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'In the [[physics]] of [[electromagnetism]], the '''Abraham–Lorentz force''' (also '''Lorentz–Abraham force''') is the [[recoil]] [[force]] on an [[acceleration|accelerating]] [[charged particle]] caused by the particle emitting [[electromagnetic radiation]]. It is also called the '''radiation reaction force''', '''radiation damping force'''<ref name="griffiths">{{cite book|last=Griffiths|first=David J.|author-link=David J. Griffiths|title=Introduction to Electrodynamics|edition=3rd|publisher=Prentice Hall|year=1998|isbn=978-0-13-805326-0|url-access=registration|url=https://archive.org/details/introductiontoel00grif_0}}</ref> or the '''self-force'''.<ref>{{cite journal |last=Rohrlich |first=Fritz |author-link=Fritz Rohrlich |date=2000 |title= The self-force and radiation reaction |journal=[[American Journal of Physics]] |volume=68 |issue=12 |pages= 1109–1112|doi=10.1119/1.1286430 |bibcode=2000AmJPh..68.1109R }}</ref> The formula predates the theory of [[special relativity]] and is not valid at velocities on the order of the speed of light. Its relativistic generalization is called the '''<!--redirect-->Abraham–Lorentz–Dirac force'''. Both of these are in the domain of [[classical physics]], not [[quantum physics]], and therefore may not be valid at distances of roughly the [[Compton wavelength]] or below.<ref name=Rohrlich>[http://www.lepp.cornell.edu/~pt267/files/teaching/P121W2006/ChargedSphereElectron.pdf Fritz Rohrlich: ''The dynamics of a charged sphere and the electron'', Am. J. Phys. '''65''' (11) p. 1051 (1997)]. "The dynamics of point charges is an excellent example of the importance of obeying the validity limits of a physical theory. When these limits are exceeded the predictions of the theory may be incorrect or even patently absurd. In the present case, the classical equations of motion have their validity limits where quantum mechanics becomes important: they can no longer be trusted at distances of the order of (or below) the Compton wavelength… Only when all distances involved are in the classical domain is classical dynamics acceptable for electrons."</ref> There is, however, an analogue of the formula that is both fully quantum and relativistic, called the "Abraham–Lorentz–Dirac–Langevin equation".<ref>{{cite journal|author=P. R. Johnson, B. L. Hu|year=2002|title=Stochastic theory of relativistic particles moving in a quantum field: Scalar Abraham–Lorentz–Dirac–Langevin equation, radiation reaction, and vacuum fluctuations|journal=[[Physical Review D]]|volume=65|issue=6|page=065015|doi=10.1103/PhysRevD.65.065015|arxiv = quant-ph/0101001 |bibcode = 2002PhRvD..65f5015J }}</ref> The force is proportional to the square of the object's [[electric charge|charge]], times the [[Jerk (physics)|jerk]] (rate of change of acceleration) that it is experiencing. The force points in the direction of the jerk. For example, in a [[cyclotron]], where the jerk points opposite to the velocity, the radiation reaction is directed opposite to the velocity of the particle, providing a braking action. The Abraham–Lorentz force is the source of the [[radiation resistance]] of a radio [[antenna (radio)|antenna]] radiating [[radio wave]]s. There are pathological solutions of the Abraham–Lorentz–Dirac equation in which a particle accelerates ''in advance'' of the application of a force, so-called ''pre-acceleration'' solutions. Since this would represent an effect occurring before its cause ([[retrocausality]]), some theories have speculated that the equation allows signals to travel backward in time, thus challenging the physical principle of [[causality (physics)|causality]]. One resolution of this problem was discussed by Arthur D. Yaghjian<ref name=Yaghjian> {{cite book |last=Yaghjian |first=Arthur D. |title=Relativistic Dynamics of a Charged Sphere: Updating the Lorentz–Abraham Model |edition=2nd |series=Lecture Notes in Physics |volume=686 |publisher= Springer |location=New York |year=2006 |page=Chapter 8 |isbn=978-0-387-26021-1 |url=https://books.google.com/books?id=bZkaJZ5htiQC |nopp=true}} </ref> and is further discussed by [[Fritz Rohrlich]]<ref name=Rohrlich /> and Rodrigo Medina.