arXiv:2107.00043v1 [physics.class-ph] 30 Jun 2021 etlycag h aueo ae-atritrcin [ interactions laser-matter of nature the change mentally eeaelsr of lasers generate [ facilities analo [ laser the force ultra-intense into Lorentz of insights the give could to it does because RR investigating as trivial, from far equation literature. parti the a of in momentum debate the affect not does no emission feels photon particle of accelerated rate uniformly a that predict RR of radiu with shell [ charged pa eliminated a for a for that generally equation more Langevin shown quantum have a of) limit classical (the electrodynamic [ classical of modified incompatibility be the to from stem need equations Maxwell’s that proposed [ (‘preacceleration’) bee violation not causality unphy and have has point-charges, solutions’) on problem based RR (LAD), Dirac the and Abraham but electrodynamics, classical of a o h nhscleet.Frhrspotcm rmFr an Ford from came support Further effects. el unphysical to the chosen for be could distribution charge (unspecified) some b is uhre al.: et Author First Introduction 1. electrodynamics acceleration uniform reaction radiation O Keywords F N I E L C I T R A Teyoun acceleration uniform undergoing sphere charged in a mass acceleration-dependent the and reaction Radiation u loi unu lcrdnmc QD [ (QED) electrodynamics quantum in also cur accel uniform and photons, emit always charges accelerated utfe aito ecin(R [ (RR) reaction radiation feel must 2 colo aua cec,UIT 0UITgl lugn U Ulju-gun, UNIST-gil, 50 UNIST, Science, Natural of School eateto hsc,SP n nvriyo Strathclyde, of University and SUPA Physics, of Department , 3 ∗ h rthnsta netne atcemgtrsleteis the resolve might particle extended an that hints first The aito ecini o utapolmo lsia electr classical of problem a just not is reaction Radiation unif a of reaction radiation the is confusion of area Another ic lcrmgei ed eefudt ar oetm i momentum, carry to found were fields electromagnetic Since ORCID [email protected]. author: Corresponding , 1 , : 4 (s): Kang ,btnn fteemdl sfe fdffiute,admn asp many and difficulties, of free is models these of none but ], 8 .Alteecnieain eersrce ononrelativi to restricted were considerations these All ]. a , 10 rpitsbitdt Elsevier to submitted Preprint Adam 24 W∕cm Noble 2 16 nest,adi hsrgm Ri xetdntol ob obs be to only not expected is RR regime this in and intensity, ,tene oudrtn Rhsrcnl eoemr urgent. more become recently has RR understand to need the ], 1 b .Svrltertclmdl aebe ugse oexplai to suggested been have models theoretical Several ]. , aulR. 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If a classically obtained theory meets this requirement, it will bridge the gap between classical and quantum electrodynamics. In this paper we presenta new calculation of the classical self-force on a uniformly accelerated charge. The particle model we adopt is not Dirac’s model (point charge), but that of Lorentz (charged sphere). Such a choice is made to avoid the inconsistencies of the point-charge that lead to unphysical behaviour. A similar work has been reported in Ref. [18], where the self-force on a homogeneously charged sphere under uniform acceleration was calculated. In contrast, in the model used in this paper, we assume that the sphere can be Lorentz-contracted and it bears an image of a point-charge on its surface so that the sphere and the corresponding point-charge are indistinguishable (when observed from outside). Remarkably, our new calculation yields the radiation reaction of a uniformly accelerated charge. Moreover, we show the self-force can be formulated in a simple closed form. Furthermore, we find that the self-force leads to an acceleration-dependent increase in the effective mass of the particle. This increased (effective) mass can be interpreted as the energy source of photons, as it returns to the original mass after the radiation is over. Throughout this paper, we employ the Einstein summation convention, and use the Minkowski metric g with signature {−+++}. SI units are used for all of the equations.
