## Radiation reaction for a uniformly accelerated charge

Regarding my paper

S. Parrott, "Radiation from a uniformly accelerated charge
and the equivalence principle",
Found. Phys. 32 (2002), 407-440,
www.arxiv.org/gr-qc/9303025,

Ted Bunn wrote:

> I don't think that the author has justified his main
> conclusion, which is that "purely local experiments"
> can distinguish a charge in uniform acceleration
> from a charge at rest in a uniform gravitational field.

The paper is long, and I see now that the "justification"
could have been more explicitly presented. I will try to
remedy this here. First, I'd like to comment on the history
of the paper, because what Bunn reasonably interprets as
the "main conclusion" was not originally intended
as the main conclusion.

Around 1993, there was a discussion on
sci.physics.research concerning whether the "fact"
(or assumption) that

(1) any accelerated charge in Minkowski space

and the "fact" (or assumption) that

(2) a charge stationary relative to the Earth (say)

contradict the "Equivalence Principle" of general relativity.

At that time, it was widely believed that this question
a famous paper of Boulware:

D. Boulware, "Radiation from a uniformly accelerated charge",
Annals of Physics 124 (1980), 169-187.

For example, R. Peirels book "Surprises in Theoretical Physics"
(Princeton Univ. Press, 1979) devoted a chapter to
an exposition of this paper.
These references were cited in the discussion as
definitive evidence for the Equivalence Principle in this context.

I was intrigued by the problem, and after thinking
about it, came to the conclusion that Boulware's analysis
did not prove what people thought.
Then I wrote up my analysis in the above paper.
The primary intention was to point out some questionable aspects
of Boulware's widely accepted analysis,
but since I had privately come to question the Equivalence Principle,
I broadened the scope to include my views on this.

The paper's Abstract is very short:

"We argue that purely local experiments can distinguish
a stationary charged particle in a
static gravitational field from an accelerated particle
in (gravity-free) Minkowski space.
Some common arguments to the contrary are
analyzed and found to rest on
a misidentification of "energy".

Note that it says "argue", rather than "show" or "prove".

Boulware's analysis is unusually mathematically detailed,
but other authors have used arguments similar to his to come to
the same conclusion. A typical example is:

A. Kovetz and G. Tauber, "Radiation from an accelerated
charge and the principle of equivalence",
Am. J. Phys. 37 (1969), 382-385.

The idea of these arguments is to consider
a network of observers uniformly accelerating
along with the particle. Such a network may be visualized as a
"uniformly accelerated" "elevator" or "rocket"
containing the particle, and we'll call the observers "elevator observers".

Then one can show that
each elevator observer sees a vanishing Poynting vector
(a vector describing the flow of electromagnetic energy radiation),
which could be interpreted as seeing zero energy radiation.
Thus it might seem that there is no energy radiation in
the "elevator frame". But this way of speaking
can be misleading because the correctness of this conclusion
depends on one's definition of "energy".

My paper points out that the (zero) "energy radiation"
seen by the uniformly accelerated observers
is *not* the same "energy radiation" which is calculated
in standard electrodynamics texts.
They are not merely the same quantity ("energy")
calculated in different coordinate systems,
but are essentially different quantities.
They are mathematically different and physically different.
Thus in discussions in which both "energies" appear,
the use of the same term for both invites confusion.

Below I will refer to the radiation "energy" given in
standard texts as "Minkowski-frame energy". This "energy" is
proportional to the integral of the square
of the particle's proper acceleration
over the accelerated portion of its worldline,
and hence is always positive
(or zero in case the particle is never accelerated).
The other (always zero) "energy" seen by
the "elevator" observers will be called the
"elevator-frame energy".

I can't fully explain here why they are different
without writing down equations too complicated for ASCII,
but the paper gives the full explanation.
The idea is that the standard way of constructing
a conserved "energy" in general relativity is to
contract a timelike Killing vector field
with an energy-momentum tensor.
Usually one is lucky to find even one timelike Killing field
in an arbitrary spacetime (which is why there is no law of
conservation of energy in general relativity),
but the so-called "Rindler wedge" in Minkowski space
has the unusual property of posessing
two essentially unrelated timelike Killing fields.
One is time translation relative to
a given Minkowski coordinatization,
and the other is related to time translation
as seen by "elevator" observers.
The nonzero "energy" radiation calculated in electrodynamics texts
comes from the former Killing field,
while the (zero) "energy" radiation corresponding to
the vanishing Poynting vectors seen by the "elevator" observers
comes from the latter Killing field.