<ref name=Medina>{{cite journal |author1=Rodrigo Medina |title=Radiation reaction of a classical quasi-rigid extended particle |doi=10.1088/0305-4470/39/14/021 |year=2006 |journal=Journal of Physics A: Mathematical and General |volume=39 |issue=14 |pages=3801–3816 |arxiv=physics/0508031|bibcode = 2006JPhA...39.3801M }}</ref> == Definition and description == Mathematically, the Abraham–Lorentz force is given in [[SI units]] by :<math>\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}} = \frac{ q^2}{6 \pi \epsilon_0 c^3} \mathbf{\dot{a}}</math> or in [[Gaussian units]] by :<math>\mathbf{F}_\mathrm{rad} = { 2 \over 3} \frac{ q^2}{ c^3} \mathbf{\dot{a}}.</math> Here '''F'''_rad is the force, <math>\mathbf{\dot{a}}</math> is the derivative of [[acceleration]], or the third derivative of [[displacement (vector)|displacement]]), also called [[Jerk (physics)|jerk]], ''μ''₀ is the [[magnetic constant]], ''ε''₀ is the [[electric constant]], ''c'' is the [[speed of light in vacuum|speed of light]] in [[free space]], and ''q'' is the [[electric charge]] of the particle. Note that this formula is for non-relativistic velocities; Dirac simply renormalized the mass of the particle in the equation of motion, to find the relativistic version (below). Physically, an accelerating charge emits radiation (according to the [[Larmor formula]]), which carries [[momentum]] away from the charge. Since momentum is conserved, the charge is pushed in the direction opposite the direction of the emitted radiation. In fact the formula above for radiation force can be ''derived'' from the Larmor formula, as shown [[#Derivation|below]]. ==Background== In [[classical electrodynamics]], problems are typically divided into two classes: # Problems in which the charge and current ''sources'' of fields are specified and the ''fields'' are calculated, and # The reverse situation, problems in which the fields are specified and the motion of particles are calculated. In some fields of physics, such as [[plasma physics]] and the calculation of transport coefficients (conductivity, diffusivity, ''etc.''), the fields generated by the sources and the motion of the sources are solved self-consistently. In such cases, however, the motion of a selected source is calculated in response to fields generated by all other sources. Rarely is the motion of a particle (source) due to the fields generated by that same particle calculated. The reason for this is twofold: # Neglect of the "[[self-energy|self-fields]]" usually leads to answers that are accurate enough for many applications, and # Inclusion of self-fields leads to problems in physics such as [[renormalization]], some of which are still unsolved, that relate to the very nature of matter and energy. These conceptual problems created by self-fields are highlighted in a standard graduate text. [Jackson] <blockquote> The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~ 1948–1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain. </blockquote> The Abraham–Lorentz force is the result of the most fundamental calculation of the effect of self-generated fields. It arises from the observation that accelerating charges emit radiation. The Abraham–Lorentz force is the average force that an accelerating charged particle feels in the recoil from the emission of radiation. The introduction of [[Quantum mechanics|quantum effects]] leads one to [[quantum electrodynamics]]. The self-fields in quantum electrodynamics generate a finite number of infinities in the calculations that can be removed by the process of [[renormalization]]. This has led to a theory that is able to make the most accurate predictions that humans have made to date. (See [[precision tests of QED]].) The renormalization process fails, however, when applied to the [[gravitational force]]. The infinities in that case are infinite in number, which causes the failure of renormalization. Therefore, [[general relativity]] has an unsolved self-field problem. [[String theory]] and [[loop quantum gravity]] are current attempts to resolve this