48 r3 d r s (n d· ˙v)(n s·
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Using (G.28) and (G.55) in (G.56),
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+ 1 8 r4 d (n d· ˙v) 2 ¨σ − 1
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+ 3 2 r dr s ˙v 2 (n s· ˙v) ˙σ
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E q 2 = r −2 d n d + 1 2 r−1 d
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− r d (n d· ˙v)( ˙v· ¨σ)n d
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+ 39 4 r s(n d· ˙v)(n d·¨v)(n d
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+ 3rd −1 r s(n d· ˙v)(n d·σ)(
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+ 1 4 r s(n s·... v) ˙σ + 15 8 r
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+ 1 8 r−2 d r3 s (n d· ˙v)(n s
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− 3 4 r−2 d r2 s (n d· ˙σ)(n
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− 9 16 r s(n d·σ)(n s·¨v)¨v
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B q 1 = −rd −1 n d× ˙v + n d
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− 39 8 r d(n d· ˙v) 2 (n d·σ)
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− 1 2 r−1 d r s(n s·¨v)n d ×
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+ 3 4 r s(n d·σ)(n s· ˙v) ˙v×
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− 5 12 r d(n d·¨v)¨v×σ − 1
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If we add the various E a n and B a
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+ 15 16 r d ˙v 2 (n d· ¨σ)n d +
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+ 3 16 r−2 d r3 s (n d· ˙v)(n d
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+ 3 8 r s(n d·¨v)(n d·σ)(n s·
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+ 1 24 r(n·¨v)(¨v·σ)n − 5 24
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λE q 2 = r −2 d n d + 1 2 r−1
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+ 27 4 (n d· ˙v) 3 (n d·σ)n d
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+ 1 2 r−1 d r s(n d·σ)(n s·¨v
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− 1 2 r−1 d r s(n s· ˙v)n d
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+ 1 8 r−1 d r s(n s· ˙v)( ˙v·
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λ(σ·∇)E d 1 = −rd −2 ¨v
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− 11rd −1 (n d· ˙v)(n d·σ)(
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+ 1 6 (¨v· ¨σ)σ − 1 2 (... v
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− 3rd −2 r2 s (n d· ˙v)(n d·
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d·... − 3rd −1 (n d·σ)(n σ)
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− 12rd −2 r s(n d·¨v)(n d·σ
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+ 101 4 (n d·¨v)(n d·σ)( ˙v·
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+ 7rd −2 (n d·σ)(¨v·σ)n d +
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− 57 2 r−1 d (n d· ˙v) 2 (n d
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+ (n d· ˙v)(n d·... σ)σ + 3(n
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− rd −2 r2 s (n s· ˙v)(n s·
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+ 1 8 r−1 d (n d· ˙v) 2 ( ˙v·
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P (n) ad (r d, r s ) = ˙σ·E a n(
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F (2) dq = 1 15 η 1 ˙v 2 σ − 1
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F (M) µd = −3η ′ 3σ× ˙σ,
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N F (1) dd = 1 10 η 1( ˙v·σ) ˙
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% ID string: Test 3-d integrations.
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+rd^{-3}(n.a)(n.b) +rd^{-3}(n.c)(n.
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+(n.q)(n.r)(ns.s)(ns.t)ns +(ns.u)v
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+\f{1}{105}(i.o)(j.m)(k.l)a +\f{2}{
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+rd^{-2}rs^2(n.u)(n.v)(ns.w)(ns.x)y
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+\f{1}{30}(o.p)a*m +\fourthirds a
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+\f{1}{105}(w.y)(x.z)a’*a +\f{1}{
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Bibliography [1] M. Abraham, Ann. P
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[39] H. J. Bhabha, Proc. Roy. Soc.
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[80] J. R. Ellis, J. Math. Phys. 7
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[124] M. Kolsrud and E. Leer, Phys.
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[167] M. Pavsic (1987): see [34]. [
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[212] I. M. Ternov, V. G. Bagrov an
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It doesn’t matter whether you suc
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