10.07.04 · em-sr / radiation

Radiation reaction: the Abraham-Lorentz force and the self-energy problem

shipped3 tiersLean: none

Anchor (Master): Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 16.1-16.4; Rohrlich, Classical Charged Particles, 3rd ed. (2007)

Intuition Beginner

When a charged particle accelerates, it radiates electromagnetic waves. By energy conservation, the radiated energy must come from somewhere — it comes from the particle's own kinetic energy. This means the radiation exerts a back-reaction force on the particle, slowing it down. This force is called the radiation reaction or the Abraham-Lorentz force.

The radiation reaction force for a non-relativistic charge is:

F_{rad} = \frac{μ _{0} q ^{2}}{6 π c} \dot{a}

where $\dot{a}$ is the rate of change of acceleration (the "jerk"). This is a peculiar force: it depends not on the velocity or acceleration, but on the derivative of the acceleration.

This leads to two famous problems. First, the runaway solutions: if a particle is released from rest with no external force, the Abraham-Lorentz equation $F_{rad} = m a$ admits solutions where the acceleration grows exponentially, violating conservation of energy. Second, the pre-acceleration problem: to avoid runaways, one must impose a boundary condition that requires the particle to start accelerating before the force is applied, violating causality.

These pathologies signal that classical electrodynamics breaks down at very small distances. Quantum mechanics resolves the problems, but the classical Abraham-Lorentz force remains the correct description for macroscopic charged particles.

Visual Beginner

Problem	Description
Self-energy divergence	Point charge has infinite electrostatic energy
Runaway solutions	Acceleration grows without bound
Pre-acceleration	Particle responds before force is applied
Resolution	Quantum mechanics at small scales

Worked example Beginner

The radiation reaction coefficient is $τ = μ_{0} q^{2} / (6 π m c)$ . For an electron:

$τ = 4 π \times 1 0^{- 7} \times (1.6 \times 1 0^{- 19})^{2} / (6 π \times 9.11 \times 1 0^{- 31} \times 3 \times 1 0^{8})$

$= 4 π \times 1 0^{- 7} \times 2.56 \times 1 0^{- 38} / (6 π \times 2.73 \times 1 0^{- 22})$

$\approx 6.3 \times 1 0^{- 24}$ s.

The characteristic time $τ \sim 1 0^{- 23}$ s is the time for light to cross the classical electron radius ( $r_{e} \sim 1 0^{- 15}$ m). On any timescale longer than $τ$ , the radiation reaction is a small perturbation. Only at sub-atomic timescales do the pathologies of the Abraham-Lorentz equation become relevant.

Check your understanding Beginner

Formal definition Intermediate+

The Abraham-Lorentz equation. The equation of motion for a non-relativistic charged particle including radiation reaction is:

m a = F_{ext} + τ m \dot{a}

where $τ = μ_{0} q^{2} / (6 π m c)$ is the radiation reaction time and $F_{ext}$ is the external force.

Derivation from energy conservation. The power radiated by the Larmor formula 10.07.01 is $P = μ_{0} q^{2} a^{2} / (6 π c)$ . The work done by the radiation reaction force must supply this energy:

\int_{t_{1}}^{t_{2}} F_{rad} \cdot v d t = - \frac{μ _{0} q ^{2}}{6 π c} \int_{t_{1}}^{t_{2}} a^{2} d t

Integrating the right side by parts: $\int a^{2} d t = - \int v \cdot \dot{a} d t + [v \cdot a]_{t_{1}}^{t_{2}}$ . For periodic motion (or motion that starts and ends at rest), the boundary term vanishes, giving:

\int_{t_{1}}^{t_{2}} (F_{rad} - \frac{μ _{0} q ^{2}}{6 π c} \dot{a}) \cdot v d t = 0

Since this holds for arbitrary $v (t)$ : $F_{rad} = (μ_{0} q^{2} /6 π c) \dot{a} = τ m \dot{a}$ .

Runaway solutions. For $F_{ext} = 0$ : $m a = τ m \dot{a}$ , giving $a (t) = a_{0} e^{t / τ}$ . The acceleration grows exponentially with timescale $τ \sim 1 0^{- 23}$ s. This is unphysical: a free charge spontaneously accelerates to infinite speed.

