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'In the [[physics]] of [[electromagnetism]], the '''Abraham–Lorentz force''' (also known as the '''Lorentz–Abraham force''') is the [[Newton's third law|recoil force]] on an accelerating [[charged particle]] caused by the particle emitting [[electromagnetic radiation]] by self-interaction. It is also called the '''radiation reaction force''', the '''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> It is named after the physicists [[Max Abraham]] and [[Hendrik Lorentz]]. The formula although predating the theory of [[special relativity]], was initially calculated for non-relativistic velocity approximations was extended to arbitrary velocities by [[Max Abraham]] and was shown to be physically consistent by [[George Adolphus Schott]]. The non-relativistic form is called '''<!--redirect-->Lorentz self-force''' while the relativistic version is called the '''<!--redirect-->Lorentz–Dirac force''' or collectively known as '''Abraham–Lorentz–Dirac force'''.<ref name=":1" /> The equations 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 are, however, two analogs of the formula that are both fully quantum and relativistic: one is 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 |s2cid=102339497}}</ref> the other is the self-force on a moving mirror. <ref>{{cite journal| author1=Aizhan Myrzakul | author2= Chi Xiong | author3 = Michael R.R. Good|year=2021|title=CGHS Black Hole Analog Moving Mirror and Its Relativistic Quantum Information as Radiation Reaction|journal=[[Entropy]]| volume=23| issue=12| page=1664| doi=10.3390/e23121664| pmid=34945970 | pmc= 8700335 |arxiv = 2101.08139| bibcode= 2021Entrp..23.1664M | doi-access= free }}</ref> The force is proportional to the square of the object's [[electric charge|charge]], multiplied by 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 |author1-link=Arthur D. Yaghjian |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 |no-pp=true}} </ref> and was 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 |s2cid=15040854 }}</ref> == Definition and description == Mathematically, the '''Lorentz-self force''' derived for non-relativistic velocity approximation <math>v\ll c</math>, is given in [[SI units]] by: <math display="block">\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}} = \frac{ q^2}{6 \pi \varepsilon_0 c^3} \mathbf{\dot{a}} = \frac{2}{3} \frac{ q^2}{4 \pi \varepsilon_0 c^3} \mathbf{\dot{a}}</math> or in [[Gaussian units]] by <math display="block">\mathbf{F}_\mathrm{rad} = { 2 \over 3} \frac{ q^2}{ c^3} \mathbf{\dot{a}}.</math> where <math>\mathbf{F}_\mathrm{rad}</math> 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. 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]]. The '''Abraham-Lorentz force''', a generalization of Lorentz self-force for arbitrary velocities is given by:<ref name=ma1>{{Cite journal |last=Abraham |first=Max |date=1 December 1906 |title=Theorie der Elektrizität. Zweiter Band: Elektromagnetische Theorie der Strahlung |journal=Monatshefte für Mathematik und Physik |volume=17 |issue=1 |pages=A39 |doi=10.1007/bf01697706 |issn=0026-9255|doi-access=free }}</ref><ref>{{Cite book |last=Barut |first=A. O. |url=https://www.worldcat.org/oclc/8032642 |title=Electrodynamics and classical theory of fields & particles |date=1980 |publisher=Dover Publications |isbn=0-486-64038-8 |location=New York |pages=179–184 |oclc=8032642}}</ref> <math display="block">\mathbf{F}_\mathrm{rad} =\frac{2kq^2}{3c^3}\left(\gamma^2\dot{a}+\frac{\gamma^4v(v \cdot \dot{a})}{c^2} + \frac{3\gamma^4a(v\cdot a)}{c^2}+\frac{3\gamma^6v(v\cdot a)^2}{c^4}\right)</math> Where γ is the Lorentz factor associated with v, velocity of particle. The formula is consistent with special relativity and reduces to Lorentz's self-force expression for low velocity limit. The covariant form of radiation reaction deduced by Dirac for arbitrary shape of elementary charges is found to be:<ref name=":0" /><ref name=":2">{{Cite book |last=Barut |first=A. O. |url=https://www.worldcat.org/oclc/8032642 |title=Electrodynamics and classical theory of fields & particles |date=1980 |publisher=Dover Publications |isbn=0-486-64038-8 |location=New York |pages=184–185 |oclc=8032642}}</ref> <math display="block">F^{\mathrm{rad}}_\mu = \frac{\mu_0 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> == History == The first calculation of electromagnetic radiation energy due to current was given by [[George Francis FitzGerald]] in 1883, in which [[radiation resistance]] appears.