Abraham–Lorentz force

In the physics of electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham force) is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. It is also called the radiation reaction force or the self force.

The formula predates the theory of special relativity and is not valid at velocities of the order of the speed of light. Its relativistic generalization is called the "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.^[1] There is, however, an analogue of the formula that is both fully quantum and relativistic, called the "Abraham–Lorentz–Dirac–Langevin equation".^[2]

The force is proportional to the square of the object's charge, times the 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.

It was thought that the solution of the Abraham–Lorentz force problem predicts that signals from the future affect the present, thus challenging intuition of cause and effect ( retrocausality ). For example, there are pathological solutions using the Abraham–Lorentz–Dirac equation in which a particle accelerates in advance of the application of a force, so-called pre-acceleration solutions. One resolution of this problem was discussed by Yaghjian^[3] and is further discussed by Rohrlich^[1] and Medina.^[4]

Definition and description

Mathematically, the Abraham–Lorentz force is given in SI units by

\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}}

or in cgs units by

\mathbf{F}_\mathrm{rad} = { 2 \over 3} \frac{ q^2}{ c^3} \mathbf{\dot{a}}.

Here F_rad is the force, $\mathbf{\dot{a}}$ is the jerk (the derivative of acceleration, or the third derivative of displacement), μ₀ is the magnetic constant, ε₀ is the electric constant, c is the 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 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-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]

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.

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 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:

P = \frac{\mu_0 q^2}{6 \pi c} \mathbf{a}^2

.

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 $\tau _{1}$ to $\tau _{2}$ :

\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

.

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:

\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

.

Clearly, we can identify

\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}}

.

A more rigorous derivation, which does not require periodic motion, was found using an Effective Field Theory formulation.^[5]^[6] An alternative derivation, finding the fully relativistic expression, was found by Dirac.

Signals from the future

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 ^[1] in the introduction concerning "the importance of obeying the validity limits of a physical theory".

For a particle in an external force $\mathbf{F}_\mathrm{ext}$ , we have

m \dot {\mathbf{v} } = \mathbf{F}_\mathrm{rad} + \mathbf{F}_\mathrm{ext} = m t_0 \ddot { \mathbf{{v}}} + \mathbf{F}_\mathrm{ext} .

where

t_0 = \frac{\mu_0 q^2}{6 \pi m c}.

This equation can be integrated once to obtain

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' .

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

\exp \left( -{t'-t \over t_0 }\right )

which falls off rapidly for times greater than $t_0$ in the future. Therefore, signals from an interval approximately $t_0$ into the future affect the acceleration in the present. For an electron, this time is approximately $10^{-24}$ 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 $r$ such that two electrons placed at rest at a distance $r$ 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 Planck's 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 $\hbar \to 0$ as a "classical limit", some speculate that even the classical theory needs renormalization, no matter how Planck's constant would be fixed.

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.^[7]

Definition

The expression derived by Dirac is given in signature (−, +, +, +) by

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].

With Liénard's relativistic generalization of Larmor's formula in the co-moving frame,

P = \frac{\mu_o q^2 a^2 \gamma^6}{6 \pi c},

one can show this to be a valid force by manipulating the time average equation for power:

\frac{1}{\Delta t}\int_0^t P dt = \frac{1}{\Delta t}\int_0^t \textbf{F} \cdot \textbf{v}\,dt.

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,^[3] and is further discussed by Rohrlich^[1] and Medina.^[4]

References

1 2 3 4 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."
↑ P. R. Johnson, B. L. Hu (2002). "Stochastic theory of relativistic particles moving in a quantum field: Scalar Abraham–Lorentz–Dirac–Langevin equation, radiation reaction, and vacuum fluctuations". Physical Review D. 65 (6): 065015. arXiv:quant-ph/0101001. Bibcode:2002PhRvD..65f5015J. doi:10.1103/PhysRevD.65.065015.
1 2 Yaghjian, Arthur D. (2006). Relativistic Dynamics of a Charged Sphere: Updating the Lorentz–Abraham Model. Lecture Notes in Physics. 686 (2nd ed.). New York: Springer. Chapter 8. ISBN 978-0-387-26021-1.
1 2 Rodrigo Medina (2006). "Radiation reaction of a classical quasi-rigid extended particle". Journal of Physics A: Mathematical and General. 39 (14): 3801–3816. arXiv:physics/0508031. Bibcode:2006JPhA...39.3801M. doi:10.1088/0305-4470/39/14/021.
↑ Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2014). "Radiation reaction at the level of the action". International Journal of Modern Physics A. 29 (24): 1450132–90. arXiv:1402.2610. Bibcode:2014IJMPA..2950132B. doi:10.1142/S0217751X14501322.
↑ Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2013). "Theory of post-Newtonian radiation and reaction". Physical Review D. 88 (10): 104037. arXiv:1305.6930. Bibcode:2013PhRvD..88j4037B. doi:10.1103/PhysRevD.88.104037.
↑ Dirac, P. A. M. (1938). "Classical Theory of Radiating Electrons". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 167 (929): 148–169.

External links

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[Rohrlich-1] 1 2 3 4 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."

[2] P. R. Johnson, B. L. Hu (2002). "Stochastic theory of relativistic particles moving in a quantum field: Scalar Abraham–Lorentz–Dirac–Langevin equation, radiation reaction, and vacuum fluctuations". Physical Review D. 65 (6): 065015. arXiv:quant-ph/0101001. Bibcode:2002PhRvD..65f5015J. doi:10.1103/PhysRevD.65.065015.

[Yaghjian-3] 1 2 Yaghjian, Arthur D. (2006). Relativistic Dynamics of a Charged Sphere: Updating the Lorentz–Abraham Model. Lecture Notes in Physics. 686 (2nd ed.). New York: Springer. Chapter 8. ISBN 978-0-387-26021-1.

[Medina-4] 1 2 Rodrigo Medina (2006). "Radiation reaction of a classical quasi-rigid extended particle". Journal of Physics A: Mathematical and General. 39 (14): 3801–3816. arXiv:physics/0508031. Bibcode:2006JPhA...39.3801M. doi:10.1088/0305-4470/39/14/021.

[5] Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2014). "Radiation reaction at the level of the action". International Journal of Modern Physics A. 29 (24): 1450132–90. arXiv:1402.2610. Bibcode:2014IJMPA..2950132B. doi:10.1142/S0217751X14501322.

[6] Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2013). "Theory of post-Newtonian radiation and reaction". Physical Review D. 88 (10): 104037. arXiv:1305.6930. Bibcode:2013PhRvD..88j4037B. doi:10.1103/PhysRevD.88.104037.

[7] Dirac, P. A. M. (1938). "Classical Theory of Radiating Electrons". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 167 (929): 148–169.