2. Uniformly accelerated charge Denoting the position and velocity of a particle by χ and u (= dχ∕d, with the proper time ), Dirac’s radiation reaction is given by
q2 d2u u du du K = 0 − , (1) rad 6c d2 c2 d d 0 1 −7 ⋅ ⋅ −2 where 0 =4 × 10 kg m C is the vacuum permeability, q is the particle’s charge, and c is the speed of light. This force was derived by the Taylor series expansion of the electromagnetic field F near a point charge [19], and it can be considered as RR because it includes a friction term with magnitude equal to the Larmor formula,
q2 du du P = 0 . (2) rad 6c d d This formula represents the total power of the radiation emitted by an accelerated point-charge. The problem is that if a point-charge is hyperbolically (or uniformly) accelerated, Dirac’s RR vanishes while the radiated power does not (Prad > 0); the hyperbolic motion is given by
2 2 2 χ (t) = zc + c t ẑ, (3) t where zc is an arbitrary constant and ẑ is the unit vector in the direction of the constant acceleration. Substitution K of Eq. (3) into Eq. (1) leads to rad = 0. This apparent contradiction is usually attributed to the reversible exchange of energy with the particle’s bound field [20]. However, since a particle with finite size has a finite bound energy, this argument cannot fully explain the origin of the radiation by an extended particle undergoing indefinite hyperbolic motion. To illuminate this problem,we revisit Lorentz’s model of a particle as a charged sphere. A homogeneouslycharged spherical surface at rest generates the static potential of a point-charge (= q∕40r) in its exterior, but the field inside the sphere vanishes. In this sense the Lorentz model could be generalized as follows: 1. Outside, the sphere generates the field of a point-charge under the same motion. 2. Inside, the field identically vanishes. Hence, in this paper we will call a particle satisfying these conditions, a ‘Lorentz particle (LP).’ A Lorentz particle cannot be distinguished from an ideal point-charge (unless the inside is observable). In contrast to the result of Dirac’s model (which is based on a point-charge), we will demonstrate that a uniformly accelerated LP gets a recoil by RR, via its charge distribution.
First Author et al.: Preprint submitted to Elsevier Page 2 of 9 Short Title of the Article
In order to find the charge and current distribution of an LP, we employ the Rindler frame [21] represented by barred coordinates ̄x = ct,̄ ̄x, ̄y, ̄z , where
−1 ct̄ = zc tanh (ct∕z) , ̄x = x, (4) ̄y = y, ̄z = z2 − c2t2.
In this frame,√ the (flat) Minkowski metric takes the form
2 2 ̄g = diag − ̄z ∕zc , 1, 1, 1 , (5)
and a point-charge satisfying Eq. (3) will remain at rest at ̄x = ct,̄ 0, 0, zc . Correspondingly, the uniform accelera- tion becomes a static problem. If we suppose ̄g is the determinant of the metric (= − ̄z2∕z2), Maxwell’s equations in 0 the Rindler frame can be written
̄ ̄ ) F − ̄g = j̄ − ̄g, 0 (6) ̄ ̄ ̄ ̄ ̄ )A −√)A = F, √
and accordingly the point-charge at the position ̄x = ct,̄ 0, 0, zc leads to
̄ ̄z j0 ( ̄x, ̄y, ̄z) =−qc ( ̄x) ( ̄y) ̄z − zc , (7) zc 2 2 qc 1 ̄r + z Ā ( ̄x, ̄y, ̄z) =− 0 c , (8) 0 4 z c 2 2 2 2 2 ̄r + zc −4zc ̄z t 2 2 2 ̄ where ̄r = ̄x + ̄y + ̄z , and spatial components of the current j̄ and potential A vanish. These results exactly correspond to the published solutions [10, 21]. √ On a spherical surface given by
