Radiation Reaction and Energy Conservation

We know that

$$\mbox{\boldmath$F$}_{\rm tot} = m \dot{\mbox{\boldmath$v$}}$$ (19.7)

is (nonrelativistic) Newton's 2nd Law for a charged particle being accelerated by a (for the moment, non-electromagnetic) given external force. The work energy theorem dictates how fast the particle can gain kinetic energy if this is the only force acting.

However, at the same time it is being acted on by the external force (and is accelerating), it is also radiating power away at the total rate:

$$P(t) = \frac{2}{3c} \frac{e^2}{4\pi\epsilon_0 c^2} \dot{\mbox{\boldmath$v$}}^2$$

$$= \frac{2}{3} \frac{m r_e}{c} \dot{\mbox{\boldmath$v$}}^2$$

$$= m\tau_r \dot{\mbox{\boldmath$v$}}^2$$ (19.8)

(the Larmor formula). These are the two pieces we've thus far treated independently, neglecting the one to obtain the other.

However, in order for Newton's law to correctly lead to the conservation of energy, the work done by the external force must equal the increase in kinetic energy plus the energy radiated into the field. Energy conservation for this system states that:

$$W_{\rm ext} = \Delta E_e + \Delta E_f$$ (19.9)

or the total work done by the external force must equal the change in the total energy of the charged particle (electron) plus the energy that appears in the field. If we rearrange this to:

$$W_{\rm ext} - \Delta E_f = \Delta E_e$$ (19.10)

and consider the electron only, we are forced to conclude that there must be another force acting on the electron, one where the total work done by the force decreases the change in energy of the electron and places the energy into the radiated field. We call that force $\mbox{\boldmath$F$}_{\rm rad}$, the radiation reaction force.

Thus (rewriting Newton's second law in terms of this force):

$$\mbox{\boldmath$F$}_{\rm ext} + \mbox{\boldmath$F$}_{\rm rad} = m \dot{\mbox{\boldmath$v$}}$$

$$\mbox{\boldmath$F$}_{\rm rad} = m \dot{\mbox{\boldmath$v$}} - \mbox{\boldmath$F$}_{\rm ext}$$ (19.11)

defines the radiation reaction force that must act on the particle in order for energy conservation to make sense. The reaction force has a number of necessary or desireable properties in order for us to not get into ``trouble''^19.2.

• We would like energy to be conserved (as indicated above), so that the energy that appears in the radiation field is balanced by the work done by the radiation reaction force (relative to the total work done by an external force that makes the charge accelerate).
• We would like this force to vanish when the external force vanishes, so that particles do not spontaneously accelerate away to infinity without an external agent acting on them.
• We would like the radiated power to be proportional to $e^2$, since the power and its space derivatives is proporotional to $e^2$ and since the force magnitude should be dependent of the sign of the charge.
• Finally, we want the force to involve the ``characteristic time'' $\tau$ (whereever it needs a parameter with the dimensions of time) since no other time-scaled parameters are available.

Let's start with the first of these. We want the energy radiated by some ``bound'' charge (one undergoing periodic motion in some orbit, say) to equal the work done by the radiation reaction force in the previous equation. Let's start by examining just the reaction force and the radiated power, then, and set the total work done by the one to equal the total energy radiated in the other, over a suitable time interval:

$$\int_{t_1}^{t_2} \mbox{\boldmath$F$}_{\rm rad} \cdot \mbox{... ... \dot{\mbox{\boldmath$v$}} \cdot \dot{\mbox{\boldmath$v$}} dt$$ (19.12)

for the relation between the rates, where the minus sign indicates that the energy is removed from the system. We can integrate the right hand side by parts to obtain

$$\int_{t_1}^{t_2} \mbox{\boldmath$F$}_{\rm rad} \cdot \mbox{... ...ox{\boldmath$v$}} \cdot \mbox{\boldmath$v$}) \mid_{t_1}^{t_2}$$ (19.13)

Finally, the motion is ``periodic'' and we only want the result over a period; we can therefore pick the end points such that $\dot{\mbox{\boldmath$v$}} \cdot {\mbox{\boldmath$v$}} = 0$. Thus we get

$$\int_{t_1}^{t_2} \left( \mbox{\boldmath$F$}_{\rm rad} - m\t... ...mbox{\boldmath$v$}} \right) \cdot \mbox{\boldmath$v$}dt = 0 .$$ (19.14)

One (sufficient but not necessary) way to ensure that this equation be satisfied is to let

$$\mbox{\boldmath$F$}_{\rm rad} = m \tau_r \ddot{\mbox{\boldmath$v$}}$$ (19.15)

This turns Newton's law (corrected for radiation reaction) into

$$\mbox{\boldmath$F$}_{\rm ext} = m\dot{\mbox{\boldmath$v$}} - \mbox{\boldmath$F$}_{\rm rad}$$

$$= m(\dot{\mbox{\boldmath$v$}} - \tau_r \ddot{\mbox{\boldmath$v$}})$$ (19.16)

This is called the Abraham-Lorentz equation of motion and the radiation reaction force is called the Abraham-Lorentz force. It can be made relativistic be converting to proper time as usual.

