Modifications of Electrodynamics Required by the Principle of Equivalence
Learner
Section 9.1 of Feynman Lectures on Gravitation reads "The Principle of
Equivalence postulates that an acceleration shall be indistinguishable
from gravity by any experiment whatsoever. In particular, it cannot be
distinguished by observing electromagnetic radiation. "There is
evidently some trouble here, since we have inherited a prejudice that
an accelerating charge should radiate, whereas we do not expect a
charge lying in a gravitational field to radiate.
This is, however, not due to a mistake in our statement of equivalence
but to the fact that the rule of the power radiated by an accelerating
charge
dW/dt=(2e^2/3c^3)a^2,
has led us astray.
[snip]
Of course, in a gravitational field the electrodynamic laws of Maxwell
need to be modified, just as ordinary mechanics needed to be modified
to satisfy the principle of relativity.......Clearly, some interaction
between gravity and electrodynamics must be included in a better
statement of the laws of electricity, to make them consistent with the
principle of equivalence.
We shall not have completed our theory of gravitation untill we have
discussed these modifications of electrodynamics, and also the
mechanics of emission, reception, and absorbtion of gravitational
waves."
Feynman does not seem to have discussed these modifications and how
they reconcile the above problem with the Equivalence Principle in
the rest of the book. However, I know that, in curved spacetime
Maxwell's equations
F\alpha\beta,\gamma+F\beta\gamma,\alpha+F\gamma\alpha,\beta=0, and
F^{\alpha\beta},\beta=4PiJ^\alpha
are modified by changing *,* (the ordinary derivative) to *;* (the
covariant derivative). What I do not is how this change from *,* to
*;* solves the above problem with the Equivalence Problem.
Regards,
Learner
eb...@lfa221051.richmond.edu
In article <335be373.0210...@posting.google.com>,
Learner <alpa...@hotmail.com> wrote:
>Section 9.1 of Feynman Lectures on Gravitation reads "The Principle of
>Equivalence postulates that an acceleration shall be indistinguishable
>from gravity by any experiment whatsoever. In particular, it cannot be
>distinguished by observing electromagnetic radiation. "There is
>evidently some trouble here, since we have inherited a prejudice that
>an accelerating charge should radiate, whereas we do not expect a
>charge lying in a gravitational field to radiate.
This is a famous puzzle. Before it can be resolved, we have to be
more precise about just what we mean by "the equivalence principle"
and "radiate."
The key thing to remember about the equivalence principle is that
it only applies locally (in an infinitesimal spacetime volume).
Over a large volume, it is possible to tell the difference between
accelerations and gravity, since gravity can produce tidal forces.
Next, we have to decide what we mean by "radiate." If you want to
determine whether a charge is radiating or not, what precisely will
you do? Since we want to talk about the equivalence principle, we'd
better adopt a criterion that can be applied in an infinitesimal
volume. Say I draw an infinitesimal sphere around the charge and
integrate the normal part of the Poynting vector over the surface of
the sphere. If it's positive, then radiation is leaving the sphere.
With the adoption of this criterion, the equivalence principle comes
out OK. If I have a charge sitting still on my desk (in a
gravitational field), it doesn't radiate. If there's a charge
accelerating upward at a constant rate through gravity-free space, and
I accelerate upward next to it at the same rate, I also don't register
any radiation according to this procedure.
Similarly, if I drop a charge in my lab, it does radiate as it falls.
On the other hand, if there's a charge sitting still in gravity-free
space and I accelerate past it, it also radiates.
In both cases, the two situations that the equivalence principle says
should match do match.
The point is that (using this criterion for radiation), the question "is
it radiating?" has a reference-frame-dependent answer.
You could use a different criterion for radiation, and then you might
get different answers. One choice, for instance, would be to check
and see whether there's a radiation reaction force on the charge. (Do
I have to exert extra forces on the charge to keep it in the desired
state of motion, compared to a neutral particle of the same mass?) In
this case, according to standard theory, there is no radiation in any
of the above cases, since the radiation reaction force vanishes for
constant acceleration.
"Standard theory" here means the Lorentz-Dirac equation, which is full
of problems. (In fact, we're discussing those problems elsewhere in
sci.physics.research right now.) In fact, Stephen Parrott argues in
http://arxiv.org/abs/gr-qc/9303025 (Found. Phys. 32, 407 (2002)) that
charged particles do violate the equivalence principle on the basis of
arguments much like the above.
Parrott's book "Relativistic Electrodynamics and Differential
Geometry" is in large part an argument against the Lorentz-Dirac
equation, by the way. Anyone who's interested in understanding the
whole business of radiation reaction forces on classical point charges
should read this book, and anyone who's interested in understanding
how all this relates to the equivalence principle should read
Parrott's Found. Phys. article.