problem, formally called the problem of [[radiation reaction]] or the problem of self-force. == Derivation == The simplest derivation for the self-force is found for periodic motion from the [[Larmor formula]] for the power radiated from a point charge: :<math>P = \frac{\mu_0 q^2}{6 \pi c} \mathbf{a}^2</math>. If we assume the motion of a charged particle is periodic, then the average work done on the particle by the Abraham–Lorentz force is the negative of the Larmor power integrated over one period from <math>\tau_1</math> to <math>\tau_2</math>: :<math>\int_{\tau_1}^{\tau_2} \mathbf{F}_\mathrm{rad} \cdot \mathbf{v} dt = \int_{\tau_1}^{\tau_2} -P dt = - \int_{\tau_1}^{\tau_2} \frac{\mu_0 q^2}{6 \pi c} \mathbf{a}^2 dt = - \int_{\tau_1}^{\tau_2} \frac{\mu_0 q^2}{6 \pi c} \frac{d \mathbf{v}}{dt} \cdot \frac{d \mathbf{v}}{dt} dt</math>. The above expression can be integrated by parts. If we assume that there is periodic motion, the boundary term in the integral by parts disappears: :<math>\int_{\tau_1}^{\tau_2} \mathbf{F}_\mathrm{rad} \cdot \mathbf{v} dt = - \frac{\mu_0 q^2}{6 \pi c} \frac{d \mathbf{v}}{dt} \cdot \mathbf{v} \bigg|_{\tau_1}^{\tau_2} + \int_{\tau_1}^{\tau_2} \frac{\mu_0 q^2}{6 \pi c} \frac{d^2 \mathbf{v}}{dt^2} \cdot \mathbf{v} dt = -0 + \int_{\tau_1}^{\tau_2} \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}} \cdot \mathbf{v} dt</math>. Clearly, we can identify :<math>\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}}</math>. A more rigorous derivation, which does not require periodic motion, was found using an [[Effective field theory|Effective Field Theory]] formulation.<ref>{{Cite journal |arxiv = 1402.2610|bibcode = 2014IJMPA..2950132B|title = Radiation reaction at the level of the action|journal = International Journal of Modern Physics A|volume = 29|issue = 24|pages = 1450132–90|last1 = Birnholtz|first1 = Ofek|last2 = Hadar|first2 = Shahar|last3 = Kol|first3 = Barak|year = 2014|doi = 10.1142/S0217751X14501322}}</ref><ref>{{Cite journal |doi = 10.1103/PhysRevD.88.104037|bibcode = 2013PhRvD..88j4037B|title = Theory of post-Newtonian radiation and reaction|journal = Physical Review D|volume = 88|issue = 10|pages = 104037|last1 = Birnholtz|first1 = Ofek|last2 = Hadar|first2 = Shahar|last3 = Kol|first3 = Barak|year = 2013|arxiv = 1305.6930}}</ref> An alternative derivation, finding the fully relativistic expression, was found by [[Paul Dirac|Dirac]]. == Signals from the future == <!--Linked to from Abraham–Minkowski controversy--> Below is an illustration of how a classical analysis can lead to surprising results. The classical theory can be seen to challenge standard pictures of causality, thus signaling either a breakdown or a need for extension of the theory. In this case the extension is to [[quantum mechanics]] and its relativistic counterpart [[quantum field theory]]. See the quote from Rohrlich <ref name=Rohrlich /> in the introduction concerning "the importance of obeying the validity limits of a physical theory". For a particle in an external force <math> \mathbf{F}_\mathrm{ext}</math>, we have :<math> m \dot {\mathbf{v} } = \mathbf{F}_\mathrm{rad} + \mathbf{F}_\mathrm{ext} = m t_0 \ddot { \mathbf{{v}}} + \mathbf{F}_\mathrm{ext} .</math> where :<math>t_0 = \frac{\mu_0 q^2}{6 \pi m c}.</math> This equation can be integrated once to obtain :<math> m \dot {\mathbf{v} } = {1 \over t_0} \int_t^{\infty} \exp \left( - {t'-t \over t_0 }\right ) \, \mathbf{F}_\mathrm{ext}(t') \, dt' .</math> The integral extends from the present to infinitely far in the future. Thus future values of the force affect the acceleration of the particle in the present. The future values are weighted by the factor :<math> \exp \left( -{t'-t \over t_0 }\right ) </math> which falls off rapidly for times greater than <math> t_0 </math> in the future. Therefore, signals from an interval approximately <math> t_0 </math> into the future affect the acceleration in the present. For an electron, this time is approximately <math> 10^{-24} </math> sec, which is the time it takes for a light wave to travel across the "size" of an electron, the [[classical electron radius]]. One way