Pre-acceleration (acausal) solution. Imposing the boundary condition $a (t) \to 0$ as $t \to \infty$ (no runaways at late times) gives:

a (t) = \frac{1}{τ m} \int_{t}^{\infty} F_{ext} (t^{'}) e^{(t - t^{'}) / τ} d t^{'}

The acceleration at time $t$ depends on the force at future times $t^{'} > t$ , weighted by $e^{- (t^{'} - t) / τ}$ . The particle anticipates the force by a time of order $τ \sim 1 0^{- 23}$ s. This violates causality, but only on timescales so short that quantum mechanics dominates.

The Lorentz-Dirac equation (relativistic). The covariant generalisation of the Abraham-Lorentz equation is the Lorentz-Dirac equation:

m \frac{d u ^{μ}}{d τ _{p}} = F_{ext}^{μ} + τ m (\frac{d ^{2} u ^{μ}}{d τ _{p}^{2}} + \frac{d u ^{ν}}{d τ _{p}} \frac{d u _{ν}}{d τ _{p}} u^{μ})

where $τ_{p}$ is proper time. The second term in the radiation reaction is the Schott term (representing the near-field energy that is temporarily stored and then released).

Key derivation Intermediate+

Derivation (Abraham-Lorentz force from energy conservation).

Theorem. A non-relativistic charge q subject to external force F_ext satisfies the equation of motion $m a = F_{ext} + τ m \dot{a}$ where $τ = μ_{0} q^{2} / (6 π m c)$ .

Proof. By Larmor's formula, the radiated power is $P = μ_{0} q^{2} a^{2} / (6 π c)$ . The work-energy theorem requires that the radiation reaction force $F_{rad}$ satisfies:

W_{rad} = \int_{t_{1}}^{t_{2}} F_{rad} \cdot v d t = - \frac{μ _{0} q ^{2}}{6 π c} \int_{t_{1}}^{t_{2}} a^{2} d t

Integrating the right side by parts (assuming periodic motion or $a (t_{1}) = a (t_{2}) = 0$ ):

\int_{t_{1}}^{t_{2}} a^{2} d t = [v \cdot a]_{t_{1}}^{t_{2}} - \int_{t_{1}}^{t_{2}} v \cdot \dot{a} d t = - \int_{t_{1}}^{t_{2}} v \cdot \dot{a} d t

Therefore:

\int_{t_{1}}^{t_{2}} (F_{rad} + \frac{μ _{0} q ^{2}}{6 π c} \dot{a}) \cdot v d t = 0

Since this holds for all trajectories, $F_{rad} = - (μ_{0} q^{2} /6 π c) \dot{a}$ up to a total time derivative (which does not contribute to the integral). Choosing the simplest form gives the Abraham-Lorentz force. $□$

Bridge. The radiation reaction problem builds toward one of the deepest unsolved problems in classical physics: the self-energy of a point charge. The foundational insight is that radiation carries energy away from the source, and this energy loss must appear as a force on the source. The central message is that the resulting equation of motion is third-order in time, leading to unphysical runaway and acausal solutions. This is exactly the signal that classical electrodynamics is incomplete at the atomic scale. Putting these together, the radiation reaction is the classical limit of the quantum vacuum polarisation effects in QED, the self-energy divergence is resolved by mass renormalisation, and the Abraham-Lorentz force determines the energy loss of particles in accelerators 10.07.05.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

A charge is released from rest at $t = 0$ and acted on by a constant external force $F_{0}$ for time $T$ , then the force is removed. Solve the Abraham-Lorentz equation and show that the pre-acceleration is confined to times $∣ t ∣ ≲ τ$ before $t = 0$ .

Hint

Use the pre-acceleration solution $a (t) = (1/ τ m) \int_{t}^{\infty} F (t^{'}) e^{(t - t^{'}) / τ} d t^{'}$ with $F (t^{'}) = F_{0}$ for $0 < t^{'} < T$ and $F (t^{'}) = 0$ otherwise.