2 2 2 2 2 2 ̄x + ̄y + ̄z − zc + Rc = Rc , (9) 0 t 1 ̄ the potential A0 satisfies
2 2 qc zc + Rc ̄ 0 A0 on the surface =− , (10) 4 tzcRc ó which indicatesó that the potential is constant on the spherical surface where Rc is the radius. From this notion, we R,̄ ̄ , ̄ Ā¨ define coordinates Θ and potential distribution 0, defined as
2 R̄ ≡ ̄x2 + ̄y2 + ̄z − z2 + R2 , v c c 0 t 1 ̄x2 + ̄y2 Θ̄ ≡ tan−1 , (11) ⎛ ̄z −√ z2 + R2 ⎞ ⎜ c c ⎟ ⎜ ̄y t ⎟ ̄ ≡ tan−1 ⎜ , ⎟ ⎝ ̄x ⎠ 0 1
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4
Shape
Field-line
|ct | = 3 z
c
3 c z
2
/ / |ct | = 3 z /2
c z
|ct | = 0
1
0
0 1 2
2 2 1/2
(x + y ) / z
c
Figure 1: Shapes and electric field-lines of a uniformly accelerated LP at three different times, where Rc = zc ∕2. The dotted lines represent the field lines which would be generated if the LP were a point-charge. and
2 2 qc zc + Rc ¨ 0 ̄ Ā ( ̄x, ̄y, ̄z) ≡ − R ≤ Rc , (12) 0 4 tzcRc ⎧ ̄ ̄ ⎪A0 ( ̄x, ̄y, ̄z) R > Rc . ⎨ This potential is equal⎪ to Eq. (8), i.e. the potential of a point charge, outside the sphere (R̄ > R ), but constant for ⎩ c R̄ R j̄¨ ≤ c, so that the field vanishes inside the sphere. The chargedensity 0 inducing this potential can be obtained from Maxwell’s equation (6): z ̄¨ 1 ̄z ̄ c ̄ ̄¨ j0 ( ̄x, ̄y, ̄z) =− ) ) A0 ( ̄x, ̄y, ̄z) 0 zc ̄z qc z =− c R̄ − R . (13) 2 c 4Rc ̄z ̄ This charge density completely meets the requirements for LP: outside the sphere (R > Rc) it induces the field of Ā¨ Ā R̄ R Ā¨ the point charge ( 0 = 0), and inside ( ≤ c), the field vanishes ( 0 = const). Consequently, external observers cannot distinguish the LP from the point-charge, as they observe an image point-charge on the surface. Although the Lorentz particle has a finite volume, we can define its position as the location of the point charge which would generate the same external field as the LP. The position of a uniformly accelerated LP defined in this way satisfies Eq. (3). Figure 1 shows the LP’s positions and actual shapes at three different times observed in the lab frame.
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As the field lines in Fig. 1 indicate, according to Gauss’s law, the total charge of the LP is always q; z j¨ d3r = j̄¨ c d3 ̄r, Ê 0 Ê 0 ̄z qc z2 =− c R̄ − R d3 ̄r, 2 Ê 2 c 4Rc ̄z 2 ̄ 2 ̄ ̄ qc z R sin Θ R − Rc =− c dR̄dΘd̄ ,̄ 4R2 Ê 2 c ̄ ̄ 2 2 R cos Θ+ zc + Rc 0 t 1 =−qc. (14)
In contrast, we show that the momentum of the LP does not coincide with that of the point-charge. The original expression of the Lorentz force is given by
m d U d3r = (E + J × B)d3r, (15) q dt Ê Ê where m is the (unrenormalized) bare mass and U is relativistic velocity of the charge distribution, defined as U ≡ cJ∕ −jj , and we assume the charge and mass distributions are proportional. In the case of the point-charge, the momentum is simply mu because = q (r − χ). On the other hand, in case of the LP, the charge is distributed over a finite√ surface, and accordingly the momentum is determined by the volume integration
m p = ¨U¨ d3r, q Ê m =− ẑ j¨ U ¨ d3r. (16) qc Ê 0 3