to define this "size" is as follows: it is (up to some constant factor) the distance <math>r</math> such that two electrons placed at rest at a distance <math>r</math> apart and allowed to fly apart, would have sufficient energy to reach half the speed of light. In other words, it forms the length (or time, or energy) scale where something as light as an electron would be fully relativistic. It is worth noting that this expression does not involve the [[Planck constant]] at all, so although it indicates something is wrong at this length scale, it does not directly relate to quantum uncertainty, or to the frequency–energy relation of a photon. Although it is common in quantum mechanics to treat <math>\hbar \to 0</math> as a "classical limit", some speculate that even the classical theory needs renormalization, no matter how the Planck constant would be fixed.<!-- This last sentence needs to cite people who believe this to replace and/or supplment the classical theory --> ==Abraham–Lorentz–Dirac force== To find the relativistic generalization, Dirac renormalized the mass in the equation of motion with the Abraham–Lorentz force in 1938. This renormalized equation of motion is called the Abraham–Lorentz–Dirac equation of motion.<ref>{{Cite journal|last=Dirac|first=P. A. M.|date=1938|title=Classical Theory of Radiating Electrons|journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences|volume=167|issue=929|pages=148–169|jstor=97128|doi=10.1098/rspa.1938.0124|doi-access=free|bibcode=1938RSPSA.167..148D}}</ref> ===Definition=== The expression derived by Dirac is given in signature (&minus;,&nbsp;+,&nbsp;+,&nbsp;+) by :<math>F^{\mathrm{rad}}_\mu = \frac{\mu_o q^2}{6 \pi m c} \left[\frac{d^2 p_\mu}{d \tau^2}-\frac{p_\mu}{m^2 c^2} \left(\frac{d p_\nu}{d \tau}\frac{d p^\nu}{d \tau}\right) \right].</math> With [[Alfred-Marie Liénard|Liénard]]'s relativistic generalization of Larmor's formula in the [[co-moving frame]], :<math>P = \frac{\mu_o q^2 a^2 \gamma^6}{6 \pi c},</math> one can show this to be a valid force by manipulating the time average equation for [[Power (physics)|power]]: :<math>\frac{1}{\Delta t}\int_0^t P dt = \frac{1}{\Delta t}\int_0^t \textbf{F} \cdot \textbf{v}\,dt.</math> ===Paradoxes=== Similar to the non-relativistic case, there are pathological solutions using the Abraham–Lorentz–Dirac equation that anticipate a change in the external force and according to which the particle accelerates ''in advance'' of the application of a force, so-called ''preacceleration'' solutions. One resolution of this problem was discussed by Yaghjian,<ref name=Yaghjian/> and is further discussed by Rohrlich<ref name=Rohrlich /> and Medina.<ref name=Medina /> == Self-interactions == The antidamping mechanism resulting from the Abraham–Lorentz force can be compensated by other nonlinear terms, which are frequently disregarded in the expansions of the retarded [[Liénard–Wiechert potential]].<ref name="Rohrlich" /> These nonlinear terms can unleash a [[Self-oscillation|self-oscillatory]] dynamics.<ref>{{Cite journal|last=López|first=Álvaro G.|date=2020-09-09|title=On an electrodynamic origin of quantum fluctuations|url=https://doi.org/10.1007/s11071-020-05928-5|journal=Nonlinear Dynamics|language=en|doi=10.1007/s11071-020-05928-5|issn=1573-269X}}</ref> This trembling motion has a frequency that is closely related to the [[zitterbewegung]] frequency appearing in Dirac's equation. Using the exact [[Liénard–Wiechert potential|Liénard-Wiechert]] potential as solutions to Maxwell's equations with sources, it can be shown that self-forces are correctly described in terms of state-dependent delay differential equations, which display [[limit cycle]] behavior. Therefore, the principle of inertia, as appearing in Newton's first law, would only hold on average, since uniform motion can become unstable through a process of [[symmetry breaking]] of the Lorentz group. Consequently, pilot waves would be necessary attached to any electrodynamic body. ==Experimental observations== While the Abraham–Lorentz force is largely neglected for many experimental considerations, it gains importance for [[localized surface plasmon|plasmonic]] excitations in larger [[Plasmonic nanoparticles|nanoparticles]] due to large local field enhancements. Radiation damping acts as a limiting factor for the [[surface plasmon|plasmonic]] excitations in [[Surface-enhanced Raman spectroscopy|surface-enhanced]] [[Raman scattering]].