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American Journal of Physics Abraham–Lorentz versus Landau–Lifshitz
Abraham–Lorentz versus Landau–Lifshitz
Griffiths, David J., Proctor, Thomas C., Schroeter, Darrell F.Jak bardzo podobała Ci się ta książka?
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Tom:
78
Rok:
2010
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english
DOI:
10.1119/1.3269900
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Abraham–Lorentz versus Landau–Lifshitz David J. Griffiths,a兲 Thomas C. Proctor,b兲 and Darrell F. Schroeterc兲 Department of Physics, Reed College, Portland, Oregon 97202 共Received 24 July 2009; accepted 9 November 2009兲 The classical Abraham–Lorentz formula for the radiation reaction on a point charge suffers from two notorious defects: runaways and preacceleration. Recently, several authors have advocated as an alternative the Landau–Lifshitz formula, which has neither fault. The latter formula is often presented as an approximation to Abraham–Lorentz, raising the delicate question of how an approximation can be considered more accurate than the original. For a spherical shell of finite size, the equation for the radiation reaction is noncontroversial. We begin there, obtain the Abraham– Lorentz and Landau–Lifshitz expressions as limiting cases, and undertake some numerical studies to determine which is superior. © 2010 American Association of Physics Teachers. 关DOI: 10.1119/1.3269900兴 冉 冊 I. INTRODUCTION v t− When a charged particle1 accelerates, it radiates. This radiation carries energy off to infinity, according to the Larmor formula: dE 0q2 2 = a , dt 6c 共1兲 where q is the charge of the particle, v is its 共instantaneous兲 velocity, and a is its acceleration.2 The energy lost to radiation comes at the expense of the particle’s kinetic energy—a charged particle accelerates less than its uncharged twin. Evidently there is a “recoil” force that acts back on the radiating particle: The radiation reaction Frad. The simplest way to calculate the radiation reaction is by exploiting conservation of energy.3 The result is the Abraham–Lorentz formula, Frad = mȧ, 共2兲 where m is the mass of the particle and = 0q 2 6mc 共3兲 sets the time scale 共for an electron = 6.2⫻ 10−24 s兲. But conservation laws alone do not explain the mechanism involved. Lorentz traced the mechanism to the breakdown of Newton’s third law within the structure of the particle. Modeling the particle as a spherical shell of rad; ius R, he calculated the net selfforce on the object—the electromagnetic force of part A on part B, integrated over the entire surface. The result can be expressed in the elegant form4 Fself = 冋冉 冊 册 2R mc2 v t− − v共t兲 . 2R2 c 共4兲 Curiously, Fself involves the difference between the velocity of the sphere at time t and the velocity an instant earlier— “instant” being the time it takes light to cross a diameter. For a point charge we presumably want the limit as R → 0, 391 Am. J. Phys. 78 共4兲, April 2010 http://aapt.org/ajp 冉 冊 1 2R 2R 2R v̇共t兲 + = v共t兲 − 2 c c c 2 v̈共t兲 + . . . , 共5兲 so Fself = − mc a + mȧ. R 共6兲 The first term in Eq. 共6兲 is the electromagnetic contribution to the mass; if we move it to the other side of Newton’s second law, it adds to the “bare” mass m0. The second term is the radiation reaction, which reproduces Eq. 