Answer

For $t < 0$ : $a (t) = (F_{0} / τ m) \int_{0}^{T} e^{(t - t^{'}) / τ} d t^{'} = (F_{0} / τ m) e^{t / τ} (1 - e^{- T / τ})$ .

For $0 < t < T$ : $a (t) = (F_{0} / τ m) \int_{t}^{T} e^{(t - t^{'}) / τ} d t^{'} = (F_{0} / m) (1 - e^{(t - T) / τ})$ .

For $t > T$ : $a (t) = 0$ (no force, and the boundary condition prevents runaways).

At $t = 0$ : $a (0) = (F_{0} / τ m) (1 - e^{- T / τ})$ . For $T ≫ τ$ : $a (0) \approx F_{0} / m$ (the expected Newtonian result).

For $t ≪ 0$ : $a (t) \approx (F_{0} / τ m) e^{t / τ} \to 0$ exponentially. The pre-acceleration is confined to times $∣ t ∣ ≲ τ$ before $t = 0$ , consistent with the claim that the acausality occurs on the timescale $τ \sim 1 0^{- 23}$ s.

Exercise 4 (medium, symbolic).

Show that the Abraham-Lorentz equation for a free particle ( $F_{ext} = 0$ ) has the general solution $v (t) = v_{0} + A e^{t / τ}$ where $A$ is an arbitrary constant. Identify the runaway mode.

Hint

The equation is $m a = τ m \dot{a}$ , i.e., $\dot{a} = a / τ$ .

Answer

For $F_{ext} = 0$ : $a = τ \dot{a}$ , giving $a (t) = A e^{t / τ}$ . Integrating: $v (t) = v_{0} + τ A e^{t / τ}$ .

For $A \neq = 0$ : $∣ v ∣ \to \infty$ as $t \to \infty$ . This is the runaway mode: a free charge spontaneously accelerates to infinite speed with timescale $τ \sim 1 0^{- 23}$ s. The only non-runaway solution is $A = 0$ , giving $v = v_{0}$ (uniform motion, as expected).

The runaway solution is unphysical and is eliminated by imposing the boundary condition $a (t) \to 0$ as $t \to \infty$ . But this boundary condition is imposed at future times, which leads to the pre-acceleration problem: the present acceleration depends on future forces.

Exercise 5 (medium, symbolic).

The energy radiated per revolution by an electron in a circular orbit of radius $R$ and speed $v$ is $Δ E = (q^{2} / (3 ϵ_{0})) (v^{4} / c^{4} R)$ per orbit. Compute this for a 1 GeV electron in a 100 m radius ring.

Hint

$Δ E = q^{2} γ^{4} / (3 ϵ_{0} R)$ (the Larmor formula integrated over one orbit, with $a = v^{2} / R$ and relativistic correction $γ^{4}$ ). For $γ \approx E / (m_{e} c^{2}) = 1 0^{9} / (511 \times 1 0^{3}) \approx 1957$ .

Answer

For $E = 1$ GeV: $γ = E / (m_{e} c^{2}) = 1 0^{9} / (0.511 \times 1 0^{6}) = 1957$ .

$Δ E = \frac{e ^{2} γ ^{4}}{3 ϵ _{0} R} = \frac{( 1.6 \times 1 0 ^{- 19} ) ^{2} \times ( 1957 ) ^{4}}{3 \times 8.85 \times 1 0 ^{- 12} \times 100}$

$= \frac{2.56 \times 1 0 ^{- 38} \times 1.466 \times 1 0 ^{13}}{2.655 \times 1 0 ^{- 9}} = \frac{3.75 \times 1 0 ^{- 25}}{2.655 \times 1 0 ^{- 9}} = 1.41 \times 1 0^{- 16}$ J $= 0.88$ keV.

The electron loses 0.88 keV per revolution to synchrotron radiation. This energy must be replaced by the RF cavities in the accelerator. For a ring of circumference $2 π \times 100 = 628$ m, the revolution frequency is $f = c / (2 π R) = 4.77 \times 1 0^{5}$ Hz, and the total radiated power is $P = f Δ E = 6.7 \times 1 0^{- 11}$ W per electron. For a beam of $1 0^{11}$ electrons: $P = 6.7$ kW — a significant power loss.

Exercise 6 (hard, symbolic).

