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?
Best Answer
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.