I think that ultimately the problems with the Lorentz-Dirac equation
are a red herring here, though. After all, we don't need the entire
machinery of the Lorentz-Dirac equation; we just need its prediction
that there is no radiation reaction force on a uniformly accelerated
charge. That prediction is surprising at first, but in fact I think
it's correct -- or at least correct enough for present purposes.
It's pretty generic: I suspect that any "reasonable" theory of
point charges would agree on it.
Here's one way to say why I think this. Instead of a point charge,
consider a small charged sphere of radius R, accelerated with uniform
acceleration a. You can calculate the radiation reaction force on
such a sphere, and you find that it's proportional to aR/c^2. So as
long as the sphere is small compared to the length scale associated
with the acceleration, radiation reaction forces are negligible. (If
the sphere accelerates for only a finite time, there are significant
radiation reactions when you start and stop the acceleration. These
have an R-independent contribution, and they do the job of making sure
that energy is conserved over the entire time the particle was
accelerating.)
Although the equivalence principle, strictly speaking, applies only to
infinitesimal regions, the fact that things work out the way it says
they should on length scales much shorter than the length scale
associated with the acceleration seems to me to be suggestive.
>This is, however, not due to a mistake in our statement of equivalence
>but to the fact that the rule of the power radiated by an accelerating
>charge
> dW/dt=(2e^2/3c^3)a^2,
>has led us astray.
I'm not sure I know what Feynman means here. I can't help but suspect
that this part
>[snip]
is the key to understanding this!
>Of course, in a gravitational field the electrodynamic laws of Maxwell
>need to be modified, just as ordinary mechanics needed to be modified
>to satisfy the principle of relativity.......Clearly, some interaction
>between gravity and electrodynamics must be included in a better
>statement of the laws of electricity, to make them consistent with the
>principle of equivalence.
>
>We shall not have completed our theory of gravitation untill we have
>discussed these modifications of electrodynamics, and also the
>mechanics of emission, reception, and absorbtion of gravitational
>waves."
This is all true, but even once you've completed this program
(rewriting Maxwell's equations in a valid curved-space form), you
still have to do some work, along the lines indicated above, to see
how and whether things square with the equivalence principle.
>Feynman does not seem to have discussed these modifications and how
>they reconcile the above problem with the Equivalence Principle in
>the rest of the book. However, I know that, in curved spacetime
>Maxwell's equations
>
>F\alpha\beta,\gamma+F\beta\gamma,\alpha+F\gamma\alpha,\beta=0, and
>F^{\alpha\beta},\beta=4PiJ^\alpha
>
>are modified by changing *,* (the ordinary derivative) to *;* (the
>covariant derivative). What I do not is how this change from *,* to
>*;* solves the above problem with the Equivalence Problem.
Exactly. Replacing derivatives with covariant derivatives is
necessary, but that alone doesn't really resolve the issue with the
equivalence principle. To finish the job, you have to go through
a song and dance like the one I was trying to do at the beginning
of this post: you have to state precisely what it means to say
something is radiating, and what precisely is meant by the
equivalence principle.
-Ted
--
[My posts come from a machine that doesn't accept incoming mail. To
e-mail me, use an address of the form user...@domain.edu, as opposed
to user...@machinename.domain.edu.]
Hans Aberg
alpa...@hotmail.com (Learner) wrote:
>Feynman does not seem to have discussed these modifications and how
>they reconcile the above problem with the Equivalence Principle in
>the rest of the book. However, I know that, in curved spacetime
>Maxwell's equations
>F\alpha\beta,\gamma+F\beta\gamma,\alpha+F\gamma\alpha,\beta=0, and
>F^{\alpha\beta},\beta=4PiJ^\alpha
>are modified by changing *,* (the ordinary derivative) to *;* (the
>covariant derivative). What I do not is how this change from *,* to
>*;* solves the above problem with the Equivalence Problem.
From the mathematical point of view the whole thing is very
simple, because one just writes up a Lagrangian involving the
covariant derivative (Levi-Civita connection) and an electromagnetic
two-form. Then the Einstein equation, that was done for EM by Hilbert
(the metric variation technique is due to him), is attained by a total
metric variation.
The funny thing is that this contains the equivalence
principle. It must also be coordinate independent on a Lorentz
manifold, as all the processes involved are coordinate independent.
So it solves the mystery discussed by Feynman of gravitational
EM compensations. I think that the book by Misner, Thorne & Wheeler
has something about this.
The new mystery is why such a simple construction can lead to
an accurate physical description of gravity and EM fields as observed
in nature.
Hans Aberg * Anti-spam: remove "remove." from email address.
* Email: Hans Aberg <remove...@member.ams.org>
* Home Page: <http://www.matematik.su.se/~haberg/>
* AMS member listing: <http://www.ams.org/cml/>