<ref>{{Cite web |title=On the Quantity of Energy transferred to the Ether by a Variable Current {{!}} WorldCat.org |url=https://www.worldcat.org/title/249575548 |access-date=2022-11-20 |website=www.worldcat.org |language=en |oclc=249575548}}</ref> However, dipole antenna experiments by [[Heinrich Hertz]] made a bigger impact and gathered commentary by Poincaré on the ''amortissement'' or damping of the oscillator due to the emission of radiation.<ref>{{Cite journal |last=Hertz |first=H. |date=1887 |title=Ueber sehr schnelle electrische Schwingungen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18872670707 |journal=Annalen der Physik und Chemie |language=de |volume=267 |issue=7 |pages=421–448 |doi=10.1002/andp.18872670707|bibcode=1887AnP...267..421H }}</ref><ref>{{Cite journal |last=Hertz |first=H. |date=1888 |title=Ueber electrodynamische Wellen im Luftraume und deren Reflexion |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18882700802 |journal=Annalen der Physik und Chemie |language=de |volume=270 |issue=8A |pages=609–623 |doi=10.1002/andp.18882700802|bibcode=1888AnP...270..609H }}</ref><ref>{{Cite book |last=Hertz |first=Heinrich |url=http://worldcat.org/oclc/672404956 |title=Electric waves : being researches on the propagation of electric action with finite velocity through space |date=1893 |publisher=Macmillan |isbn=978-1-144-84751-5 |oclc=672404956}}</ref> Qualitative discussions surrounding damping effects of radiation emitted by accelerating charges was sparked by [[Henri Poincaré|Henry Poincaré]] in 1891.<ref>{{Cite book |last=Poincaré |first=Henri |url=https://commons.wikimedia.org/wiki/File:Poincar%C3%A9_-_La_th%C3%A9orie_de_Maxwell_et_les_oscillations_hertziennes,_1904.djvu#file |title=La théorie de Maxwell et les oscillatiions Hertziennes: La télégraphie sans fil |date=1904 |publisher=C. Naud |series=Scientia. Phys.-mathématique ;no.23 |location=Paris}}</ref><ref>{{Cite journal |last=Pupin |first=M. I. |date=1895-02-01 |title=Les oscillations électriques .—H. Poincaré, Membre de l'Institut. Paris, George Carré, 1894. (concluded) |url=https://www.science.org/doi/10.1126/science.1.5.131 |journal=Science |language=en |volume=1 |issue=5 |pages=131–136 |doi=10.1126/science.1.5.131 |issn=0036-8075}}</ref> In 1892, [[Hendrik Lorentz]] derived the self-interaction force of charges for low velocities but did not relate it to radiation losses.<ref>{{Citation |last=Lorentz |first=H. A. |title=La Théorie Électromagnétique de Maxwell et Son Application Aux Corps Mouvants |date=1936 |url=http://dx.doi.org/10.1007/978-94-015-3447-5_4 |work=Collected Papers |pages=164–343 |place=Dordrecht |publisher=Springer Netherlands |doi=10.1007/978-94-015-3447-5_4 |isbn=978-94-015-2215-1 |access-date=2022-11-20}}</ref> Suggestion of a relationship between radiation energy loss and self-force was first made by [[Max Planck]].<ref>{{Cite journal |last=Planck |first=Max |date=1897 |title=Ueber electrische Schwingungen, welche durch Resonanz erregt und durch Strahlung gedämpft werden |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18972960402 |journal=Annalen der Physik und Chemie |language=de |volume=296 |issue=4 |pages=577–599 |doi=10.1002/andp.18972960402|bibcode=1897AnP...296..577P }}</ref> Planck's concept of the damping force, which did not assume any particular shape for elementary charged particles, was applied by [[Max Abraham]] to find the radiation resistance of an antenna in 1898, which remains the most practical application of the phenomenon.<ref>{{Cite journal |last=Abraham |first=M. |date=1898 |title=Die electrischen Schwingungen um einen stabförmigen Leiter, behandelt nach der Maxwell'schen Theorie |url=http://dx.doi.org/10.1002/andp.18983021105 |journal=Annalen der Physik |volume=302 |issue=11 |pages=435–472 |doi=10.1002/andp.18983021105 |bibcode=1898AnP...302..435A |issn=0003-3804|hdl=2027/uc1.$b564390 |hdl-access=free }}</ref> In the early 1900s, Abraham formulated a generalization of the Lorentz self-force to arbitrary velocities, the physical consistency of which was later shown by [[George Adolphus Schott|Schott]].