共2兲. We write F = Fext + Fself = m0a and obtain 冉 Fext + Frad = m0 + 冊 mc a = ma, R 共7兲 where m is the “physical” mass and Fext represents any external force acting on the particle.5,6 For more than a century the Abraham–Lorentz formula 共Eq. 共2兲兲 has generally been regarded as the “correct” 共nonrelativistic兲 expression for the radiation reaction. But the resulting equation of motion 共Eq. 共7兲兲 suffers from two problems, runaways and preacceleration. Even if the external force is zero, Eq. 共7兲 admits solutions that accelerate exponentially. These runaway solutions can be avoided by imposing suitable boundary conditions, but the cure is arguably worse than the disease, for now the particle begins to accelerate before the force is applied. Runaways violate conservation of energy, and preacceleration violates causality; both offend common sense. This is the skeleton in the closet of classical electrodynamics. Generations of physicists 共Dirac, Feynman, and Rohrlich, to name a few兲 have agonized over the apparent inconsistency. Some have suggested that it presages quantum mechanics; others have argued that it is an artifact of the point limit, indicating that a true point charge is impossible in classical electrodynamics. Indeed, if Eq. 共4兲 is used in place of Eq. 共6兲, no pathologies occur provided R exceeds the “classical” radius,7 Rclass = c . 共8兲 But it does seem peculiar that classical electrodynamics should harbor such a counterintuitive constraint on its validity. One would have thought that the only restriction on the © 2010 American Association of Physics Teachers 391 sources 共 and J兲 is the continuity equation, expressing local conservation of charge. What is more, the electron is known to be much smaller than its classical dimension. In recent years there has been a flurry of interest in the Landau–Lifshitz formula, Frad = Ḟext , 共9兲 as an alternative to Abraham–Lorentz. Landau and Lifshitz8 introduced this expression as an approximation to Abraham– Lorentz. In the first instance we ignore the radiation reaction altogether, so Eq. 共7兲 says Fext = ma, and hence ȧ = 共1 / m兲Ḟext. We then put this into Eq. 共2兲 to calculate the radiation reaction.9 It is difficult to argue that an approximation can be more accurate than what it is approximating,10 but there are other ways to obtain the Landau–Lifshitz equation.11 Moreover, it makes no reference to the size of the particle, so the point limit is not problematic, and best of all it does not admit runaways or preacceleration. Of course, if the Landau–Lifshitz equation is exact, there must be something wrong with the conventional derivation of Abraham–Lorentz. However, the fact that we can obtain the latter by two radically different routes 共conservation laws and the limit of the selfforce兲 suggests that we should not abandon it lightly. But the conservation law argument is manifestly ambiguous because not all of the energy that leaves the particle eventually escapes in the form of radiation—some of it may be temporarily stored in the “nearby” fields and later reabsorbed by the particle. Only in the long term average 共and perhaps not even then because the fields of a point charge carry infinite energy兲 can we be sure that the energy lost to radiation corresponds to the work done by Frad. Dirac, confronting this ambiguity, remarked wryly that other formulas are possible, “but they are all much more complicated… so that one would hardly expect them to apply to a simple thing like an electron.”12 As for the selfforce argument, it is conceivable that the expansion in Eq. 