Abraham–Lorentz force

In the physics of electromagnetism, the Abraham–Lorentz 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.

The formula is in the domain of classical physics, not quantum physics, and therefore, may not valid at distances of roughly the Compton wavelength (λ_C ≈ 2.43 pm) or below. There is, however, an analogue of the formula which is both fully quantum and relativistic, called the "Abraham-Lorentz-Dirac-Langevin equation". See Johnson and Hu.

The force is proportional to the square of the object's charge, times the so-called "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. 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 preacceleration solutions! One resolution of this problem was discussed by Yaghjian, and is further discussed by Rohrlich, Medina., and Ribarič and Šušteršič.

Definition and description

Mathematically, the Abraham–Lorentz force is given by:

$\mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} ={\frac {q^{2}}{6\pi \epsilon _{0}c^{3}}}\mathbf {\dot {a}}$ (SI units)

or

$\mathbf {F} _{\mathrm {rad} }={2 \over 3}{\frac {q^{2}}{c^{3}}}\mathbf {\dot {a}}$ (cgs units)

where:

F is the force,

$\mathbf {\dot {a}}$ is the jerk (the derivative of acceleration, or the third derivative of displacement),

μ₀ and ε₀ are the permeability and permittivity of free space,

c is the speed of light in free space

q is the electric charge of the particle.

Note that this formula is non-relativistic velocities, although both the Abraham-Lorentz and Abraham-Lorentz-Dirac formulas were both fully relativistic. Dirac simply renormalized the mass of the particle in the equation of motion.

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:

1. Problems in which the charge and current sources of fields are specified and the fields are calculated, and
2. 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:

1. Neglect of the "self-fields" usually leads to answers that are accurate enough for many applications, and
2. Inclusion of self-fields leads to problems in physics such as renormalization, some of which still unsolved, that relate to the very nature of matter and energy.

This 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 unsolved self-field problems. String theory is a current attempt to resolve these problems for all forces.

Derivation

We begin with the Larmor formula for radiation of a point charge:

$P={\frac {\mu _{0}q^{2}a^{2}}{6\pi c}}$.

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 $\tau _{2}$:

$\int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=\int _{\tau _{1}}^{\tau _{2}}-Pdt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}a^{2}}{6\pi c}}dt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot {\frac {d\mathbf {v} }{dt}}dt$.

Notice that we can integrate the above expression 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

$\mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}}$.

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 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

$m{\dot {\mathbf {v} }}=\mathbf {F} _{\mathrm {rad} }+\mathbf {F} _{\mathrm {ext} }=mt_{0}{\ddot {\mathbf {v} }}+\mathbf {F} _{\mathrm {ext} }.$

where

$t_{0}={\frac {\mu _{0}q^{2}}{6\pi mc}}.$

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.

Abraham–Lorentz–Dirac Force

Dirac renormalized the mass in the equation of motion with the Abraham–Lorentz force 1938 and this renormalized equation of motion is called the Abraham–Lorentz–Dirac equation of motion.

Definition

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

$F_{\mu }^{\mbox{rad}}={\frac {\mu _{o}q^{2}}{6\pi mc}}\left({\frac {d^{2}p_{\mu }}{d\tau ^{2}}}-{\frac {p_{\mu }}{m^{2}c^{2}}}\,\left({\frac {dp_{\nu }}{d\tau }}{\frac {dp^{\nu }}{d\tau }}\right)\right)$

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.

${\frac {1}{\Delta t}}\int _{0}^{t}Pdt={\frac {1}{\Delta t}}\int _{0}^{t}{\textbf {F}}\cdot {\textbf {v}}dt$

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, and is further discussed by Rohrlich and Medina.

Ribarič and Šušteršič interpreted the Lorentz-Abraham-Dirac equation as the lowest order, asymptotic differential relation for the velocity of the radiating pointlike mass. Such a relation can be used to determine experimentally its kinetic constants, but not to compute its trajectory.

See also

• Max Abraham
• Hendrik Lorentz
• Cyclotron radiation
• Electromagnetic mass
• Radiation resistance
• Radiation damping
• Synchrotron radiation
• Wheeler–Feynman absorber theory

Further reading

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X. See sections 11.2.2 and 11.2.3
• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.\
• Donald H. Menzel, Fundamental Formulas of Physics, 1960, Dover Publications Inc., ISBN 0-486-60595-7, vol. 1, page 345.

External links

• MathPages - Does A Uniformly Accelerating Charge Radiate?
• Feynman: The Development of the Space-Time View of Quantum Electrodynamics