Note that this is not necessarily the only way to satisfy the integral constraint above. Another way to satisfy it is to require that the difference be orthogonal to ${\mbox{\boldmath$v$}}$. Even this is too specific, though. The only thing that is required is that the total integral be zero, and short of decomposing the velocity trajectory in an orthogonal system and perhaps using the calculus of variations, it is not possible to make positive statements about the necessary form of $\mbox{\boldmath$F$}_{\rm rad}$.

This ``sufficient'' solution is not without problems of its own, problems that seem unlikely to go away if we choose some other ``sufficient'' criterion. This is apparent from the observation that they all lead to an equation of motion that is third order in time. Now, it may not seem to you (yet) that that is a disaster, but it is.

Suppose that the external force is zero at some instant of time $t = 0$. Then

$$\dot{\mbox{\boldmath$v$}} \approx \tau \ddot{\mbox{\boldmath$v$}}$$ (19.17)

or

$$\dot{\mbox{\boldmath$v$}}(t) = \mbox{\boldmath$a$}_0 e^{t/\tau}$$ (19.18)

where $\mbox{\boldmath$a$}_0$ is the instantaneous acceleration of the particle at $t = 0$.

Recalling that $\mbox{\boldmath$v$}\cdot \dot{\mbox{\boldmath$v$}} = 0$ at $t_1$ and $t_2$, we see that this can only be true if $\mbox{\boldmath$a$}_0 = 0$ (or we can relax this condition and pick up an additional boundary condition and work much harder to arrive at the same conclusion). Dirac had a simply lovely time with the third order equation. Before attacking it, though, let us obtain a solution that doesn't have the problems associated with it in a different (more up-front) way.

Let us note that the radiation reaction force in almost all cases will be very small compared to the external force. The external force, in addition, will generally be ``slowly varying'', at least on a timescale compared to $\tau_r \approx 10^{-24}$ seconds. If we assume that $\mbox{\boldmath$F$}_{\rm ext}(t)$ is smooth (continuously differentiable in time), slowly varying, and small enough that $\mbox{\boldmath$F$}_{\rm rad} \ll \mbox{\boldmath$F$}_{\rm ext}$ we can use what amounts to perturbation theory to determine $\mbox{\boldmath$F$}_{\rm rad}$ and obtain a second order equation of motion.

Under these circumstances, we can assume that $\mbox{\boldmath$F$}_{\rm ext} \approx m\dot{\mbox{\boldmath$v$}}$, so that:

$$\mbox{\boldmath$F$}_{\rm ext} = m(\dot{\mbox{\boldmath$v$}} - \tau_r \ddot{\mbox{\boldmath$v$}})$$

$$\approx m \dot{\mbox{\boldmath$v$}} - \tau_r \frac{d \mbox{\boldmath$F$}_{\rm ext}}{dt}$$ (19.19)

or

$$m \dot{\mbox{\boldmath$v$}} = \mbox{\boldmath$F$}_{\rm ext} + \tau_r \frac{d \mbox{\boldmath$F$}_{\rm ext}}{dt}$$

$$= \mbox{\boldmath$F$}_{\rm ext} + \tau_r \left\{\frac{\partial }{\p... ...oldmath$v$}\cdot \mbox{\boldmath$\nabla$})\right\}\mbox{\boldmath$F$}_{\rm ext}$$ (19.20)

This latter equation has no runaway solutions or acausal behavior as long as $\mbox{\boldmath$F$}_{\rm ext}$ is differentiable in space and time.

We will defer the discussion of the covariant, structure free generalization of the Abraham-Lorentz derivation until later. This is because it involves the use of the field stress tensor, as does Dirac's original paper -- we will discuss them at the same time.

What are these runaway solutions of the first (Abraham-Lorentz) equation of motion? Could they return to plague us when the force is not small and turns on quickly? Let's see...