共5兲 is unreliable, and we shall see that a different plausible approximation to Eq. 共4兲 yields the Landau–Lifshitz equation. The purpose of this paper is to consider whether Landau– Lifshitz is superior to Abraham–Lorentz. In Sec. II we rehearse the proof that Eq. 共4兲 gives rise to no anomalies as long as R ⬎ Rclass, and we show how Eq. 共4兲 leads to the Landau–Lifshitz formula. In Sec. III we conduct some numerical studies to see which approximation better matches the “exact” result 共from Eq. 共4兲兲, and in Sec. IV we draw some tentative conclusions. A. Runaways Setting Fext = 0, we look for solutions of the form v共t兲 = v0et 共11兲 for constant v0 and . If Re共兲 ⬎ 0, the solution describes a runaway. Substituting this ansatz into Eq. 共10兲, we find 冉 冊 共12兲 z = k共e−z − 1兲, 共13兲 1− c c 2  = 2 共e−2R/c − 1兲 R 2R or where z ⬅ 2R / c and k ⬅ 1 / 关共R / c兲 − 1兴. Writing z = x + iy and equating the real parts, we obtain If x is positive 共the condition for a runaway兲, the term in parentheses is negative, and hence k ⬍ 0, which means that the bare mass is negative and therefore 共see Eq. 共7兲兲 m⬍ mc . R 共15兲 That is, if R ⬎ c = Rclass, then there are no runaways. B. Preacceleration Equation 共10兲 can be solved 共formally兲 as follows. Let f̃共兲 denote the Fourier transform of the function f共t兲, f̃共兲 = 1 冑2 冕 ⬁ f共t兲e−itdt; then f共t兲 is the inverse Fourier transform of f̃共兲, f共t兲 = 1 冑2 冕 ⬁ f̃共兲eitd . c R II. MOTION OF A CHARGED SPHERICAL SHELL − The equation of motion for a charged spherical shell 共Eqs. 共4兲 and 共7兲兲 is 冉 冊 冋冉 冊 册 − 共10兲 We first examine Eq. 共10兲 for evidence of runaways and preacceleration.13 392 Am. J. Phys., Vol. 78, No. 4, April 2010 共17兲 −⬁ In terms of Fourier transforms, Eq. 共10兲 becomes 1− 2R 1 c dv c − 2 v t− 1− − v共t兲 = Fext . R dt 2R c m 共16兲 −⬁ 冉 冊冑 冕 2 共14兲 x = k共e−x cos y − 1兲. = 1 2 c 2 2R2 冋 1 冑2 1 1 m 冑2 ⬁ ṽ共兲ieitd −⬁ 1 冑2 冕 冕 ⬁ 冕 ⬁ ṽ共兲eite−i2R/cd −⬁ ṽ共兲eitd −⬁ ⬁ 册 F̃ext共兲eitd , 共18兲 −⬁ from which it follows that Griffiths, Proctor, and Schroeter 392 Im(ω) Im(ω) ∞ 4 3 2 Re(ω) 1 Fig. 1. Closing the contour for Eq. 共21兲. Re(ω) − 40 F̃ext共兲/m . i共1 − c/R兲 + 共c2/2R2兲关1 − e−2iR/c兴 ṽ共兲 = 共19兲 − 20 20 40 Fig. 2. The zeroes of D共兲 for R = 2c, in units of 1 / 4. Evidently 1 v共t兲 = 冑2 冕 ⬁ −⬁ F̃ext共兲/m i共1 − c/R兲 + 共c2/2R2兲关1 − e−2iR/c兴 it ⫻e d 1 2m = 共20a兲 冕冕 ⬁ ⬁ −⬁ −⬁ Fext共t⬘兲e−it⬘dt⬘ i共1 − c/R兲 + 共c2/2R2兲关1 − e−2iR/c兴 it ⫻e d . 共20b兲 It is more convenient to work with the acceleration 共a = dv / dt兲 and express the solution in terms of the Green’s function G, a共t兲 = 1 m 冕 ⬁ G共t − t⬘兲Fext共t⬘兲dt⬘ , D共兲 = 1 − 共21兲 冕 ⬁ −⬁ it e d . 1 − 共c/R兲 − i共c2/2R2兲关1 − e−2iR/c兴 Preacceleration occurs if a共t兲 is sensitive to Fext共t⬘兲 at later times 共t⬘ ⬎ t兲; causality 共the absence of preacceleration兲 requires that for t ⬍ 0. 共23兲 We can turn Eq. 共22兲 into a contour integral in the complex plane by adjoining a semicircle at infinity 共see Fig. 1兲. In order that this “extra” piece contribute nothing, we must close the contour above 共Im共兲 ⬎ 0兲 if t ⬎ 0 and below 共Im共兲 ⬍ 0兲 if t ⬍ 0. By Cauchy’s theorem, the integral vanishes if the integrand has no singularities inside the contour. Thus there will be no preacceleration provided the denominator, D共兲 ⬅ 1 − 再 冋 冉 冊 冉 冊 册冎 2iR c 2 c −i 2 1− 1− 2R R c 1 2iR 2 c 2 + ... =1 − i = − i共 − 0兲, 共22兲 c 2 c − i 2 关1 − e−2iR/c兴, 2R R Am. J. Phys., Vol. 78, No. 4, April 2010 共25a兲 共25b兲 where 0 ⬅ −i / . Evidently D共兲 has a single zero at 0 共in the lower halfplane兲. In this case we can calculate G共t兲 analytically, G共t兲 = i 2 冕 ⬁ −⬁ e it d . 