̄ ¨ As the LP remains at rest in the Rindler frame, we obtain U = −c ̄z∕zc, 0, 0, 0 , and the Lorentz boost yields
m ct z2 p = ẑ j̄¨ Ū ¨ c d3 ̄r, qc Ê 0 0 ̄z ̄z2 mc2t z2 = ẑ c R̄ − R d3 ̄r, 2 Ê 3 c 4Rc ̄z 2 ̄ 2 ̄ ̄ mc2t z R sin Θ R − Rc = ẑ c dR̄dΘd̄ ,̄ 4R2 Ê 3 c ̄ ̄ 2 2 R cos Θ+ zc + Rc 0 t 1 2 mc2t R = 1+ c ẑ. (17) z y z2 c x c x w 2 2 Variables in this equation can be substituted by the velocity and acceleration using u = ẑc t∕zc and du∕dt = ẑc ∕zc, leading to the final form of the momentum,
2 R du 2 p = mu 1+ c . (18) v c4 dt ó ó ó ó What is indicated byó Eq.ó (18) is interesting: the LP’s momentum is always greater than the momentum of the corresponding point-charge.ó ó This is the consequence of the change in the shape of the LP every moment as shown in Fig. 1; to be Lorentz-contracted, the rear and front surfaces of the LP need to move faster and slower than the corresponding point-charge, and Eq. (13) indicates that the charge density on the rear surface is higher than that on the
First Author et al.: Preprint submitted to Elsevier Page 5 of 9 Short Title of the Article
j̄¨ > j̄¨ u front surface ( 0 0 ), which means the average velocity of the LP is greater than . It leads to the same ̄z
F = (¨E¨ + J¨ × B¨)d3r, self Ê
= ẑ gj¨ F ¨ d3r. (19) Ê 3 In a similar way to the momentum calculation, the integral is calculated as
z2 F =−ẑ j̄¨ )̄ Ā¨ c d3 ̄r, self Ê 0 3 0 ̄z2
4 2 2 q2c2 z ̄z − zc + Rc = 0 ẑ c R̄ − R d3 ̄r, 2 4 Ê 4 t c 32 Rc ̄z Rc 2 2 4 ̄ 3 ̄ ̄ q c z R sin 2Θ R − Rc = 0 ẑ c dR̄dΘd̄ ,̄ 642R5 Ê 4 c ̄ ̄ 2 2 R cos Θ+ zc + Rc 0 t 1 q2 c2 R2 =− 0 1+ c ẑ. (20) 6R z y z2 c c x c x w By the same substitution of zc by acceleration used in the momentum calculation, the self-force is written
2 q2 du R du 2 F =− 0 1+ c . (21) self 4 6Rc dt v c dt ó ó ó ó As mentioned, the self-force also showsó ó the same square root factor as the momentum. This factor tends to unity as ó ó the radius Rc approaches zero, and the diverging Fself for a small Rc can be treated as the mass renormalization the same way as in Dirac’s theory for a point-charge. However, for finite Rc an important feature of Eq. (21) is that the renormalized mass depends on the acceleration. Since the self-force can be absorbed into a mass renormalization it might seem that, like the point-charge, the LP under uniform acceleration does not feel radiation reaction. However, interestingly, if the acceleration is very large Eq. (21) converges to the Larmor formula:
2 du c Prad ≫ ⟹ Fself ≃− ẑ. (22) dt Rc c ó ó ó ó The remarkableó ó point is that even if the radius is extremely small, we can obtain the above result if the acceleration is largeó enoughó 1. Thus the self-force of the uniformly accelerated LP includes radiation reaction. Furthermore, while previoustheorieshave been approximatedby Taylor series or other approximations[19, 2],Eq.(21) is exact. Therefore, a mathematically accurate equation of motion for a uniformly accelerated LP can be derived from Eq. (21).
1Here we ignore quantum effects which may become significant when the proper acceleration approaches that induced by the Sauter-Schwinger 2 3 field, ES = me c ∕e`, with me the electron mass and e the elementary charge. We discuss limitations imposed by ES at the end of this Section.