<ref name=ma1/><ref>{{Cite book |first=Max |last=Abraham |url=http://worldcat.org/oclc/257927636 |title=Dynamik des Electrons |date=1902 |oclc=257927636}}</ref><ref>{{Cite journal |last=Abraham |first=Max |date=1904 |title=Zur Theorie der Strahlung und des Strahlungsdruckes |url=http://dx.doi.org/10.1002/andp.19043190703 |journal=Annalen der Physik |volume=319 |issue=7 |pages=236–287 |doi=10.1002/andp.19043190703 |bibcode=1904AnP...319..236A |issn=0003-3804}}</ref> [[George Adolphus Schott|Schott]] was able to derive the Abraham equation and attributed "acceleration energy" to be the source of energy of the electromagnetic radiation. Originally submitted as an essay for the 1908 [[Adams Prize]], he won the competition and had the essay published as a book in 1912. The relationship between self-force and radiation reaction became well-established at this point.<ref>{{Cite book |last=Schott |first=G.A. |url=http://worldcat.org/oclc/1147836671 |title=Electromagnetic Radiation and the Mechanical Reactions, Arising From It, Being an Adams Prize Essay in the University of Cambridge |date=2019 |publisher=Forgotten Books |isbn=978-0-243-65550-2 |oclc=1147836671}}</ref> [[Wolfgang Pauli]] first obtained the covariant form of the radiation reaction<ref>{{Cite book |url=https://link.springer.com/book/10.1007/978-3-642-58355-1 |title=Relativitätstheorie |year=2000 |language=en |doi=10.1007/978-3-642-58355-1|last1=Pauli |first1=Wolfgang |isbn=978-3-642-63548-9 |editor-first1=Domenico |editor-last1=Giulini }}</ref><ref>{{Cite book |last=Pauli |first=Wolfgang |url=http://worldcat.org/oclc/634284762 |title=Theory of relativity: Transl. by G. Field. With suppl. notes by the author. |date=1967 |publisher=Pergamon Pr |oclc=634284762}}</ref> and in 1938, [[Paul Dirac]] found that the equation of motion of charged particles, without assuming the shape of the particle, contained Abraham's formula within reasonable approximations. The equations derived by Dirac are considered exact within the limits of classical theory.<ref name=":0" /> ==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 that moves with velocity much lower than that of speed of light: <math display="block">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 display="block">\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 display="block">\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 the Lorentz self-force equation which is applicable to slow moving particles as: <math display="block">\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}.}</math> Note: There are two problems with this derivation: 1. The equality of two integrals rarely means that the two integrands are equal. 2. Because of the Larmor power radiated, the boundary term will not vanish. A more rigorous derivation, which does not require periodic motion, was found using an [[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| s2cid = 118541484}}</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|s2cid = 119170985}}</ref> A generalized equation for arbitrary velocities was formulated by Max Abraham, which is found to be consistent with special relativity. An alternative derivation, making use of theory of relativity which was well established at that time, was found by [[Paul Dirac|Dirac]] without any assumption of the shape of the charged particle.<ref name=":1">{{Cite web |last=Kirk |first=McDonald |date=6 May 2017 |title=On the History of the Radiation Reaction 1 |url=http://kirkmcd.princeton.edu/examples/selfforce.pdf |url-status=live |archive-url=https://web.archive.org/web/20221017154015/http://kirkmcd.princeton.edu/examples/selfforce.pdf |archive-date=17 October 2022 |access-date=20 November 2022 |website=Princeton}}</ref> == 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 display="block"> 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 display="block">t_0 = \frac{\mu_0 q^2}{6 \pi m c}.