共 − 0兲 共26兲 For t ⬎ 0 we close the contour above, and because it encloses no singularities, G共t兲 = 0. For t ⬍ 0 we close below 共picking up a minus sign because the contour now runs clockwise兲, G共t兲 = − 1 i 2ieit兩=0 = et/ . 2 共27兲 Putting this back into Eq. 共21兲, we obtain a共t兲 = 1 m 冕 ⬁ e共t−t⬘兲/Fext共t⬘兲dt⬘ . 共28兲 t D. The Landau–Lifshitz formula 共24兲 has no zeros in the lower halfplane. But D共兲 = 0 is precisely Eq. 共12兲, with  → i, and we already found that it 393 If R ⬍ c, all bets are off. In the limit R → 0, we have −⬁ 1 2 G共t兲 = 0 C. The point limit + where G共t兲 ⬅ admits no solutions for Re共兲 ⬎ 0 共which is to say, Im共兲 ⬍ 0兲, provided R ⬎ c. For example, if R = 2c, the zeroes of D共兲 are shown in Fig. 2. Thus the same condition that excludes runaways also prevents preacceleration: the spherical shell has to be larger than a certain critical size. We now derive the Landau–Lifshitz formula not as an approximation to Abraham–Lorentz but directly from Eq. 共10兲, the equation of motion for a finite spherical shell—or rather from its solution, Eq. 共21兲 共Ref. 14兲 with Griffiths, Proctor, and Schroeter 393 1 G共t兲 = 2 冕 ⬁ −⬁ R G c e it d . 1 − 共c/R兲 − i共c2/2R2兲关1 − e−2iR/c兴 R y c 共29兲 We change variables 共t⬘ → t − t⬘兲 and write 冕 ma共t兲 = ⬁ 1 G共t⬘兲Fext共t − t⬘兲dt⬘ 共30a兲 y 1 −⬁ =Fext共t兲 冕 ⬁ G共t⬘兲dt⬘ + −⬁ 冕 ⬁ G共t⬘兲关Fext共t − t⬘兲 (a) γ > 0 −⬁ − Fext共t兲兴dt⬘ . R G c 共30b兲 R y c Now 冕 ⬁ 4 G共t兲dt −⬁ = 冕 1 ⬁ it 2 兰−⬁e dt 2 2 ⬁ −⬁ 1 − 共c/R兲 − i共c /2R 兲关1 − e−2iR/c兴 y 1 d 共31a兲 冕 (b) γ < 0 ⬁ ␦共兲 = 2 2 −2iR/c d 兴 −⬁ 1 − 共c/R兲 − i共c /2R 兲关1 − e 共31b兲 = 1, and thus ma共t兲 = Fext共t兲 + 冕 ⬁ Fig. 3. 共a兲 The Green’s function for ␥ = 1 / 2 共solid line兲, ␥ = 1 共dashed line兲, and ␥ = 2 共dotted line兲. These positive values of ␥ correspond to R ⬎ c. 共b兲 The Green’s function for ␥ = −3 / 2 共solid line兲, ␥ = −2 共dashed line兲, and ␥ = −3 共dotted line兲. Note the preacceleration that occurs when R ⬍ c. The open arrow indicates a delta function at t = 0. G共t⬘兲关Fext共t − t⬘兲 − Fext共t兲兴dt⬘ . 共32兲 −⬁ If the external force varies slowly, we may expand the term in square brackets, Fext共t − t⬘兲 − Fext共t兲 ⬇ − t⬘Ḟext共t兲, and write 共approximately兲 ma共t兲 = Fext共t兲 − Ḟext共t兲 冕 ⬁ 冕 t⬘G共t⬘兲dt⬘ . 共34兲 ⬁ tG共t兲dt −⬁ = 冕 ⬁ −⬁ 1 ⬁ it 2 兰−⬁te dt 2 2 共36兲 共33兲 −⬁ But ma共t兲 = Fext共t兲 + Ḟext共t兲, 1 − 共c/R兲 − i共c /2R 兲关1 − e−2iR/c兴 d which is to say that in this approximation the radiation reaction force is Ḟext共t兲, the Landau–Lifshitz formula, Eq. 共9兲. The approximation here 共Eq. 共33兲兲 is open to criticism because t⬘ is not necessarily small in Eq. 共32兲. What we must hope is that G共t兲 is sharply peaked around t = 0. In Fig. 3 we plot 共R / c兲G as a function of y ⬅ ct / R for several values of the parameter ␥ ⬅ 1 / 共r − 1兲, where r ⬅ R / c.15 For these curves, at least, G is close to zero beyond about y = 2. So the Landau–Lifshitz formula can be expected to hold as long as the next term in Eq. 共33兲 is small over this range, 共35a兲 = 冕 ⬁ −⬁ − i d ␦ 共 兲 d 1 − 共c/R兲 − i共c2/2R2兲关1 − e−2iR/c兴 F̈ext Ⰶ Ḟext . d 共37兲 共35b兲 =i d 兵1 − 共c/R兲 − i共c2/2R2兲关1 − e−2iR/c兴其−1兩=0 d 共35c兲 =− . We have discussed three different equations of motion for a charged particle. 