Radiation reaction and the equivalence principle (again!)

It's taken me a long time, but I've finally written up what I think
about the radiation reaction force on charged particles, the
equivalence principle, and all that. This is largely a response to
some posts by Stephen Parrott a while back.

This is a really long article, and probably few people (if any)
will bother to read it all the way through. I divided
it into sections so that you can find just the bits (if any)
that interest you.

PART 1 summarizes the problem that I'm interested in, which is an
apparent contradiction between the equivalence principle and the
radiation produced by a charged particle. Specifically, I'm concerned
with the part of this problem that has to do with the radiation
reaction force: either it's zero for uniform acceleration (which seems
to contradict energy conservation), or it's not (which seems to
contradict the equivalence principle). Parrott argues for the
latter position. I want to claim that the former position has not
been shown to be wrong, and that there are some reasons to think
it's right.

PART 2 states my opinions about what we do and don't know about
the radiation reaction force on a charged particle.

In PART 3 I try to argue in favor of the possibility that
the radiation reaction force vanishes for uniform acceleration.
To be specific, I just want to argue that this possibility
is not obviously wrong. This part has some grungy
calculation.

PART 4 tries to make the same argument from another point of view.
There's less calculation here.

PART 5 looks at specific arguments made in Parrott's paper gr-qc/9303025 .
This is the only place in the literature, as far as I know, where
this issue is addressed.

Enjoy!

-Ted

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PART 1. The Problem.

Suppose you wake up in a room with no windows. You're sitting on the
floor, held there by what seems to you like a gravitational force. As
everyone knows, you can't tell whether there really is a gravitational
force or whether the room is uniformly accelerating. That's the
equivalence principle.

Or can you? In gr-qc/9303025, Parrott says you can, as long as you
have a charged particle with you. If the room is accelerating, then
the charge will be steadily radiating energy off into the surrounding
Minkowski space. If you're in a static gravitational field, it won't.
Parrott puts it like this:

"Imagine that the acceleration of a charged particle in Minkowski
space is produced by a tiny rocket engine attached to the particle.
Since the particle is radiating energy which can be detected and
used, conservation of energy suggests that the radiated energy must
be furnished by the rocket -- we must burn more fuel to produce a
given accelerating worldline than we would to produce the same
worldline for a neutral particle of the same mass. Now consider a
stationary charge in Schwarzschild space-time, and suppose a rocket
holds it stationary relative to the coordinate frame (accelerating
with respect to local inertial frames). In this case, since no
radiation is produced, the rocket should use the same amount of fuel
as would be required to hold stationary a similar neutral particle.
This gives an experimental test by which we can determine *locally*
whether we are accelerating in Minkowski space or stationary in a
gravitational field -- simply observe the rocket's fuel consumption."

Before I go on, let me say that Parrott's article says much more than
just this! The bulk of the article is a careful and mathematically
rigorous analysis of what one means by "energy" and "radiation" in
this sort of situation. A number of things have been written over the
years on this subject that are muddled and even just plain wrong, and
Parrott's paper is an excellent antidote. Although I want to express
doubt in the argument quoted above, I want to emphasize that Parrott's
analysis of these topics is extremely lucid, and anyone who wants to
understand this topic absolutely must study this paper.

The above argument depends crucially on the following claim:

A. A uniformly accelerating rocket carrying a charged particle must
burn more fuel than a uniformly accelerating rocket carrying a
neutral particle of the same rest mass and acceleration.

(That's not a quote; it's my paraphrase.)

Let's make the following assumptions:

1. Energy is conserved, in some appropriate sense.
2. Uniformly accelerated charges radiate.

It's not completely obvious what one means by assumption 1 when there
are charged particles running around, since the field energy of a
charged particle is infinite. At the very least, it should mean that
you can't use charged particles to make a perpetual motion machine: if
you push a charged particle around for a while, and end up with it
back in its initial state, the work you did pushing it around had
better be equal to the extra energy in the fields.

Some people seem to reject one or the other of these two assumptions
as a way of "rescuing" the equivalence principle. I don't want to do
that. I believe in Maxwell's equations, which make it quite clear
that uniformly accelerated charges radiate, and that
Minkowski-dwellers can gather that radiation up and use it for
whatever they like. And I certainly don't want to give up energy
conservation! So the question I have is this: Do assumptions 1 and 2
together imply the claim A? It seems intuitively obvious that the
answer is yes, but as everyone who's done physics for a while knows,
intuition is sometimes wrong!