</math> This equation can be integrated once to obtain <math display="block"> 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 display="block"> \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{{Who|date=November 2020}} 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 name=":0">{{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><ref>{{Cite journal|last1=Ilderton|first1=Anton|last2=Torgrimsson|first2=Greger|date=2013-07-12|title=Radiation reaction from QED: Lightfront perturbation theory in a plane wave background|url=https://link.aps.org/doi/10.1103/PhysRevD.88.025021 |journal=Physical Review D |volume=88|issue=2 |pages=025021 |doi=10.1103/PhysRevD.88.025021 |arxiv=1304.6842|bibcode=2013PhRvD..88b5021I |s2cid=55353234 }}</ref> ===Definition=== The expression derived by Dirac is given in signature (&minus;,&nbsp;+,&nbsp;+,&nbsp;+) by<ref name=":0" /><ref name=":2" /> <math display="block">F^{\mathrm{rad}}_\mu = \frac{\mu_0 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 display="block">P = \frac{\mu_0 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 display="block">\frac{1}{\Delta t}\int_0^t P dt = \frac{1}{\Delta t}\int_0^t \textbf{F} \cdot \textbf{v}\,dt.</math> == Paradoxes == === Pre-acceleration === 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 /> === Runaway solutions === Runaway solutions are solutions to ALD equations that suggest the force on objects will increase exponential over time. It is considered as an unphysical solution. === Hyperbolic motion === {{See also|Paradox of radiation of charged particles in a gravitational field}} The ALD equations are known to be zero for constant acceleration or hyperbolic motion in [[Minkowski space|Minkowski space-time diagram]]. The subject of whether in such condition electromagnetic radiation exists was matter of debate until [[Fritz Rohrlich]] resolved the problem by showing that hyperbolically moving charges do emit radiation. Subsequently the issue is discussed in context of energy conservation and equivalence principle which is classically resolved by considering "acceleration energy" or Schott energy. == Self-interactions == However 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" /> ==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 |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 |bibcode=2002PhRvL..88g7402S }}</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 |bibcode=2006PCCP....8.3540N }}</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=1 |pages=93.1–93.8 |doi= 10.1088/1367-2630/4/1/393|bibcode=2002NJPh....4...93S |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|bibcode=1954PhRv...95....8B }}</ref> The Abraham–Lorentz force has been observed in the semiclassical regime in experiments which involve the scattering of a relativistic beam of electrons with a high intensity laser.<ref>{{Cite journal|last1=Cole|first1=J. M.| last2=Behm|first2=K. T.|last3=Gerstmayr|first3=E.|last4=Blackburn|first4=T. G.|last5=Wood|first5=J. C.| last6=Baird|first6=C. D.|last7=Duff|first7=M. J.| last8=Harvey|first8=C.| last9=Ilderton|first9=A.| last10=Joglekar|first10=A. S.| last11=Krushelnick|first11=K.| date=2018-02-07|title=Experimental Evidence of Radiation Reaction in the Collision of a High-Intensity Laser Pulse with a Laser-Wakefield Accelerated Electron Beam| url=https://link.aps.org/doi/10.1103/PhysRevX.8.011020| journal=Physical Review X| volume=8| issue=1| pages=011020| doi=10.1103/PhysRevX.8.011020|arxiv=1707.06821 |bibcode=2018PhRvX...8a1020C |hdl=10044/1/55804|s2cid=3779660|hdl-access=free}}</ref><ref>{{Cite journal | last1=Poder|first1=K.| last2=Tamburini|first2=M.|last3=Sarri|first3=G.|last4=Di Piazza|first4=A.| last5=Kuschel|first5=S.| last6=Baird|first6=C. D.| last7=Behm|first7=K.| last8=Bohlen|first8=S.| last9=Cole|first9=J. M. | last10=Corvan|first10=D. J.| last11=Duff|first11=M.| date=2018-07-05|title=Experimental signatures of the quantum nature of radiation reaction in the field of an ultra-intense laser | journal=Physical Review X| volume=8|issue=3 |pages=031004 | doi=10.1103/PhysRevX.8.031004| arxiv=1709.01861|bibcode=2018PhRvX...8c1004P | issn=2160-3308| hdl=10044/1/73880|hdl-access=free}}</ref> In the experiments, a supersonic jet of helium gas is intercepted by a high-intensity (10¹⁸–10²⁰&nbsp;W/cm²) laser. The laser ionizes the helium gas and accelerates the electrons via what is known as the “laser-wakefield” effect. A second high-intensity laser beam is then propagated counter to this accelerated electron beam. In a small number of cases, inverse-Compton scattering occurs between the photons and the electron beam, and the spectra of the scattered electrons and photons are measured. The photon spectra are then compared with spectra calculated from Monte Carlo simulations that use either the QED or classical LL equations of motion. ==See also== *[[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]]'