共1兲 The spherical shell equation, Thus 394 共35d兲 III. NUMERICAL STUDIES Am. J. Phys., Vol. 78, No. 4, April 2010 Griffiths, Proctor, and Schroeter 394 冋冉 冊 册 冉 冊 1− 2R 1 c c 2 a共t兲 − 2 v t − − v共t兲 = Fext , R 2R c m 共38兲 with the solution given in Eqs. 共21兲 and 共22兲. 共2兲 The Abraham–Lorentz equation, which results from taking the limit R → 0, a − ȧ = 1 Fext , m 共39兲 a共t兲 = 1 Fext + Ḟext . m m F 共1 − e共t−T兲/兲, m 0, 共0 ⱕ t ⱕ T兲 共t ⱖ T兲. 冧 a共t兲 = 共40兲 冦 共t ⬍ 0兲 0, F , 共0 ⬍ t ⬍ T兲 m 0, 共t ⬎ T兲, 冧 共43兲 In this section we take the differentialdifference Eq. 共38兲 to be correct and ask which of the others is the better approximation for a range of radii above and below the critical value and for a variety of external forces. together with delta functions at 0 and T, A. Rectangular force The spherical shell result is given by Eq. 共21兲, F 关␦共t兲 − ␦共t − T兲兴. m Suppose 冦 0, 共t ⱕ 0兲 Fext共t兲 = F, 共0 ⬍ t ⬍ T兲 0, 共t ⱖ T兲. 冧 a共t兲 = 共41兲 The Abraham–Lorentz equation gives16 a共t兲 = 冦 F m 冕 共44兲 T G共t − t⬘兲dt⬘ = 0 F m 冕 t G共t⬘兲dt⬘ 共45兲 t−T 共or you can solve Eq. 共10兲 analytically in steps of size 2R / c兲. The acceleration depends on the parameter r = R / c. For example, if r = 3 / 2 and T = 6,17 共t ⱕ 0兲 0, F −2t/3 共0 ⱕ t ⱕ 3兲 , 3e m F t 共3 ⱕ t ⱕ 6兲 3 + 2e2 − 3 e−2t/3 , m t F 2 4 t 2 + 2e2共1 − 4e2兲 + 3共1 − 2e2 + 7e4兲 e−2t/3 , 共6 ⱕ t ⱕ 9兲. e m 3 冋 冉 冊册 冋 冉冊 共42兲 The Landau–Lifshitz equation says with the solution in Eq. 共28兲. 共3兲 The Landau–Lifshitz equation, which results from assuming the external force is slowly varying, a= 冦 F 共1 − e−T/兲et/ , 共t ⱕ 0兲 m 册 冧 共46兲 Several examples are shown in Fig. 4, above and below the critical value r = 1. B. Trapezoidal force Suppose Fext共t兲 = 冦 0, 共t ⱕ 0兲 共F/⑀T兲t, 共0 ⱕ t ⱕ ⑀T兲 F, 共⑀T ⱕ t ⱕ 共1 + ⑀兲T兲 − 共F/⑀T兲关t − 共1 + 2⑀兲T兴, 共共1 + ⑀兲T ⱕ t ⱕ 共1 + 2⑀兲T兲 0, 共t ⱖ 共1 + 2⑀兲T兲 冧 共47兲 for some constant ⑀. 共Note that Eq. 共47兲 reduces to Eq. 共41兲 in the limit ⑀ → 0.兲 The Landau–Lifshitz equation says 395 Am. J. Phys., Vol. 78, No. 4, April 2010 Griffiths, Proctor, and Schroeter 395 maF maF 1 t t 1 γ= 1 Τ 1 2 Τ γ = −3 maF maF 1 t 1 t Τ 1 γ=1 Τ γ = −2 maF maF 1 t 1 Τ t 1 γ=2 γ=− Τ 3 2 maF 1 maF 1 t 1 Τ t 1 Landau−Lifshitz 6 Τ Abraham−Lorentz Fig. 4. Acceleration under a rectangular force with T = 6. The open arrows in the Landau–Lifshitz plot indicate the two delta functions, one at t = 0 and one at t = T. The shaded areas indicate the external force. a共t兲 = 冦 0, 共t ⱕ 0兲 共F/⑀mT兲共t + 兲, 共0 ⱕ t ⱕ ⑀T兲 F/m, 共⑀T ⱕ t ⱕ 共1 + ⑀兲T兲 − 共F/⑀mT兲关t + − 共1 + 2⑀兲T兴, 共共1 + ⑀兲T ⱕ t ⱕ 共1 + 2⑀兲T兲 0, 共t ⱖ 共1 + 2⑀兲T兲. 冧 共48兲 Examples are plotted in Fig. 5. 396 Am. J. Phys., Vol. 78, No. 4, April 2010 Griffiths, Proctor, and Schroeter 396 maF maF 1 1 t 1 Τ γ= t 1 1 2 maF Τ γ = −3 maF 1 1 t 1 Τ t 1 γ=1 maF Τ γ = −2 maF 1 1 t 1 Τ t 1 γ=2 γ=− Τ 3 2 maF 1 maF 1 t 1 Τ t 1 Landau−Lifshitz 8 Τ Abraham−Lorentz Fig. 5. Acceleration under a trapezoidal force with T = 6. C. Gaussian force = Suppose Fext共t兲 = Fe−t 2/T2 共49兲 . The Abraham–Lorentz equation gives F t/ e a共t兲 = m 397 冕 ⬁ e−t⬘/e −t⬘2/T2 dt⬘ t Am. J. Phys., Vol. 78, No. 4, April 2010 冑FT 2m 2/42 冋 冉 冊册 1 − Erf t T + T 2 et/ . 