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PART 2: What do we know for sure?

So is claim A true or false? The only answer that is completely
unassailable is this: WE DON'T KNOW. The reason is simple: Claim A is
a statement about the radiation reaction force on a charged particle,
and there is no acceptable theory of the radiation reaction force on
charged particles. To be specific, there is no known theory that (a)
conserves energy-momentum in some suitable renormalized sense and (b)
has non-pathological solutions for generic physically reasonable
initial data.

The main candidate for the radiation reaction force is something
called the Lorentz-Dirac equation (sometimes with Abraham's name
thrown in). This equation is famously pathological: it generically
leads to the particle zooming off with unbounded acceleration forever
once the external force has gone away. Some people seem to think that
only special, unrealistic initial conditions lead to those runaway
solutions, but those people are wrong: the pathological solutions are
completely generic. Once again, Parrott has written very lucidly on
this. Check out math-ph/0505042 .

People have written down other candidates for a theory of radiation
reaction, but as far as I can tell they're all unsatisfactory in one
way or another. Often, these are "effective" theories of point
charges, presumed valid in some approximation for sufficiently gradual
accelerations. For instance, they may be approximations based on an
expansion in the presumed-to-be-small quantity (classical electron
radius) / (some length scale associated with charge's motion). Both
the order-reduced Lorentz-Dirac equation and the
differential-difference equation advocated by Rohrlich fall into this
category.

I don't think there's anything wrong with this approach, as long as
everyone involved clearly understands that these are just
approximations. The danger is that someone might think they're
dealing with an exact theory, when really it's just one that conserves
energy to some finite order in an expansion in one of those small
quantities. In the case of the order-reduced Lorentz-Dirac equation,
I've never read a treatment that's really rigorous about the nature of
the approximations being made. Specifically, I've never read anything
that explains precisely what conditions have to be satisfied for the
errors to be small. Some friendly mathematical physicist should do
that, if it hasn't been done.

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PART 3: Come on! Isn't it obvious?

OK, so there's no satisfactory theory of radiation reaction on point
charges. Still, surely it's obvious that any reasonable theory would
satisfy Claim A: the little rocket has got to put out extra energy to
compensate for the radiated energy, doesn't it? It's clearly
impossible for the rocket to accelerate for a billion years, pouring
energy out into Minkowski space the whole time, without burning any
more fuel than it would have done if it weren't radiating!

Well, I claim it's not obvious. Let's see why. Let me emphasize here
that I'm not going to prove that claim A is false. I just want to
show that claim A is not obvious, by showing a theory in which you get
approximately the right total energy out, even without any radiation
reaction force during the time of uniform acceleration. I'll
say what I mean by "approximately" as we go along.

In order to discuss energy conservation at all, we need to consider an
acceleration that occurs for only a finite period of time. (If the
rocket has been accelerating ever since t=-infinity, then its energy
has been infinite the whole time, even after you do any needed mass
renormalization on the point charge.) So let's consider a particle
that

-starts at rest in some "lab frame,"
-is gently nudged into accelerated motion over a (proper) time delta,
-accelerates uniformly with proper acceleration A for a long time T,
-is gently nudged back into uniform motion over a (proper) time delta.

Parrott considers this situation in Appendix 1 of gr-qc/9303025. He
also imagines that the particle subsequently does the same steps in
reverse, so that at the very end it's at rest in the lab frame again.
Let's not bother with that part yet.

I claim that it's possible to conserve energy with a theory in which
the radiation reaction force is applied only during the "nudges" at
the beginning and the end.

Let's define a time constant

t0 = (2/3) e^2 / m.

Here e is the particle's charge and m is its mass. We're working in
Gaussian units with c=1. The constant t0 is just (2/3) of the
classical electron radius (if the particle is an electron, of course).
It's a really really small number.

I want to assume that the acceleration is not too ridiculously large.
Specifically, I want to assume that

A t0 << 1

For an electron, this means the acceleration must be less
than about 10^{32} m/s^2.

I also want to assume that the nudges are brief enough that

A delta << 1.

Physically, this just means that the particle doesn't go from rest to
a relativistic speed during the initial nudge. (I'm not assuming that
the time of uniform acceleration is small -- just the nudging time.)
Finally, I want to assume that the nudges are brief compared
to the duration of the acceleration:

delta << T.