These two "energies" are not only not numerically equal
(one is zero and the other nonzero for a uniformly accelerated particle)
but are conceptually and physically different quantities.
Boulware calculates *both* energies (for different purposes),
but does not distinguish between them.

So far as I know, the mathematics
leading to these conclusions has never been questioned.
Any differences of opinion concern physical interpretations,
not mathematics.

____________________________________________________________________

Nothing in the above questions
the Equivalence Principle (EP), *per se*.
It only questions Boulware's analysis leading to his conclusion
that "facts" (1) and (2) above do *not* contradict the EP.
My paper proper (i.e., before appendices) concludes:

"Does Einstein's Equivalence Principle hold for
charged particles? We cannot definitively answer this
because a mathematically precise statement of the
equivalence principle'' seems elusive ---
most statements in the literature are
not sufficiently definite to be susceptible of
proof or disproof. However, we do conclude that most
usual formulations seem not to hold in
any direct and obvious way for charged particles. ..."

Thus no claim is made that the paper "proves" or "shows"
that the Equivalence Principle is unequivocally false.

However, the paper does present a thought experiment
which, if carried out physically, could in principle decide the issue.
I see only one likely outcome from the experiment
(the outcome which would falsify the EP), but others disagree.
Bunn summarizes this experiment as follows:

> The purely local experiment he has in mind is a measurement
> of the force required to support the charge in its motion.
> If it's at rest in a uniform gravitational field, the force
> should be equal to the charge's weight. If it's accelerating
> (he argues), it should be more than that,
> because of the radiation reaction force.

This is a good summary,
reaction force" does *not* necessarily mean
the Lorentz-Dirac expression for radiation reaction.
It should mean the additional force (whatever it may be)
necessary to "pay" for the radiated energy
(as compared to an identically accelerated uncharged particle
of the same mass.)

It does seem to me "obvious" that
if the charged particle radiates (Minkowski-frame) energy,
then ordinary ideas of conservation of energy
demand that an additional force must be necessary
And, for commonsense reasons other than conservation of energy,
it also seems to me "obvious" that this additional force must be active
during the uniform acceleration.
But since there is disagreement that the latter is "obvious",
let's put this question on hold for the moment.
My answer to it is given in Appendix 2 of the paper, which
I will later summarize.
___________________________________________________________________

Meanwhile, let me attempt to explain why
I regard it as "obvious" by rephrasing the essence
of the controversy in everyday language.

I often drift off to sleep while listening
to a late-night radio talk show dealing with the paranormal,
e.g., extrasensory perception, unidentified flying objects,
and the like. It features guests ranging from
obvious wackos to fringe "scientists" to
reputable physicists.

Sometimes it can be hard to tell the difference
between them if one misses the introduction
giving their credentials.
One frequent guest who is a reputable physicist
presents some of the wilder speculations of some string theorists
(things like multiple universes) as if they were established physics.
He is fond of prefacing the speculations
with "we physicists believe that ...", where "..."
consists of some far-out speculation which
some string theorist may have published,
but which is hardly a mainstream view.
It is an amusing way to occupy the mind as sleep creeps in.

There have been guests who claim to have worked in secret
government laboratories reverse engineering flying saucers.
A typical interview might go something like
the following:

HOST: So, Bob, you were the leader of a research group
transferring flying saucer technology to automobiles?

BOB: Yes, we used it to build a car that will
run as long as you want without refueling.

HOST (feigning unbelief): How can there be a car that will run forever?

BOB: Well, it doesn't actually run forever,
just as long as you want.
For example, if you want it to run for one month, it will.
Or it can run for a year if you want.
You choose how long you want it to run, and
it will run that long.
We scientists say that it will run *arbitrarily long*,
but that's not the same as running *forever*.

HOST: Why doesn't it run out of gas?