共50b兲 The Landau–Lifshitz equation says a共t兲 = 共50a兲 eT 冉 冊 2t F 2 2 1 − 2 e−t /T . m T 共51兲 Examples are plotted in Fig. 6. Griffiths, Proctor, and Schroeter 397 maF maF 1 1 2 2 t 5 γ= 1 2 5 Τ t 5 γ = −3 maF 5 Τ 5 Τ 5 Τ 5 Τ maF 1 1 2 2 t 5 γ=1 5 Τ t 5 γ = −2 maF maF 1 1 2 2 t 5 γ=2 5 Τ t 5 γ=− 3 2 maF maF 1 1 2 2 t 5 5 Τ Landau−Lifshitz t 5 Abraham−Lorentz Fig. 6. Acceleration under a Gaussian force with T = 6. D. Exponential force Suppose Fext共t兲 = Fe −兩t兩/T 共52兲 . The Abraham–Lorentz equation yields a共t兲 = 398 F t/ e m 冕 ⬁ e−t⬘/e−兩t⬘兩/Tdt⬘ t Am. J. Phys., Vol. 78, No. 4, April 2010 共53a兲 冦 冉 冊 冧 共53b兲 Griffiths, Proctor, and Schroeter 398 2 t/ 1 et/T − e , 共t ⬍ 0兲 T+ FT T − = 1 −t/T m 共t ⬎ 0兲. e , T+ The Landau–Lifshitz equation gives maF maF 1 1 2 2 t 5 γ= 1 2 5 t 5 Τ γ = −3 maF 5 Τ 5 Τ 5 Τ 5 Τ maF 1 1 2 2 t 5 γ=1 5 t 5 Τ γ = −2 maF maF 1 1 2 2 t 5 γ=2 5 t 5 Τ γ=− 3 2 maF maF 1 1 2 2 t 5 5 t Τ 5 Landau−Lifshitz Abraham−Lorentz Fig. 7. Acceleration under an exponential force with T = 6. g a共t兲 = 1 tc R 再 共t ⬍ 0兲 F 共T + 兲et/T mT 共T − 兲e−t/T , 共t ⬎ 0兲. 冎 共54兲 Examples are plotted in Fig. 7. IV. CONCLUSION Fig. 8. The parameter ␥ ⬅ 1 / 关共R / c兲 − 1兴 as a function of R. 399 Am. J. Phys., Vol. 78, No. 4, April 2010 What are we to make of these graphs? For each external force 共rectangular, trapezoidal, Gaussian, and exponential兲, Griffiths, Proctor, and Schroeter 399 maF 5 t 1 Τ maF 1 t 1 Τ maF 1 2 t 5 5 Τ maF Naturally, Abraham–Lorentz does better in the latter case; indeed, as R → 0 共␥ → −1兲 the exact result converges to Abraham–Lorentz. But this regime is presumably unphysical 共preacceleration is clearly visible in Figs. 4 and 5兲. Suppose we exclude R ⬍ c. Then the question becomes: Which 共Landau–Lifshitz or Abraham–Lorentz兲 is the better approximation for positive ␥ and in particular for large positive ␥ 共R approaching the critical value兲? By eye, the graphs for ␥ = 2 look closer to Landau–Lifshitz—it avoids preacceleration and is more sensitive to kinks in Fext. But we need to examine larger ␥. In Fig. 9 we superimpose the finite sphere and Landau– Lifshitz results for ␥ = 10. It would be nice to report that Landau–Lifshitz does a good job in this regime, but that is plainly not true—except in the Gaussian case. The rectangle is arguably an unfair test because Ḟ blows up at the discontinuities.18 Perhaps the discontinuity in Ḟ 共which produces a “ringing” to which Landau–Lifshitz is insensitive兲 accounts for the problems with the trapezoid and the exponential 共an exact match occurs in the latter case for t ⬍ 0 but only when Ⰶ T兲. But is it legitimate to impose conditions on the allowable external forces? Ideally, we would like a formula for the radiation reaction that holds for any external force. What do we really mean by a “point charge” in classical electrodynamics? If the mathematical point limit is excluded on physical grounds, perhaps what we should mean is “as small as possible without violating causality.” In that case the differentialdifference Eq. 共10兲 共with R ⬎ c兲 is presumably the correct 共nonrelativistic兲 expression. In some cases the Landau–Lifshitz formula is a good approximation. But it seems to us an overstatement to represent the Landau– Lifshitz equation as “physically correct.”19 1 2 t 5 5 Τ Fig. 9. Comparison of the exact result with ␥ = 10 共solid curve兲 to the Landau–Lifshitz result 共dashed curve兲 for the forces discussed in the text. As before, shading indicates the external force, and open arrows represent delta functions. APPENDIX: EVALUATING THE GREEN’S FUNCTION Here we discuss the numerical evaluation of the Green’s function as given in Eq. 