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To understand how the radiation reaction in Eq. (21) can be associated with the mass increase in (18), it is useful to write the equation of motion of a constantly accelerating LP. The momentum conservation is given by dp = F + F . (23) dt self ext Uniform acceleration of an LP can be generated by a homogeneous electric and vanishing magnetic field, yielding the external force
F = (¨E + J¨ × B )d3r, ext Ê ext ext E =− ext j¨ d3r, c Ê 0
= qEext. (24) Finally, by substituting Eqs. (18),(21)and (24)inEq.(23), we obtain the equation of motion for the uniformly accel- erated LP:
2 q2 du R du 2 m + 0 1+ c = qE . (25) 6R dt v c4 dt ext 0 c 1 ó ó Following Dirac, we define the renormalizedó ó mass as ó ó 2 ó ó 0q mq ≡ m + . (26) 6Rc Note that, unlike the case of the point-charge, this is a finite mass renormalization, and the last term of Eq. (26) originates from the self-energy of a finite charge distribution. It is also notable that the last term of Eq. (26) has dimensions of mass. The square root term remains as the only difference between Eq. (25) and the well-known form of Lorentz force. This term increased the effective mass in Eq.(18) and yielded the radiation reaction in Eq. (21). This strongly implies that the mass variation and radiation reaction are closely related phenomena. As the role and form of 2 2 this term are analogous to the Lorentz factor (= 1+ u∕c ), we define a new factor a as t 2 2 Rc du ð ð ≡ 1+ , (27) a v c4 dt ó ó leading to ó ó ó ó du ó ó m = qE . (28) a q dt ext Indeed, by this factor the particle becomes harder to accelerate, which implies that a stronger uniform field should be applied to the LP in order to get the conventional constant acceleration (= qEext ∕mq). In other words, to get the expected acceleration, more energy should be provided to the particle. This extra energy compensates for the energy loss by radiation, as indicated by Eq. (22). It is interesting to note that the factor a is a priori independent of the charge q and mass mq, depending only on 2 the radius Rc, which is undetermined by the theory. A ‘natural’ choice of radius is given by Rc = 0q ∕6mq, the minimum radius required to eliminate the problems of runaways and preacceleration [8]. In the case of electrons, this −15 radius is roughly 2×10 m, so that even the Sauter-Schwinger electric field ES would induce a relative mass shift −5 a −1 of only 10 . However, there are a number of conjectures referring to a finite radius for the electron, ranging from 10−22 m to 2×10−12 m [22, 23, 24]. In the context of radiation reaction, it has been argued [25] that quantum uncertainty should imbue the electron with an effective radius equal to the Compton wavelength, C = ℎ∕mec ≃ −12 2×10 m. In this case, the effective mass increases by 5% in a field strength E = 0.05ES. For the intensity of 26 2 a laser pulse, 0.05ES corresponds to 6×10 W∕cm , which is expected to be attained within ten years [16]. This suggests that it may be possible to observe the acceleration-dependentmass increase in the near future, if the electron is an LP, while also explaining why this effect was not observed at recent experiments on radiation reaction [26, 27], which operated at intensities of 4×1020 W∕cm2.
2The Lorentz factor can be also given by = 1∕ 1 − v∕c 2, where v = dχ∕dt = u∕ . t First Author et al.: Preprint submitted to Elsevier ð ð Page 7 of 9 Short Title of the Article
3. Conclusion In this paper we have determined the self-force of a uniformly accelerated and Lorentz-contracted charged sphere, or ‘Lorentz particle (LP),’ which to an external observer is indistinguishable from a correspondingly moving point- charge, while the field inside vanishes. This self-force, Eq.(21), was found, for extremely strong acceleration, to have magnitude equivalent to the Larmor formula (see Eq. (22)). Based on momentum conservation (23), the equation of motion of the uniformly accelerated LP was obtained as Eq. (28), and the new factor a was found to give an acceleration-dependent increase to the effective mass of the LP. Despite its simplicity, the LP model overcomes a notable limitation of the pointparticle: Eq. (21) demonstrates that the uniformly accelerated LP does experience radiation reaction, and Eq. (28) reveals how the energy of the radiation originates in the reduced acceleration produced by an external force. It is also important to note that these results are mathematically rigorous. More general results remain unknown, and we will seek further support for the LP model by exploring other trajectories in future work.
Acknowledgment This work is supported by the Basic Research in Science & Engineering Program through the National Research Foundation (NRF) of Korea (under Grant Nos. NRF-2016R1A5A1013277, NRF-2017M1A7A1A03072766, NRF- 2020R1I1A1A01067536, and NRF-2020R1A2C1102236). Adam Noble, Samuel R. Yoffe, and Dino A. Jaroszyn- ski are supported by the UK EPSRC (Grant No. EP/N028694/1) and the EC’s Laserlab-Europe H2020 EC-GA No. 871124.
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