Does GR provide a maximum electric field limit? Does GR provide a limit to the maximum electric field? 
I've gotten conflicting information regarding this, and am quite confused.  I will try to quote exactly when possible so as not to confuse things more with my paraphrasing.
The author of the Motion Mountain physics textbook claims in his book there is a limit, and clarifies on his site that ( http://www.motionmountain.net/wiki/index.php?title=Dislike_Page ): 

"electromagnetic fields are limited in
  magnitude. Now, every electromagnetic
  field contains energy, and energy
  density is limited by general
  relativity: if energy density is too
  high, a black hole appears. The
  smallest possible black hole then
  leads to a field limit. If you deny an
  upper field limit, you deny general
  relativity. However, general
  relativity has been confirmed in every
  experiment so far."

This sounds very obvious and intuitive to me. However one of my physics TA's got very upset when I used this in a thought experiment when discussing some limits in physics.  When I told him the textbook I got that from, he looked it up, and commented on the Motion Mountain wiki website his argument:

"I'd like to add something here.
  Suppose for a moment the energy
  density limit is correct, then if an
  object is sped up until it length
  contracts enough such that the energy
  density is greater than this limit,
  does it turn into a black hole? No,
  quoting John Baez "The answer is that
  a black hole does not form"*
  http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html
  *The claim that "if energy density is too high, a black hole appears" is an
  incorrect oversimplification. The
  issue is that in GR, gravity depends
  on more than just the energy density
  (component T^00 of the stress energy
  tensor). So we can't just look at the
  "relativistic mass" (E/c^2) to judge
  whether a black hole forms. It
  actually isn't even enough to look at
  the "invariant mass", for while the
  trace T of the stress energy tensor
  for a particle is just its invariant
  mass, for an electromagnetic field it
  is identically zero even though
  electromagnetic fields curve spacetime
  in GR. So none of these concepts of
  mass are sufficient when discussing
  gravity using GR (especially when
  considering electromagnetic fields),
  because gravity couples to the entire
  stress energy tensor. I hope this was
  helpful."

This makes much less sense to me, and I don't understand how energy density tending to infinity could EVER avoid being a black hole. No offense to my TA, but I'm skeptical as he's disagreeing with a textbook author.  Plus, the author's response was that my TA is another Einstein denier, and not worth responding to.
So I'd like a third party's answer on this.  Does GR provide a theoretical limit to the strength of an electric field?  Is it best to just ignore my TA on this one?
 A: All the answers to this question are wrong. They are right that there is no limit to the magnitude of the electric field limit in classical GR, but they are wrong to claim that this is because electric fields by themselves can't make a black hole.


*

*There is no bound to the magnitude of the electric field in GR:


It is not enough to have a high energy density to make a black hole, you need an total energy greater than R in a sphere of radius R. If the sphere is arbitrarily small, the energy density can be arbitrarily big. For a specific example of a strong field without collapse, an extremal black hole solution in general relativity has a metric:
$$ds^2 = (1-Q/r)^2 dt^2 - {dr^2 \over \left(1-{Q/r}\right)^2 } + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2)$$
And an electric field in the direction of r of magnitude $Q/r^2$. This electric field is maximum at $r=Q$ (at least outside the black hole), and the maximum value is 1/Q, which is as big as you like. To make a non singular solution with the same properties, just replace the black hole horizon with a slightly bigger sphere with the same mass and the same surface charge.


*

*An electric field which has a large enough magnitude, all by itself, in a large enough region of space, causes gravitational collapse.


The easiest way to see this is to consider a parallel plate capacitor of area $L^2$ and charge Q with no net momentum, and pull apart the plates to distance L, then you have order $|Q/L^2|^2 L^3$ amount of mass in a region of radius L, and when Q is bigger than L (which happens with small charge densities when L is large) you get a total energy at large distances which is too big to avoid collapse just by considering the total mass and zero momentum.


*

*There is nothing wrong with John Baez's examples of highly boosted noncollapsing stress, but they are irrelevant.


These are not like a static electic field because they have a large net mometum. The static electric field has zero field momentum.
A: The answer is: the Motion Mountain book is wrong, your TA and John Baez are correct
A couple answers here are discussing specific solutions in GR.  I think there is a much easier and more general way to answer this.
In special relativity, we can choose to use a different coordinate system (we don't need to physically change our motion as some posters seem to claim), and by merely changing the coordinate system the electric field changes.  In fact, the electric field can be made arbitrarily large this way. All that this relies on is the Lorentz symmetry in special relativity.
This is possible because the value of the electric field is just a coordinate system dependent component value of a tensor - the electromagnetic tensor.
Now the global Lorentz symmetry of SR doesn't survive when transitioning to GR, but there is still local Lorentz symmetry.  This means we can still change our coordinate system locally, to have the electromagnetic field tensor change (at a point at least) the same way as before.  So what the Motion Mountain book is claiming it can derive from GR, is actually violating GR.  So we don't even need to look at any specific solutions in GR.  The local Lorentz symmetry of GR is enough to show that the textbook is wrong.
As other posters have mentioned, the same goes with energy density.  So the textbook author's starting comment of "energy density is limited by general relativity: if energy density is too high, a black hole appears" is wrong.  In my opinion, if this is an example display of the textbook author's understanding of GR, it is probably best to avoid his books altogether.
A: Your TA is right that energy density alone does not trigger black hole formation. Consider a ball that's sitting still. Now speed up and look at the ball again. It will have gained (kinetic) energy. Relativistically, you can make the ball's energy density arbitrarily large by moving sufficiently near the speed of light. But the ball hasn't done anything in this process. It's you that has been changing speed. The notion of a black hole is not observer-dependent, so energy density alone cannot make a black hole form. 
That said, there are senses in which electromagnetic fields can form black holes in general relativity. Two colliding electromagnetic plane waves (which are necessarily gravitational plane waves as well) can do this. The field strength for a single plane wave can be arbitrarily large, however. Any "limits" are highly dependent on the specifics of the configuration.
A: Trust John Baez over your TA.  Another thing that you need in addition to a large mass density in order to form a black hole is a matter distribution that behaves in a certain way--in particular, you can fill the whole universe with radiation in a certain Robertson-Walker spacetime, and the density of radiation, and thus the magnitude of the RMS E-field is just a parameter with no upper limit, and the Robertson-Walker spacetime contains no black hole for any value of its parameters.
A: I don't know about what Baez is talking about, but i assure you that speeding up an object (relative to my "funny" frame) definitely increases the relative kinetic energy of the object in that frame, but not all energy transforms the same;
kinetic energy transforms as the 0-component of a cuadrivector, while rest-energy ($mc^2$) transforms as a scalar under Lorentz transformations. Also $T^{00}$ transforms as a tensor component, because it is the integral density of a 0-component vector against a oriented 3-form defining flow over a 3D volume element. Once you integrate $T^{00}$ over a volume you get a scalar density. Gravitational contributions only come from energy tensors like densities or rest-energies integrated over volumes. A binding energy in an atom would also behave as a scalar so it couples directly to gravity. A high-speed electron has an unbounded kinetic-energy but doesn't not contribute to any tensor energy density in the same unbounded way
hope this clarifies the issue 
