Single-Particle Electrodynamics - Assassination Science

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6.3.3 Sending and receiving constituents It follows from the discussion of Section 6.1 that a calculation of the selfinteraction equations of motion involves two steps: Firstly, we need we need to consider the fields generated by all of the constituents, as given by the retarded field expressions of Chapter 5. Secondly, we need to compute the forces that these fields exert on all of the other constituents of the body. (We shall consider the term “forces” to encompass the three related concepts of “force on”, “power into” and “torque on” a particle, where unambiguous; there does not appear to be another word covering this concept with any less ambiguity). To link the two parts of our calculation, we must essentially determine the path travelled by the retarded fields in going from “sender” to “receiver” (as we shall henceforth call them); from this information, we can compute the appropriate kinematical quantities of the sending and receiving constituents, which appear in the generated field and received force expressions respectively. From the considerations of Chapter 3, it is clear that the forces on the body are most simply considered at the instant that the body is at rest with respect to the Lorentz frame that one chooses. We shall therefore choose precisely such a frame in which to base our considerations. The centre of the sphere is, as in Chapter 3, considered to be at the origin of spatial and temporal coördinates, and the receiving constituents are labelled by their three-position r in this rest frame: |r| ≤ ε. Now, consider a particular constituent r. This constituent will “see” all of the other constituents of the body, at their various retarded times: a short time ago for those constituents near to r; a longer time ago for those constituents far from r; and, from the discussion of Section 6.1, the 235
particular times depend on the acceleration, etc., of the body as a whole. Now, the forces on the receiving constituent r, due to the retarded fields generated by all of the other constituents, are to be summed together to obtain the net forces on the constituent r; and then, in turn, the forces on all of the constituents r are to be weighted by the accelerative redshift factor λ(r), and then summed together to obtain the forces on the body as a whole. Clearly, in carrying out this procedure, we must have some way of labelling the generating constituents of the body. We are not talking about the task of finding the actual retarded time, retarded position, etc.,—this will be considered in the following sections; rather, we simply need some way to identify the generating constituents. This may seem trivial, but the situation is somewhat subtle. To perform the first-mentioned sum above— namely, that of all of the forces on some particular constituent r due to the retarded fields of all of the other constituents—we need to perform some sort of integral over the “sending” sphere. But the various constituents of this “sending” sphere are all “seen” as they were at different times in the past; we must effectively integrate the “sending” sphere over a quite complicated spacetime hypersurface, not over one of its rest-hypersurfaces. And then we must ensure that we have correctly calculated the relevant transformation properties of the source densities, over this complicated hypersurface. The way out of this conceptual nightmare is to label the sending constituents by the simple three-vector r ′ , which represents the three-position of the sending constituent in the rest frame of the body: |r ′ | ≤ ε. One then computes the particular retarded four-position of this sending constituent in order that one can compute the generated fields; but one does not try to use this four-position as the variable for integrating over. Rather, the 236
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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
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has precessed by an amount ˙σ dt.