共22兲. We define ␥ ⬅ 1 / 共r − 1兲, where r = R / c, and write the Green’s function as G共t兲 = we have plotted the exact 共spherical shell兲 result for six values of the radius. For comparison, we then plot the Landau– Lifshitz and Abraham–Lorentz approximations 共which do not depend on R兲. Note that ␥ is positive when R exceeds c and negative when it is below this critical value 共see Fig. 8兲. 冦 G共t兲 = 共1 + ␥兲 ␦共t兲 + 1 2 冕 ⬁ −⬁ 冤 1+␥ 2 冕 ⬁ −⬁ e it c 1 − i␥ 共1 − e−2iR/c兲 2R d . 共A1兲 The integral diverges when t = 0; this may be handled by subtracting off the portion of the integrand that corresponds to a delta function, 冥 冧 1 − 1 e itd . c −2iR/c 1 − i␥ 共1 − e 兲 2R 共A2兲 Because the imaginary portion of the integrand is odd, the integral itself is real. This fact can be used to express the integral in manifestly real form 400 Am. J. Phys., Vol. 78, No. 4, April 2010 Griffiths, Proctor, and Schroeter 400 冉冊 冕 1 ct G共t兲 = ␦ − ␥ R ⬁ 冋冉 冊 册 ␥ sinc z 0 冉 冊 ct ct z + ␥ sinc z cos z R R dz, 1 + ␥ sinc z关2 cos z + ␥ sinc z兴 cos 1− 共A3兲 where we have made the substitution z ⬅ R / c and sinc z ⬅ sin z / z. The remaining integrand falls off as 1 / z for large z. To handle this long tail, we expand the integrand as a power series in ␥ and treat the first two terms separately, 冉冊 冕 ct ␥ G共t兲 = ␦ − ␥ R ␥3 − 冕 冋冉 冊 册 冋冉 冊 册 ⬁ sinc z cos 0 cos ⬁ sinc3 z 0 1− ␥2 ct z dz + R 冕 ⬁ 冋冉 冊 册 sinc2 z cos 0 冋冉 冊 册 2− ct z dz R ct ct z + ␥ sinc z cos 2 − z R R dz. 1 + ␥ sinc z关2 cos z + ␥ sinc z兴 3− 共A4兲 The first two integrals can be done by hand to give 冉 冊 冋 冉 冊 冉 冊册 冉冏 冏 冏 冏 冏 冏冊 冋冉 冊 册 冋冉 冊 册 冕 ct ␥ ct ct G共t兲 = ␦ + 1−H −H 2− ␥ R 2 R R ␥3 − cos ⬁ sinc3 z 0 + ␥2 8 ct ct ct −2 2− + 4− R R R ct ct z + ␥ sinc z cos 2 − z R R dz, 1 + ␥ sinc z关2 cos z + ␥ sinc z兴 3− 共A5兲 where H共z兲 is the Heaviside step function 共1 for z ⬎ 0, else 0兲. The remaining integral is straightforward to evaluate numerically; the integrand decreases faster than 1 / z2 as z approaches infinity, and we can therefore use the standard substitution20 冕 ⬁ 0 冕冋 1 K共z兲dz = 0 K共z兲 + 冉 冊册 1 1 2K z z a兲 Electronic mail: griffith@reed.edu Electronic mail: proctortc@gmail.com c兲 Electronic mail: schroetd@reed.edu 1 For the moment we take the word “particle” to mean an ideal point object. As it turns out, this is highly problematic, and in due course we will be obliged to extend the term to structures of finite size. 2 We shall assume that the motion of the particle is nonrelativistic; the Larmor expression is correct to the lowest two orders in v / c. The relativistic theory is more complicated, but the essential issues are unchanged. See A. D. Yaghjian, Relativistic Dynamics of a Charged Sphere 共SpringerVerlag, Berlin, 1992兲; F. Rohrlich, Classical Charged Particles 共AddisonWesley, Reading, MA, 1965兲; and P. A. M. Dirac, “Classical theory of radiating electrons,” Proc. R. Soc. London, Ser. A 167, 148– 169 共1938兲. 3 See, for example, D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. 共Pearson, Upper Saddle River, NJ, 1999兲, Sec. 11.2.2. 4 J. D. Jackson, Classical Electrodynamics, 3rd ed. 