Show that the Lorentz-Dirac equation conserves four-momentum by verifying that the radiation reaction four-force $K_{rad}^{μ} = τ m (\overset{u}{¨}^{μ} + \overset{u}{˙}^{ν} \overset{u}{˙}_{ν} u^{μ})$ is orthogonal to the four-velocity: $K_{rad}^{μ} u_{μ} = 0$ .

Hint

Differentiate $u^{μ} u_{μ} = c^{2}$ twice in proper time and use the result to show the Schott term cancels the $\overset{u}{˙}^{2}$ term.

Answer

From $u^{μ} u_{μ} = c^{2}$ : $\overset{u}{˙}^{μ} u_{μ} + u^{μ} \overset{u}{˙}_{μ} = 2 \overset{u}{˙}^{μ} u_{μ} = 0$ , so $\overset{u}{˙} \cdot u = 0$ . Differentiating again: $\overset{u}{¨} \cdot u + \overset{u}{˙} \cdot \overset{u}{˙} = 0$ , so $\overset{u}{¨} \cdot u = - \overset{u}{˙} \cdot \overset{u}{˙}$ .

Compute $K_{rad}^{μ} u_{μ} = τ m (\overset{u}{¨}^{μ} u_{μ} + \overset{u}{˙}^{ν} \overset{u}{˙}_{ν} u^{μ} u_{μ}) = τ m (- \overset{u}{˙}^{2} + \overset{u}{˙}^{2} c^{2} / c^{2}) = τ m (- \overset{u}{˙}^{2} + \overset{u}{˙}^{2}) = 0$ .

The orthogonality $K^{μ} u_{μ} = 0$ ensures that the radiation reaction preserves the mass-shell condition $p^{μ} p_{μ} = m^{2} c^{2}$ . The first term ( $\overset{u}{¨}^{μ}$ , the Schott term) represents the near-field energy that is temporarily stored and then released as radiation. The second term ( $\overset{u}{˙}^{2} u^{μ}$ ) is the net radiation loss.

Exercise 7 (hard, symbolic).

The classical electron radius $r_{e} = e^{2} / (4 π ϵ_{0} m c^{2}) \approx 2.82 \times 1 0^{- 15}$ m is obtained by setting the electrostatic self-energy equal to the rest energy. Show that $τ = r_{e} / (2 c)$ and discuss why this relation connects the radiation reaction to the electron's spatial extent.

Hint

$τ = μ_{0} e^{2} / (6 π m c)$ . $r_{e} = e^{2} / (4 π ϵ_{0} m c^{2})$ . Use $μ_{0} ϵ_{0} = 1/ c^{2}$ .

Answer

$τ = μ_{0} e^{2} / (6 π m c) = μ_{0} ϵ_{0} c^{2} \cdot e^{2} / (6 π ϵ_{0} m c^{3}) = e^{2} / (6 π ϵ_{0} m c^{3})$ .

$r_{e} = e^{2} / (4 π ϵ_{0} m c^{2})$ .

$τ / r_{e} = (e^{2} / (6 π ϵ_{0} m c^{3})) / (e^{2} / (4 π ϵ_{0} m c^{2})) = (4 π) / (6 π c) = 2/ (3 c)$ . Wait: $τ = r_{e} \cdot (2/ (3 c))$ . More precisely: $τ = (2/3) (r_{e} / c)$ .

This means $τ$ is the light-travel time across the classical electron radius (up to a factor of 2/3). The radiation reaction time is the time it takes a light signal to cross the electron. This connects the radiation reaction to the electron's spatial structure: a finite-size electron would not have the self-energy divergence, and the radiation reaction would arise from the finite propagation time of the field across the electron's extent.

In QED, the electron is treated as a point particle, and the divergences are handled by renormalisation. The classical electron radius $r_{e}$ is not the actual size of the electron (which is pointlike to $< 1 0^{- 18}$ m experimentally), but a characteristic scale where classical electrodynamics breaks down.

Exercise 8 (hard, symbolic).