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is in terms of the quantities of Sp
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2.6.10 The FitzGerald three-spin In
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such systems. On the other hand, fo
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equations, one then finds, for eith
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i.e., the torque. Clearly, the corr
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The component ∆L 0 is our warning
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vanishes, for v = 0: P | v=0 = 0; (
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wish of them—not least of which b
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The author will, in the remaining c
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(see equation (A.58) of Section A.8
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6(c 1·c 3 ) + 4c 2 2 = 0, 8(c 1·c
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and reverting t(τ) from (2.91), on
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Chapter 3 Relativistically Rigid Bo
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than considering the rest frame of
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We also note Pearle’s proof of hi
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properties of the constituent r are
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where we set u(τ) = r (3.7) becaus
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+ 1 { [1 ] .... + (r· ˙v) v + 2 [
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applied than the former.) We shall
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which varies with time. We deem tha
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desirable properties—but unfortun
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the magnitude d of d describes the
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shall compute the net force on the
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we shall make the analysis relativi
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Now, let us consider such a spin-ha
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adreact of Appendix G; the expressi
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4.2.2 The magnetic-charge dipole Ma
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ut its employment of moving constit
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point of the loop at angle θ to th
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Now, the power into each positive c
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or, on reärranging the terms, −(
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Again noting the relations (4.23) a
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its discovery. Mathematically, the
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direction: E ≡ jE y . Now, let us
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(i.e., 2πε/v orb with v = 1, the
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a sufficient amount of inertia, the
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The fact that the Penfield-Haus eff
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tum of ±µ×E. Firstly, one can sh
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concepts precisely and correctly, a
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on the structure of spacetime, one
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4.3.4 Relativistic Lagrangian deriv
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we know that the operator equations
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invariance requirements. They sough
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That (4.73) is an amazing result is
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is an important example. Clearly, t
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let us therefore add a Pauli term (
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On the other hand, for a Pauli mome
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moment actually cancels with the Di
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Chapter 5 The Retarded Fields I hav
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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,
Page 185 and 186: Equation (5.28) is, in the above no
Page 187 and 188: this is Jackson’s equation (14.13
Page 189 and 190: moments of a particle. There are a
Page 191 and 192: It can be noted that equation (5.51
Page 193 and 194: the overall Lorentz-gauge four-pote
Page 195 and 196: Applying (5.61) to the delta functi
Page 197 and 198: The results above are for a particl
Page 199 and 200: (One should carefully note that the
Page 201 and 202: under differentiation, the latter e
Page 203 and 204: Because the field expressions diver
Page 205 and 206: expression (5.81) in the direction
Page 207 and 208: We now note that the non-zero elect
Page 209 and 210: clearly, for any ε, this parametri
Page 211 and 212: For a point electric dipole moment,
Page 213 and 214: where the convenient constant η 1
Page 215 and 216: the former generates solely an elec
Page 217 and 218: The position of the mechanical cent
Page 219 and 220: when the particle is in arbitrary m
Page 221 and 222: Chapter 6 Radiation Reaction Accord
Page 223 and 224: perhaps, fortunately,—such a “f
Page 225 and 226: Lorentz performed this calculation
Page 227 and 228: 6.2 Previous analyses To the author
Page 229 and 230: other than pedagogy, in the past fi
Page 231 and 232: immediate practicality, since then
Page 233 and 234: 6.3 Infinitesimal rigid bodies We n
Page 235: enefit that, by Gauss’s law, ther
Page 239 and 240: ground our understanding of the qua
Page 241 and 242: whence the reverse transformation i
Page 243 and 244: We define η 0 ≡ 1 ( ) 4 −2 ∫
Page 245 and 246: and ending values of each loop are
Page 247 and 248: = 1 8( 4 3 π(2ε)3 ) 2 , (6.23) wh
Page 249 and 250: = 4 3 π(2ε)5 { 1 5 − 3 4 α +
Page 251 and 252: where by the notation ∫ d 2 n d w
Page 253 and 254: 6.4.6 Angular outer integrals We no
Page 255 and 256: 6.4.7 Is there an easier way? It mi
Page 257 and 258: 6.5.3 Final expression for the reta
Page 259 and 260: Using the fact that t ret ≡ −R,
Page 261 and 262: 6.5.9 Modified retarded normal vect
Page 263 and 264: 6.6 Divergence of the point particl
Page 265 and 266: “godlike” parameter, i.e., one
Page 267 and 268: Using the chain rule on (6.74), we
Page 269 and 270: Our first use of (6.84) is to compu
Page 271 and 272: The significance of this result is
Page 273 and 274: lead to delta-function contribution
Page 275 and 276: (6.99) then gives { (a·b) − 5(n
Page 277 and 278: 6.7.1 The bare inverse-cube integra
Page 279 and 280: 6.7.2 Inverse-cube integral with tw
Page 281 and 282: we immediately find that the r d -i
Page 283 and 284: 6.8.4 Duality symmetry consideratio
Page 285 and 286: 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,
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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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