共Wiley, New York, 1999兲, Sec. 16.3; H. Levine, E. J. Moniz, and D. H. Sharp, “Motion of extended charges in classical electrodynamics,” Am. J. Phys. 45, 75–78 共1977兲; P. Pearle, “Classical electron models,” in Electromagnetism: Paths to Research, edited by D. Teplitz 共Plenum, New York, 1982兲. 5 The electromagnetic correction to the mass 共mc / R兲 blows up as R → 0 as expected because the electrostatic energy E of a point charge is infinite. The actual expression obtained from the selfforce differs from E / c2 by a factor of 4/3. See, for example, D. J. Griffiths and R. E. Owen, “Mass renormalization in classical electrodynamics,” Am. J. Phys. 51, 1120– b兲 401 共A6兲 dz. Am. J. Phys., Vol. 78, No. 4, April 2010 1126 共1983兲. A new and intriguing resolution of the 4/3 puzzle has been suggested by R. Medina, “Radiation reaction of a classical quasirigid extended particle,” J. Phys. A 39, 3801–3816 共2006兲. 7 This is the radius such that the physical mass is equal to the electromagnetic contribution, which is to say that the bare mass is zero. 8 L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields 共Pergamon, Oxford, 1971兲, Sec. 75. The current revival of interest in the Landau–Lifshitz formula was inspired by G. W. Ford and R. F. O’Connell, “Relativistic form of radiation reaction,” Phys. Lett. A 174, 182–184 共1993兲; H. Spohn, “The critical manifold of the Lorentz–Dirac equation,” Europhys. Lett. 50, 287–292 共2000兲; F. Rohrlich, “The selfforce and radiation reaction,” Am. J. Phys. 68, 1109–1112 共2000兲; “The correct equation of motion of a classical point charge,” Phys. Lett. A 283, 276–278 共2001兲; “Dynamics of a classical quasipoint charge,” Phys. Lett. A 303, 307–310 共2002兲; “Dynamics of a charged particle,”Phys. Rev. E 77, 0466091–4 共2008兲; Medina 共Ref. 6兲; and J. D. Jackson, “Comment on ‘Preacceleration without radiation: The nonexistence of preradiation phenomenon,’ by J. A. Heras 关Am. J. Phys. 74, 1025–1030 共2006兲兴,” Am. J. Phys. 75, 844 共2007兲. 9 This amounts to assuming that Frad has a relatively small influence on the motion. One can think of the Landau–Lifshitz formula as the first term in a perturbative expansion of the Abraham–Lorentz formula; to find the next order we put Eq. 共9兲 into Eq. 共7兲, solve for ȧ, and so on: Frad = Ḟext + 2F̈ext + . . .. 6 Griffiths, Proctor, and Schroeter 401 10 See, however, Spohn, Ref. 8. See, for example, Ref. 6. 12 Dirac, Ref. 2, p. 154. 13 This analysis follows Levine, Moniz, and Sharp, Ref. 4, and Medina, Ref. 6. 14 This argument follows Ref. 6. 15 Persuading MATHEMATICA to graph this function is far from trivial; in the Appendix we show how to set it up. 16 You can get it either by piecewise solution to the differential equation 共Ref. 3, Problem 11.19兲 or from Eq. 共28兲. 17 This case was treated by Levine et al., Ref. 4. 11 402 Am. J. Phys., Vol. 78, No. 4, April 2010 18 For restrictions on the applicability of the Landau–Lifshitz formula, see Rohrlich, Ref. 8, and W. E. Baylis and J. Huschilt, “Energy balance with the Landau–Lifshitz equation,” Phys. Lett. A 301, 7–12 共2002兲. 19 Rohrlich, Ref. 8. Jackson 共Ref. 8兲 is more cautious; while maintaining that the Abraham–Lorentz equation has been “superseded,” he describes Landau–Lifshitz as “the correct causal classical equation of motion in the domain in which it is applicable” 共which may be circular but is otherwise unobjectionable兲. 20 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. 共Cambridge U. P., Cambridge, 1992兲, p. 148. Griffiths, Proctor, and Schroeter 402