<ref name="plasmon1">{{cite journal |last1=Wokaun |first1=A. |last2= Gordon |first2= J. P.|author2-link=James P. Gordon|last3=Liao |first3= P. F. |date=5 April 1952 |title=Radiation Damping in Surface-Enhanced Raman Scattering |url= |journal=[[Physical Review Letters]] |volume=48 |issue=14 |pages=957–960 |doi=10.1103/PhysRevLett.48.957 }}</ref> The damping force was shown to broaden surface plasmon resonances in [[Colloidal gold|gold nanoparticles]], [[nanorod]]s and [[Cluster (physics)|clusters]].<ref>{{cite journal |last1= Sönnichsen |first1=C. |display-authors=etal |date=February 2002 |title=Drastic Reduction of Plasmon Damping in Gold Nanorods |journal=[[Physical Review Letters]] |volume=88 |issue=7 |page= 077402|doi= 10.1103/PhysRevLett.88.077402|pmid=11863939 }}</ref><ref>{{cite journal |last1= Carolina |first1=Novo |display-authors=etal|date=2006 |title= Contributions from radiation damping and surface scattering to the linewidth of the longitudinal plasmon band of gold nanorods: a single particle study |journal=[[Physical Chemistry Chemical Physics]] |volume=8 |issue=30 |pages= 3540–3546 |doi= 10.1039/b604856k|pmid=16871343 }}</ref><ref>{{cite journal |last1= Sönnichsen |first1=C. |display-authors=etal |date=2002 |title=Plasmon resonances in large noble-metal clusters |journal=[[New Journal of Physics]] |volume=4 |issue= |pages=93.1–93.8 |doi= 10.1088/1367-2630/4/1/393|doi-access=free }}</ref> The effects of radiation damping on [[nuclear magnetic resonance]] were also observed by [[Nicolaas Bloembergen]] and [[Robert Pound]], who reported its dominance over [[Spin–spin relaxation|spin–spin]] and [[spin–lattice relaxation]] mechanisms for certain cases.<ref>{{cite journal |last1= Bloembergen |first1=N. |last2=Pound |first2=R. V. |author1-link=Nicolaas Bloembergen |author2-link=Robert Pound |date=July 1954 |title=Radiation Damying in Magnetic Resonance Exyeriments |url=http://mriquestions.com/uploads/3/4/5/7/34572113/radiation_damping_physrev.95.8.pdf |journal=[[Physical Review]] |volume=95 |issue=1 |pages=8–12 |doi= 10.1103/PhysRev.95.8|access-date= }}</ref> ==See also== *[[Max Abraham]] *[[Hendrik Lorentz]] *[[Lorentz force]] *[[Cyclotron radiation]] **[[Synchrotron radiation]] *[[Electromagnetic mass]] *[[Radiation resistance]] *[[Radiation damping]] *[[Wheeler–Feynman absorber theory]] *[[Magnetic radiation reaction force]] ==References== {{Reflist}} == Further reading == * {{cite book|last=Griffiths|first=David J.|author-link=David J. Griffiths|title=Introduction to Electrodynamics|edition=3rd|publisher=Prentice Hall|year=1998|isbn=978-0-13-805326-0|url-access=registration|url=https://archive.org/details/introductiontoel00grif_0}} See sections 11.2.2 and 11.2.3 * {{cite book |author=Jackson, John D.|author-link= John David Jackson (physicist)|title=Classical Electrodynamics|edition=3rd|publisher=Wiley|year=1998|isbn=978-0-471-30932-1}} * Donald H. Menzel (1960) ''Fundamental Formulas of Physics'', Dover Publications Inc., {{ISBN|0-486-60595-7}}, vol. 1, page 345. * Stephen Parrott (1987) ''Relativistic Electrodynamics and Differential Geometry'', § 4.3 Radiation reaction and the Lorentz-Dirac equation, pages 136–45, and § 5.5 Peculiar solutions of the Lorentz-Dirac equation, pages 195–204, Springer-Verlag {{ISBN|0-387-96435-5}} . ==External links== * [http://www.mathpages.com/home/kmath528/kmath528.htm MathPages – Does A Uniformly Accelerating Charge Radiate?] * [http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html Feynman: The Development of the Space-Time View of Quantum Electrodynamics] * [http://airccse.com/ijel/papers/1116ijel05.pdf EC. del Río: Radiation of an accelerated charge] {{DEFAULTSORT:Abraham-Lorentz force}} [[Category:Electrodynamics]] [[Category:Electromagnetic radiation]] [[Category:Radiation]] [[Category:Hendrik Lorentz]]'