With those assumptions, I can tell you a story that explains how the
particle can accelerate for as long as you like, even though the
radiation reaction force is zero during the period of uniform
acceleration. It goes like this:

1. During the initial nudge, the radiation reaction force
causes an impulse I = m A t0, as measured in the lab frame.
That is, at the end of the nudge, the particle is going *faster*
in the lab frame than if the same nudge had been applied to an
uncharged particle, by an amount A t0.

2. During the time of uniform acceleration, the radiation reaction
force is zero -- that is, the force required to cause the particle to
accelerate is the same as would be required for an uncharged particle
with the same world line.

3. During the final nudge, the same thing happens in reverse: there's
an impulse I = -m A t0, as measured in the particle's instantaneous
rest frame.

In this model, the radiation reaction force lest you "borrow" a bit of
energy when you start accelerating, but you have to "pay it back" at
the end.

Incidentally, I didn't just make up this model out of nowhere. It's
what you get from the Lorentz-Dirac equation if you play the standard
runaway-solution-avoiding trick. It's also what you get from the
order-reduced Lorentz-Dirac equation, and also from various models of
small charged spheres, to leading order in the small quantity
(acceleration times radius). I don't want to defend any of those
procedures particularly vigorously at the moment, and I never want to
defend the runaway-avoiding Lorentz-Dirac procedure, which involves
nasty pre-acceleration. I just figured I'd let you know where this
model came from.

It's easiest to say exactly what this means in terms of the particle's
rapidity, defined to be

eta = arctanh(v),

where v is the particle's speed as measured in a particular frame (say
the lab frame). Let's say that the nudges are from tau=-delta to
tau=0 and from tau=T to tau=T+delta, so that T is the proper time of
uniform acceleration. The rapidity is

eta(tau) = 0 for tau<-delta
= A (t0 + tau) for 0 < tau < T
= A T for tau > T+delta

I don't care to say exactly what happens during the two nudges.
Nothing too dramatic -- in particular, the proper accelerations during
those times are always at most of order A.

It's straightforward to check, using the Larmor formula, that energy
radiated (as measured in the lab frame) is

U_{rad} = m [A t0 sinh(A T) + O(A t0)^2+ O(A^2 t0 delta)]

The last two terms are down from the first by the small ratios
t0/T and delta/T.

To calculate the work done by the rocket engine, you integrate f^0
dtau, where f^0 is the lab-frame time component of the four-force.
Again, the integral is straightforward, and you get

W = m[(cosh(AT)-1) + A t0 sinh(AT)+ O(A t0)^2 + O(A delta)^2]

The first term is the change in the particle's mechanical energy
(mc^2*gamma - mc^2). The second term is the radiated energy. So up
to second order in those small quantities, everything works out just
as it should.

Of course I'd rather have an exact theory, instead of one with all
these approximations. But I don't. I do think, however, that this
provides a strong argument for the non-obviousness (although not for
the falsehood) of Claim A. Let me belabor this point a bit.

Consider the following scenario: Suppose you start with an electron at
rest. You nudge it over the course of a picosecond into accelerated
motion with an acceleration of 10^{20} m/s^2. You then make it
accelerate with uniform proper acceleration of 10^{20} m/s^2 for a
year by applying only the force that would be required for an
uncharged particle, and then nudge it back into uniform motion over a
picosecond.

Claim A is that it's impossible to do this while conserving energy.
But in the above model, energy is conserved to a part in 10^{11} of
the radiated energy. If you think that it's intuitively obvious that
energy can't be conserved, but it's OK for it to be 99.999999999%
conserved, all I can say is that you've got a much more refined
intuition for this stuff than I do!

I hope it's clear what I'm getting at here. I'm not saying that
energy really is only 99.999999999% conserved. Energy is 100%
conserved. I'm just saying that if a model with no radiation reaction
during uniform acceleration can conserve energy to 99.999999999%
accuracy (or some similarly ridiculously-close-to-100% value), then
it's not obvious that no theory could do it to 100% accuracy.
Depending on exactly what happens during the nudges, maybe you can
make a theory that's 100%-energy-conserving.

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PART 4: Another way of saying the same thing.

Here's Claim A:

A. A uniformly accelerating rocket carrying a charged particle must
burn more fuel than a uniformly accelerating rocket carrying a
neutral particle of the same rest mass and acceleration.