The Landau-Lifshitz equation is an approximation to the Lorentz-Dirac equation that avoids the third-order structure by treating the radiation reaction as a perturbation:

m \overset{u}{˙}^{μ} = F_{ext}^{μ} + τ (\partial_{ν} F_{ext}^{μν} u^{ν} + F_{ext}^{μν} \overset{u}{˙}_{ν})

Show that this is a second-order ODE (no $\overset{u}{¨}$ term) by substituting $\overset{u}{˙}^{ν} = F_{ext}^{ν} / m$ in the radiation reaction term.

Hint

The Landau-Lifshitz approach iterates once: replace $\overset{u}{˙}$ in the radiation reaction by the zeroth-order (Lorentz force) value, eliminating the third derivative.

Answer

Starting from the Lorentz-Dirac equation: $m \overset{u}{˙}^{μ} = F_{ext}^{μ} + τ m (\overset{u}{¨}^{μ} + \overset{u}{˙}^{2} u^{μ})$ .

The key idea: for $τ$ small, the radiation reaction is a perturbation. To first order in $τ$ , replace $\overset{u}{˙}^{μ}$ by $F_{ext}^{μ} / m$ (the zeroth-order Lorentz force) everywhere it appears in the radiation reaction:

τ m \overset{u}{¨}^{μ} \approx τ \frac{d}{d τ _{p}} F_{ext}^{μ} = τ (\partial_{ν} F^{μν} u^{ν} + F^{μν} \overset{u}{˙}_{ν}) \approx τ (\partial_{ν} F^{μν} u^{ν} + F^{μν} F_{ν} / m)

The result is:

m \overset{u}{˙}^{μ} = F_{ext}^{μ} + τ (\partial_{ν} F^{μν} u^{ν} + \frac{1}{m} F^{μν} F_{ν}) + τ \overset{u}{˙}^{2} u^{μ}

This is a second-order ODE: the right side contains $u^{μ}$ and $F_{ext} (x)$ (no derivatives of $u$ beyond first order). The Landau-Lifshitz equation is free of runaways and pre-acceleration, and agrees with the Lorentz-Dirac equation to order $τ$ . It is the preferred equation for practical calculations in accelerator physics.

Lean formalization Intermediate+

Mathlib has ODE theory but does not contain the Abraham-Lorentz equation, the runaway solutions, the pre-acceleration boundary condition, the Lorentz-Dirac equation, or the Landau-Lifshitz approximation. The self-energy divergence and mass renormalisation are beyond Mathlib's current scope. lean_status: none.

Advanced results Master

The self-energy and mass renormalisation. The electrostatic self-energy of a sphere of charge $q$ and radius $R$ is $U = q^{2} / (8 π ϵ_{0} R)$ . For a point charge ( $R \to 0$ ), $U \to \infty$ . The total mass is $m = m_{0} + U / c^{2}$ where $m_{0}$ is the "bare" mechanical mass. As $R \to 0$ : $m_{0}$ must be adjusted (renormalised) to keep $m$ finite. This is the classical analogue of QED mass renormalisation.

The Dirac derivation (1938). Dirac derived the Lorentz-Dirac equation by analysing the force on a spherical shell of charge, carefully separating the retarded and advanced fields at the shell surface. The difference $F_{ret} - F_{adv}$ gives the radiation reaction force, free of divergences. This approach avoids the self-energy problem by never computing the self-field energy.

Experimental tests. The radiation reaction has been directly measured in laser-electron interactions (Cole et al., 2018; Poder et al., 2018). Ultra-intense laser pulses ( $I > 1 0^{20}$ W/cm $^{2}$ ) collide with GeV electrons, and the radiation loss modifies the electron trajectories in a way that is consistent with the Landau-Lifshitz equation. At even higher intensities ( $I > 1 0^{23}$ W/cm $^{2}$ ), quantum radiation reaction effects (photon emission probabilities replacing continuous energy loss) are expected to become important.

Synthesis. The radiation reaction problem reveals the limits of classical electrodynamics: a consistent description of a point charge interacting with its own field leads to unphysical solutions. The foundational insight is that the back-reaction from radiation is a real physical effect that modifies the motion of charged particles. The central message is that the Abraham-Lorentz and Lorentz-Dirac equations are the best classical descriptions, valid on timescales much longer than $τ \sim 1 0^{- 23}$ s. Putting these together, the radiation reaction determines the energy loss in particle accelerators 10.07.05, the Landau-Lifshitz approximation provides the practical computational tool, and the resolution of the pathologies requires the quantum field theory treatment where the electron is a point particle with renormalised mass.

