Single-Particle Electrodynamics - Assassination Science

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In Einsteinian mechanics, the electromagnetic field satisfies Maxwell’s equations: ∇·B ≡ 0, (2.8) ∇×E + ∂ t B ≡ 0, (2.9) ∇·E ≡ ρ, (2.10) ∇×B − ∂ t E ≡ J. (2.11) The homogeneous Maxwell equations (2.8) and (2.9) tell one that the six components of the fields E and B may in fact be completely specified in terms of four quantities A µ ≡ (ϕ, A): E ≡ −∇φ − ∂ t A, B ≡ ∇×A. (2.12) However, the A µ are not directly manifested physically. They are also arbitrary up to a gauge transformation: A µ −→ A µ + ∂ µ Λ. The electromagnetic field also possesses mechanical energy, mechanical momentum and mechanical angular momentum densities. These are given by W ρ (x) = 2{ 1 E 2 (x) + B 2 (x) } , (2.13) p ρ (x) = E(x)×B(x), (2.14) s ρ (x) = x×p ρ (x). (2.15) The former two densities are a part of the electromagnetic mechanical stressenergy tensor, T µν : T 00 = 2{ 1 E 2 + B 2} , 45
T 0i ≡ T i0 = E×B, T ij = 1 2 (E2 + B 2 )δ ij − EE − BB. (2.16) The mechanical stress-energy tensor T µν is always symmetric—not just for the electromagnetic field, but in complete generality. We shall always use the symbol T µν for the mechanical stress-energy tensor, and not for any other quantity; and we shall always use the adjective “mechanical”. The mechanical angular momentum density (2.15) is likewise contained in the mechanical angular momentum density rank-3 tensor: M αβγ ≡ T α[β x γ] . We shall always use the symbol M αβγ for the mechanical angular momentum tensor, and not for any other quantity; and shall always use the adjective “mechanical”. 2.3.6 Electrically charged particles If a particle possesses an electric charge q, then the corresponding Law of Nature for its Newtonian force is the Lorentz force law: F = q(E + v×B), (2.17) where E and B are the electric and magnetic field at the position of the particle respectively. The power into the electric charge, P ≡ d t W, is, for an electric charge, simply given by P = q(v·E), (2.18) 46
Page 1 and 2: Single-Particle Electrodynamics Mr.
Page 3 and 4: Copyright Declaration I understand
Page 5 and 6: I apologise to Dr. Geoff Taylor for
Page 7 and 8: Contents 1 Overview of this Thesis
Page 9 and 10: 3.2.2 Einsteinian rigidity . . . .
Page 11 and 12: 6 Radiation Reaction 220 6.1 Introd
Page 13 and 14: A Notation and Conventions 314 A.1
Page 15 and 16: A.8.18 MCLF and CACS components . .
Page 17 and 18: E.2 Quantum field theory . . . . .
Page 19 and 20: G.7 test3int: Testing of 3-d integr
Page 21 and 22: lood pressure: such inane things ar
Page 23 and 24: 1.2 What do I need to read? If the
Page 25 and 26: Section 2.7: Classical limit of qua
Page 27 and 28: Section 5.2: History of the retarde
Page 29 and 30: Section 6.7: Inverse-cube integrals
Page 31 and 32: Section A.7: Matrices Notation and
Page 33 and 34: Appendix C: Supplementary Proofs Se
Page 35 and 36: Section E.2: Quantum field theory T
Page 37 and 38: esults, not the establishment of ne
Page 39 and 40: e forgiven by their respective auth
Page 41 and 42: tron. The proton and neutron clearl
Page 43 and 44: most appropriate in the task of rem
Page 45: (see Section A.8): f ≡ d τ p. (2
Page 49 and 50: densities and the conservation laws
Page 51 and 52: 2.4 Lagrangian mechanics An alterna
Page 53 and 54: canonical momentum: b a ≡ ∂ ˙q
Page 55 and 56: to be the appropriate degrees of fr
Page 57 and 58: task. Firstly, we need to select ap
Page 59 and 60: canonical energy, canonical momentu
Page 61 and 62: Let us return to the electric charg
Page 63 and 64: should also be carefully noted agai
Page 65 and 66: For an elementary particle, the mas
Page 67 and 68: 2.6.7 Unit four-spin It is often ne
Page 69 and 70: the case. The reason, pointed out b
Page 71 and 72: period it appeared that the magnitu
Page 73 and 74: 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
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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
Page 325 and 326:
typographically, the mathematical n
Page 327 and 328:
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
Page 359 and 360:
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]
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
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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
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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
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) dependence to extract the three-g
Page 415 and 416:
mately chosen, both for simplicity,
Page 417 and 418:
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), (
Page 431 and 432:
+ 2γ 5 (γ + 1) −2 ( ˙v·σ)¨v
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( Σ ... ) 0∣ ∣ ∣v=0 = 0, (
Page 435 and 436:
+ 4γ 8 (v· ˙v) 2 (v·σ ′ ), (
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fields. These products are labelled
Page 439 and 440:
+ 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
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
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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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