Let me modify it to the following:

A'. A uniformly accelerating rocket is carrying a charged sphere of
radius r. Assume the acceleration a is such that ar << 1. The
rocket must burn more fuel than it would have to if it were
carrying a neutral particle of the same rest mass.

If A is obvious, then surely A' is obvious, right? After all, there's
no important difference between the two cases: the little sphere is
radiating energy away into Minkowski space, and that energy's got to
come from somewhere, right?

Well, there's no satisfactory theory of point particles, but there are
perfectly satisfactory theories of little charged spheres. The
details depend on exactly how you model the sphere -- how the charge
is distributed, the stresses holding it together, etc. But I know
that in at least some of those models, and I presume in all of them,
the detailed calculation works out the way it does in the argument I
gave above: if you neglect the self-force during the time of uniform
acceleration, energy comes out to be conserved to an excellent
approximation: the error is smaller than the radiated energy by a
factor of ar, which is very small in realistic scenarios and which can
be made as small as you like.

As a specific example, take the delay-differential equation advocated
in various places by Rohrlich. Rohrlich likes to argue that this
equation is in some sense "the correct equation for a quasi-point
charge." I'm not sure what that means, and I don't want to assume the
burden of arguing for that claim. But viewed as just what it is,
namely an equation of motion derived for a charged sphere of finite
radius, it's a perfectly well-behaved theory. The only approximation,
as far as I can tell, that went into its construction is that ar << 1.
For a charged sphere in this approximation, you can calculate that the
self-force during the time of uniform acceleration is of order m t0 r
a^3. The rate at which power is being radiated is m t0 a^2. So the
self-force during the time of uniform acceleration is too small to
explain the radiation, by a factor of ar. In this situation, it is
unambiguously true that nearly all of the radiated energy is supplied
during the two nudges.

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PART 5: Parrott's arguments.

I know it's taken a long time to get here!

In Appendices 1 and 2 of gr-qc/9303025, Parrott argues for Claim A, in
particular as it relates to the Lorentz-Dirac equation. I'm a big fan
of Parrott's writings on this topic in general, and I want to give his
arguments serious attention, although as you'll see I find them
unconvincing.

Appendix 1 starts by describing the Lorentz-Dirac equation and shows
that the radiation reaction force vanishes for uniform acceleration.
Parrott then says,

"It is tempting to interpret this as implying that there is no
physical radiation reaction for a uniformly accelerated charged
particle, by which we mean that a rocket-driven uniformly
accelerated charge requires no more energy from the rocket than an
otherwise identical neutral charge. However, we believe such
an interpretation is unlikely to be correct.

"An obvious flaw in the argument just given is that it is
inconsistent with usual ideas about conservation of energy. If we
grant that the uniformly accelerated charge *does* radiate energy
into Minkowski space ... then this radiated energy must be furnished
by some decrease in energy of other parts of the system."

The response to this depends on whether we're considering uniform
acceleration for all time or acceleration of finite duration.
In the former case, the energy is always infinite anyway (even if we
renormalize the charge's field energy somehow), so it's not clear what
it would even mean to worry about energy conservation. In the latter
case, we need to do the detailed energy accounting to see how the
energy stored in the field and the mechanical energy of the particle
change with time. Without such a calculation, the "obvious flaw" is
not obvious at all.

Later in Appendix 1, he specializes to the "nudging" situation that I
described above. He states correctly that the Lorentz-Dirac equation
in this case "implies that all the radiation energy must be furnished
at the beginning and ending of the trip" during the two nudges.

"In effect, we can 'borrow' an arbitrarily large amount of radiated
energy (which in principle can meanwhile be collected and used by
other observers in Minkowski space), so long as we pay it back at
the end of the trip. Although there is no logical contradiction
here, this is hard to accept physically, and seems one of many good
reasons to question the Lorentz-Dirac equation."

I heartily agree with the "many good reasons" part: Lorentz-Dirac
theory is thoroughly pathological. But this isn't really an example
of that. The reason is that this conclusion isn't specific to the
Lorentz-Dirac equation: the same thing happens (to an excellent
approximation) even for finite-sized balls of charge, where there's no
doubt about the correct physics to use.

