Single-Particle Electrodynamics - Assassination Science

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in the MCLF of the particle; in a frame in which it moves with three-velocity v, application of the Lorentz transformation (G.1) yields U 0 = γ, U = γv. (G.4) One may verify that U 2 = 1. G.4.4 Four-spin of a particle The four-spin, Σ , of a particle has components Σ ≡ (0, σ) (G.5) in the MCLF of the particle; in a frame in which it moves with three-velocity v, application of the Lorentz transformation yields Σ 0 = γ(v·σ), Σ = σ + γ 2 (γ + 1) −1 (v·σ)v. One may verify that Σ 2 = −1. G.4.5 The FitzGerald three-spin We [65] introduce a further quantity, derived from the three-spin σ, that considerably simplifies a number of explicit algebraic expressions: the FitzGerald three-spin, σ ′ = σ − γ(γ + 1) −1 (v·σ)v. (G.6) 421
The reason for us naming it after FitzGerald can be seen by computing its three-magnitude: σ ′ 2 = 1 − (v·σ) 2 . The magnitude of σ ′ is (like that of σ) unity, if σ lies in a plane perpendicular to the three-velocity v; but it is contracted by a factor of √ 1 − v2 ≡ 1 γ if σ lies parallel or antiparallel to the direction of v; in other words, σ ′ acts like a FitzGerald–Lorentz contracted [87] version of σ. It is curious, but probably not fundamentally meaningful, that a concept from the pre-relativistic days should be a useful tool in a completely relativistic analysis. G.4.6 Proper-time and lab-time derivatives The proper-time rate of change experienced by a particle moving with fourvelocity U, of an arbitrary quantity external to the particle (e.g., an external field), is computed by means of the relativistic convective derivative operator, d τ ≡ (U ·∂) ≡ γ∂ 0 + γ(v·∇). (G.7) The proper-time rate of change of a component of a kinematical property of a particle (i.e., the partial proper-time derivative), as seen in some given lab frame, is defined to be related to the lab-time rate of change of that component by means of the time-dilation formula: [d τ ] ≡ γd t . (G.8) G.4.7 Partial kinematical derivatives We now take successive proper-time derivatives of the components of U and Σ (i.e., partial proper-time derivatives). To do so, we first take d t of (G.2) 422
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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,
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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,
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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
Page 371 and 372: Also, (a×b)·(c×d) ≡ (a·c)(b·
Page 373 and 374: { (σ·∇)rd m r s (n s·A)n d = r
Page 375 and 376: For two electric charges q 1 and q
Page 377 and 378: where z is the position of the char
Page 379 and 380: Appendix D Retarded Fields Verifica
Page 381 and 382: (−2) ≡ − e { 3 a U mρ 2 2 ρ
Page 383 and 384: F µν + 1 2ȧUM [να U α R µ]
Page 385 and 386: Comparing (D.11) and (D.12), we see
Page 387 and 388: efore neutral particles were discov
Page 389 and 390: Appendix E The Interaction Lagrangi
Page 391 and 392: G µ ≡ G 1 k/k µ + G 2 k/B µ +
Page 393 and 394: u 2 σ µν u 1 −→ ˜Σ µν ,
Page 395 and 396: powers of ∂ 2 acting on the field
Page 397 and 398: International Journal of Modern Phy
Page 399 and 400: we shall not claim any predictive c
Page 401 and 402: least, if one wants to derive the e
Page 403 and 404: and electric charge interaction ter
Page 405 and 406: way that the “Lagrangian-based”
Page 407 and 408: Θ ≡ ζ 2 (E·B) , ∂ ′ ≡
Page 409 and 410: prospective solution of the combine
Page 411 and 412: Appendix G Computer Algebra Mrs. Du
Page 413 and 414: ) dependence to extract the three-g
Page 415 and 416: mately chosen, both for simplicity,
Page 417 and 418: appendical reference. In following
Page 419 and 420: Program Filename Description kinema
Page 421: G.4 kinemats: Kinematical quantitie
Page 425 and 426: [ ˙ Σ 0 ] = γ 2 (v· ˙σ) + γ
Page 427 and 428: − 25γ 10 (γ + 1) −2 ˙v 2 (v
Page 429 and 430: + 124γ 11 (v· ˙v) 2 (v·¨v), (
Page 431 and 432: + 2γ 5 (γ + 1) −2 ( ˙v·σ)¨v
Page 433 and 434: ( Σ ... ) 0∣ ∣ ∣v=0 = 0, (
Page 435 and 436: + 4γ 8 (v· ˙v) 2 (v·σ ′ ), (
Page 437 and 438: fields. These products are labelled
Page 439 and 440: + 3γ 4 (γ + 1) −1 (v· ˙σ)¨v
Page 441 and 442: G.5.2 Covariant field expressions S
Page 443 and 444: B ′d 1 = κ 2 ¨σ ′ ×n + κ 3
Page 445 and 446: Expanding out these expressions exp
Page 447 and 448: G.6 radreact: Radiation reaction G.
Page 449 and 450: Firstly, iterating the differential
Page 451 and 452: we define the redshift factor λ
Page 453 and 454: G.6.8 Sum and difference variables
Page 455 and 456: − 1 16 r d(r d· ˙v) 3 − 1 8 r
Page 457 and 458: − 1 120 r4 d (n d·.... v)n d + 1
Page 459 and 460: + 1 4 r2 d (n d· ˙v) 2 ˙v − 1
Page 461 and 462: + 1 8 r2 d r s ˙v 2 (n s· ˙v)n d
Page 463 and 464: − 1 48 r3 d r s (n d· ˙v)(n s·
Page 465 and 466: Using (G.28) and (G.55) in (G.56),
Page 467 and 468: + 1 8 r4 d (n d· ˙v) 2 ¨σ − 1
Page 469 and 470: + 3 2 r dr s ˙v 2 (n s· ˙v) ˙σ
Page 471 and 472: 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
Page 537 and 538:
− rd −2 r2 s (n s· ˙v)(n s·
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+ 1 8 r−1 d (n d· ˙v) 2 ( ˙v·
Page 541 and 542:
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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