BOB: An ordinary car would, but
this car can run on what we scientists call "negative gas".
When the car uses "negative gas",
it borrows gas from what we scientists call the
"zero point quantum energy field".
At the end of the trip, the user has to
replace the borrowed gas. Otherwise,
there would be a violation of what we scientists call the
"Law of Conservation of Energy".
That's why the car can run arbitrarily long,
but not forever.

The gas gauge in this car has not only positive numbers
like 1,2,3 gallons, but also negative numbers
like -1,-2,-3 gallons. If at the end of the trip
the gauge reads -100 gallons, then
you have to put more than 100 gallons in the tank before
you can start the car. Then the gas gauge will indicate a
positive number of gallons, and you can start the car.

HOST: Wow, so you can drive as far as you want
without stopping for gas!

BOB: That's right. That's how flying saucers have been flying
around the Earth for hundreds of years without refueling.
they will have to put enough fuel in their tanks to cancel
the negative fuel debt which they have built up.

HOST: Thank you for your courage in revealing this
amazing new secret technology to our audience.

Nothing that Bob said directly violates
any of the great physical conservation principles,
but even the drowsy mind can realize that it sounds fishy.
If we jazz up the story a little
to "explain" how the "zero point quantum energy field"
gets converted into gasoline and vice versa,
it might be hard to "prove" that this car can't exist.

There do exist experimentally verified
quantum-mechanical effects that seem almost as strange,
so one should be careful about being too dogmatic.
Nevertheless, I would hope that some significant fraction
of listeners would question Bob's claims.
Similarly, I hope that some significant fraction of the readers
of my paper will question the Equivalence Principle.

Now let's get back to what we scientists call "physics".
___________________________________________________________________

Bunn considers the thought experiment in greater detail:

> Start with a charge at rest in the lab, gently nudge it into
> accelerated motion, let it uniformly accelerate for a while,
> and then gently reduce the acceleration to zero,
> leaving it in uniform motion relative to the lab.
> Lorentz-Dirac theory says that there's a radiation reaction force
> only during the two "nudges" at the beginning and the end --
> during the uniform acceleration, the total force
> required is the same as it would be for an uncharged particle
> of the same mass. And if you calculate the work done by the
> external force during the whole time, it works out right:
> The work equals the final energy (mc^2 + KE of the charge +
> radiated energy) minus the initial energy (just mc^2).

This is exactly right, as regards
what the Lorentz-Dirac (LD) equation implies.
And I would bet a lot of money that
if the experiment could be performed,
a nonzero force would have to be applied
during the time *between* the nudges,
i.e., during the time of uniform acceleration
when the LD equation says that the radiation reaction vanishes.
It's for basically the reason that I don't believe in Bob's car,
but perhaps the following may be more persuasive to those who
distrust argument by analogy (as do I).

Suppose a small rocket ship furnishes the force
to which Bunn refers. For brevity,
I'll call the ship a "charged rocket".
For comparison we'll also consider an "uncharged rocket"
which gives the same acceleration
to a particle of the same mass as the charge in the charged rocket.
We'll assume that the two rockets have the same worldlines
which satisfy the conditions which Bunn describes: they start
unaccelerated, are gently nudged into uniform acceleration which
continues for a long time, and finally are nudged back into
inertial motion (i.e., constant velocity, no acceleration).

For this situation, assuming Lorentz-Dirac theory,
one can write down the equations of motion for the two rockets,
and these equations can be solved *exactly*,
with no approximations whatever.
This is carried out in full detail in Appendix 2 of my paper.

The result is that the uncharged rocket
can accelerate *forever*, assuming that
all its rest mass can be used as fuel.
In contrast, if the period of uniform acceleration is long enough,
the charged rocket will eventually
run out of fuel (i.e., rest mass).
>From that point on, its rest mass is negative (i.e., negative fuel),

so its rest mass is negative at the end of the trip.
It's just like Bob's car.

This is not a surprising result.
It could anticipated without calculation from the fact that
the Lorentz-Dirac equation conserves (Minkowski-frame) energy.
(Minkowski-frame) electromagnetic energy,
for a long enough journey,
the initial fuel supply (i.e., rest mass).
Hence the final rest mass must be negative
to "pay" for the radiated energy.
The detailed solution of the equations merely confirms
this intuitive expectation.