Appendix 2 has a detailed calculation of a charged rocket accelerating
in accordance with the Lorentz-Dirac equation. It begins by
describing a "noted expert"'s arguments for why it's OK for the
radiation reaction force to occur only during the nudges. The first
key observation is that "the energy required to accomplish the final
deceleration, as measured in the rest frame at t=tf, is modest and
independent of the period of uniform acceleration." That's completely
correct. He goes on to observe that "the modest amount of energy used
in the final frame ... can Lorentz-transform into a large amount of
energy in the initial rest frame ... so there is no apparent violation
of conservation of energy from the standpoint of the initial frame."
Absolutely right.

But, he says, we may run into trouble if we extend the thought
experiment a bit: after the particle has accelerated for a while and
then been nudged back into inertial motion, we reverse the whole
process: nudge it into acceleration in the opposite direction, let it
decelerate until it's at rest in the initial lab frame, and then nudge
it to a final stop. He seems to think that this gives a contradiction
somehow:

"The expert's estimate shows that the excess fuel used over the
entire trip from t=ti to t=2tf is modest and independent of the
duration of the uniform acceleration. At the beginning and end of
the trip the rocket is at rest in the initial frame, so the energy
of the radiation plus the exhaust should equal the rest-mass loss
of the rocket (fuel used). If the loss of rest mass is finite and
independent of the duration of the uniform acceleration (hence
independent of the arbitrarily large energy radiation), we have a
violation of energy in the initial frame."

Reluctant as I am to argue with an unnamed noted expert, the first
statement is false. Consider the "nudging" at the end of the period
of constant acceleration (i.e., when the acceleration is smoothly
changed from positive to negative). Say that requires a kilogram of
fuel. When the pilot turned in his flight plan, he didn't have to
budget just an extra kilogram of fuel -- he had to budget a kilogram
of fuel, plus enough extra fuel to accelerate that kilogram along with
the rocket until he was ready to burn it. Say that the rocket's
maximum speed has a Lorentz factor gamma of 1000. Then during the
time he's accelerating, he's going to have to supply an extra 1000 kg
of energy to carry that 1 kg of fuel along with him. If he wants to
reach a final Lorentz factor of a million, he'd have to lay in enough
fuel to give him a million extra kg of energy. So the "excess fuel"
required for the whole trip is not independent of the duration of the
acceleration -- it's exponentially sensitive to it, just as energy
conservation requires.

Parrott goes on to solve the Lorentz-Dirac equation in detail for this
situation. He shows that an uncharged rocket can accelerate for an
arbitrarily long time on a finite amount of fuel, with the rest mass
asymptotically decreasing to zero as the fuel is burned, but that a
charged rocket cannot. I haven't worked through every step of this
calculation in detail, but I'm quite prepared to believe it's right.
This is apparently meant to show that it's impossible to get
arbitrarily long periods of acceleration with a fixed amount of excess
fuel. Since the uncharged rocket can accelerate forever, and since
during uniform acceleration there's no excess force on the rocket
(according to the LD equation), this looks like a contradiction.

Well, there are a few problems here. First, since the rocket is
carrying a charged particle along with it, we shouldn't have expected
its rest mass to be able to go to zero. The charged particle itself
has positive (renormalized) rest mass, which will remain even when all
the fuel is burned. So a fair comparison is with a neutral rocket
that also has a little nugget of mass left over at the end that it
can't burn. Both of those can accelerate for only a finite duration
on any given amount of fuel.

At worst this is thought experiment is just a symptom of the known
pathology of the Lorentz-Dirac equation. The LD equation is only
supposed to work when the particle is asymptotically unaccelerated in
the future. If you want to know what the equation predicts for any
particular rocket-fuel-burning scenario you care to dream up, you can
work it out. For a given force vs. time profile, the particle's
motion is given by integrating over the *future* of the particle.
This means that the solution will in general involve
pre-accelerations.

In this case, as the rocket starts to get low on fuel, shortly before
the time when, in Parrott's calculation, its rest mass would threaten
to go negative, it'll spontaneously start to "pre-decelerate." At
least, I think that's what'd happen. I'm not particularly interested
in working it out, because I think the Lorentz-Dirac equation's
pathology makes it silly to use it in situations like this (and
perhaps in all situations).

In a better theory, assuming one to exist, I'd predict that the rocket
will eventually run out of fuel, and that when it does, it'll undergo
a bit of extra deceleration "paying back" what it "borrowed" when it
got started.

.

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