Full proof set Master

Proposition (Energy balance). For periodic motion, the time-averaged power delivered by the Abraham-Lorentz force equals the time-averaged Larmor radiated power.

Proof. The radiation reaction power is $P_{rad} = F_{rad} \cdot v = τ m \dot{a} \cdot v$ . Time-averaging over a period $T$ : $⟨ τ m \dot{a} \cdot v ⟩ = τ m ⟨ \dot{a} \cdot v ⟩ = τ m ⟨ d (a \cdot v) / d t - a^{2} ⟩ = - τ m ⟨ a^{2} ⟩$ (the total derivative averages to zero for periodic motion). The Larmor power is $⟨ P_{Larmor} ⟩ = μ_{0} q^{2} ⟨ a^{2} ⟩ / (6 π c) = τ m ⟨ a^{2} ⟩$ . So $⟨ P_{rad} ⟩ = - ⟨ P_{Larmor} ⟩$ : the radiation reaction removes energy at exactly the rate predicted by Larmor. $□$

Connections Master

Larmor formula 10.07.01 gives the radiated power; the radiation reaction force is derived from energy conservation with this power.
Poynting vector 10.03.05 describes the energy flux that carries away the radiated power.
Synchrotron radiation 10.07.05 is the relativistic generalisation where radiation reaction is strongest.
Green's functions 10.08.01 provide the retarded field formalism underlying the self-force derivation.
Stress-energy tensor 10.06.03 gives the covariant framework for the Lorentz-Dirac equation.

Historical & philosophical context Master

The radiation reaction problem has a long and controversial history. The Abraham-Lorentz force was derived independently by Abraham (1904) and Lorentz (1909) using different methods. The runaway solutions were immediately recognised as a problem, and various resolutions were proposed over the following decades.

Dirac (1938) gave the most elegant derivation of the relativistic equation by using the difference between retarded and advanced fields to isolate the radiation reaction, avoiding the self-energy divergence. However, the pre-acceleration problem remained.

The Landau-Lifshitz equation (1951) provided a practical resolution by reducing the third-order equation to a second-order perturbative approximation. This is now the standard equation used in accelerator physics.

The philosophical significance is that the radiation reaction problem is the first place where classical field theory encounters a genuine inconsistency (as opposed to a mere inaccuracy). The resolution in QED through renormalisation shows that the classical theory is an effective theory that breaks down at short distances, analogous to how Newtonian mechanics breaks down at high velocities.

Bibliography Master

Abraham, M., Theorie der Elektrizitaet, Vol. 2 (Teubner, 1904).
Lorentz, H. A., The Theory of Electrons (Teubner, 1909).
Dirac, P. A. M., "Classical theory of radiating electrons," Proc. Roy. Soc. A 167, 148-169 (1938).
Landau, L. D. and Lifshitz, The Classical Theory of Fields, 2nd ed. (Pergamon, 1951).
Rohrlich, F., Classical Charged Particles, 3rd ed. (World Scientific, 2007).
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Prerequisites

10.07.01
10.03.05

Used in

10.07.05

Tier anchors

beginner: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 11.2
intermediate: Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 16.1-16.3
master: Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 16.1-16.4; Rohrlich, Classical Charged Particles, 3rd ed. (2007)

References

Jackson — Classical Electrodynamics, 3rd ed. (Wiley, 1999) · Ch. 16.1-16.3 Radiation reaction; the Abraham-Lorentz force; runaway solutions and preacceleration
Griffiths — Introduction to Electrodynamics, 4th ed. (Cambridge, 2017) · Ch. 11.2 Radiation reaction; the Abraham-Lorentz formula; the self-energy problem
Rohrlich — Classical Charged Particles, 3rd ed. (World Scientific, 2007) · Ch. 6 The self-force and the equation of motion; the Lorentz-Dirac equation

Estimated time

beginner: 12m
intermediate: 35m
master: 60m