Thus the Equivalence Principle
combined with the Lorentz-Dirac radiation reaction expression
seems to demand the existence of negative mass,
which has never been observed.
(Of course, I can't "prove" the nonexistence of negative mass.)

___________________________________________________________________

This post is already longer than I would like,
but I'd like to make one more comment before closing.
Bunn writes:

> Granted, Lorentz-Dirac theory is sick, but this is the sort of
> situation in which it's expected to be approximately correct
> (acceleration for only a finite period,
> all nudges smooth and gentle).

That "this is the sort of situation in which
[Lorentz-Dirac theory] is expected to be approximately correct"
is a common misconception. To explain why,
I need first to discuss one of the lesser-known ways
in which Lorentz-Dirac theory is "sick".

Eliezer proved in 1943 (while a student of Dirac!)
that the Lorentz-Dirac equation implies that
under some physically reasonable conditions,
charges of opposite sign *repel* each other

C. J. Eliezer, "The hydrogen atom and the classical theory
of radiation", Proc. Camb. Phil. Soc 39 (1943),
173 ff.

This is a rigorous mathematical theorem,
whose proof requires no approximations or "hand-waving".
It was occasionally questioned in the literature around 1960,
but it has stood the test of time and
today seems universally accepted.
Eliezer's original proofs were not easy to read,
and subsequently
several authors have published more accessible proofs,
though all known to me are based on Eliezer's original ideas.

Naturally, people were reluctant to believe that
real particles would behave as Eliezer's theorems stated,
but they were also reluctant to conclude that
Lorentz-Dirac theory might be fundamentally incorrect.
So, they looked for ways around Eliezer's conclusion.

Eliezer's original proofs assumed point particles,
which simplifies the analysis.
For example, one of his theorems states that
if a negatively charged particle (an electron, say)
starts at rest in the Coulomb field of
a stationary positively charge particle (a proton, say),
then although the electron may move toward the proton for a while,
it *cannot* be attracted to a collision as expected,
before it hits the proton and
will thereafter be repelled away from the proton
with ever-increasing velocity.

Unfortunately, his original theorems didn't estimate
how close the electron would get to the proton before turning around.
So, many physicists assumed that
it would have to get close enough for the proton's field
to be so strong that "pair production" would occur,
which would need to be described by quantum mechanics.
There was no evidence for this assumption, but it was a way around
the unpalatable conclusions of Eliezer's theorem.

However, the *ideas* of Eliezer's proofs
(as opposed to the statements of his theorems)
provide ways of estimating how close the electron gets,
and it can be shown that there are situations in which
the electron turns around before it encounters strong fields.
For example, under physically reasonable auxiliary hypotheses,
one can recover Eliezer's conclusions using
a Coulomb field cut off at both
large and small distances from the proton, and these cutoffs can be
chosen so that the field is arbitrarily weak.
One can also smooth the cutoffs so that
the field is infinitely differentiable.
Carefully stated theorems with full proofs can be found in:

S. Parrott, "Variant forms of Eliezer's theorem,
www.arxiv.org/gr-qc/0505042.

[I do not intend to submit this paper for publication
because the ideas of the proofs are only variants of Eliezer's,
and subsequent simplifiers.
It was written to convince, in private correspondence,
a proponent of the Lorentz-Dirac equation that
it did predict unphysical effects
in situations with no mathematical singularities.]

The point is that Lorentz-Dirac theory is *really* "sick".
If it predicts that unlike charges repel in situations in which
no mathematical pathologies are evident
(e.g., all fields small and infinitely smooth, etc.)
then how can one confidently identify
a class of situations in which it is
"expected to be approximately correct"?

The point of constructing physical theories is
to be able to *predict* results of experiments which
have not yet been performed.
A physical theory which predicts effects
which have never been observed and
which seem in contradiction to common experience
(such as unlike charges repelling each other)
should be viewed with extreme suspicion.
To use such a theory to draw firm conclusions
about experimentally unverified situations invites error.

Stephen Parrott

 PhysOrg.com physics news on PhysOrg.com >> Iron-platinum alloys could be new-generation hard drives>> Lab sets a new record for creating heralded photons>> Breakthrough calls time on bootleg booze