Limits of General Relativity

Does GR provide a limit to the maximum electric field? The author of the Motion Mountain physics textbook claims on his site that "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 my physics TA strongly disagrees and when I asked emailed me "The value of an electric field is actually the component of a field tensor, which is a coordinate system dependent quantity. Lorentz invariance, even just the local Lorentz invariance in GR, shows that changing between (local) inertial frames can make the electric field value arbitrarily large. See http://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity So that textbook is clearly wrong, as it amounts to claiming GR contradicts itself by predicting violation of local Lorentz symmetry. The textbook author's mistake is that energy density alone isn't enough http://www.weburbia.com/physics/black_fast.html " He was patient, but with only a Phys 102 background I don't really understand, as I thought an electric field was a vector, 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. Is my TA wrong? Is the textbook wrong? Are they talking about different things and both correct somehow?
- John (age 20)
Urbana, IL, USA

John- Tough question. I'll have a first pass at it, then we can change or add to it as smarter people get a look.

That field density limit does not come from GR alone. That's because bare GR puts up with arbitrarily small distances. So you could have an arbitrarily large field if it were over too small a volume to make a black hole. GR by itself doesn't have  a lower limit on the size of a black hole. What your TA wrote looks good to me. Remember- those transforms change not only the energy density but also the volume, shape, etc.

The book author may have been including some version of quantum issues. These lead to a smallest meaningful distance (the Planck length) before quantum effects become very important. If you then assume that the field must extend over at least that distance cubed, you get a maximum field value before triggering a black hole. Does that mean that something then goes wrong with those transforms?

What you're seeing is a tail of a big problem. Integrating GR and QM doesn't work in a straightforward way. We don't know what sort of symmetry is applicable to spacetime on extremely small distance scales. It's quite likely that the whole picture of 3 spatial dimensions breaks down.

BTW on E being a vector- even in plain old Special Relativity, to get a Lorentz-invariant object you need to also include the magnetic B field into a tensor object. The 3-vector doesn't suffice. See

Mike W.

(published on 02/28/2011)

Thank you very much for replying to my question. I'm trying to figure out how these (now three) people's explanations fit together. If I understand your answer, the author's phrase "The smallest possible black hole" assumes some kind of quantum gravity which we don't really know yet, and the TA's answer was discussing this without quantum mechanics. So at first I thought I could separate it into quantum vs. non-quantum, and everyone be correct. But the TA's answer only used Lorentz symmetry, which is true in quantum mechanics and non-quantum mechanics. And then there is the author's statement "energy density is limited by general relativity" which seems like a classical statement due to black holes, which the TA and the link he provided seems to say no, but you (at least with a finite volume) and the author seem to say yes. So to help me sort this out, I'd like to follow up by asking "Is the total energy in a finite volume limited by General Relativity? And does the answer depend on quantum mechanics?"
- John (age 20)
Urbana, IL, USA

John- You're right that my previous answer wasn't complete.

GR sets a limit on energy in a given volume, due to the formation of black holes. (Let's assume here we're talking about spherical volumes, just to keep things well-defined.) However, for smaller black holes that maximal density is higher. So if you don't include some quantum limit on how small the black hole can be, there's no overall limit on density in GR. Your TA was right about that.

If you do include QM, then it seems there would be a minimum size, which then would translate to a maximum density. However,  the straightforward combination of QM and GR is inconsistent, in a way that can't just be patched up for those very small distances. If GR is wrong on those scales, that means that we don't know what the relevant symmetry group is. The Lorentz transforms may not be part of it.

So there's a domain in which your TA was right. I don't know enough about what that author was saying to know if he somehow was also right. I think he's the same guy who wrote something about entropy in the same "I can derive it all from a simple limit" form. So far as I could tell, it was gibberish. Don't take my view as gospel.

Mike W.

(published on 03/02/2011)

Now I know why this came up in discussion with other gradstudents yesterday. While I don't want to comment on quantum gravity issues, I would like to point out a mistake in the statement "GR sets a limit on energy in a given volume, due to the formation of black holes." Suppose this limit is correct, and a particular given box can only have at most E energy in it before becoming a black hole. Now if we have an object at rest of energy E/2 that fits in the box, everything is fine. However, if we give the box to an observer moving at 9/10th the speed of light with respect to the object, he can still fit the object into the box, but will say the object has energy > E. So does one observer claim a black hole and a singularity forms while the other observer doesn't? No. 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. As a final comment, I've heard about this Motion Mountain "book" before. I looked at it after a Phys101 student brought up questions from it ... Let's just say it is self-published, "non-mainstream", and should not be considered a "textbook" by students.
- physics gradstudent (age 24)
Urbana, IL, USA

Thanks very much for these clear and careful remarks. I don't know who you are, but would be delighted if you would join our team here at this Q&A site. Please get in touch.

Mike W.

(published on 03/03/2011)

Thank you very much everyone for replying to my question. To make sure I understand, it sounds like the response can be summarized to: It is hard to comment on the gravity from electric fields in GR, but there are strong hints that a quantum gravity would limit the electric field? Without understanding all the detailed physics, is that at least a good summary to take away? Searching more on maximum electric fields, I think this discussion can be resolved by noting Motion Mountain in its electrodynamics textbook lists 1.9 x 10^62 V/m as having been already observed to be the maximum possible electric field in nature. Based on your comments, this measurement might be one of the first measurements of quantum gravity effects! That is a neat way of looking at it. I'd also like to comment (or defend) why I and other students like the Motion Mountain textbooks. Unlike any other physics textbook I've seen, they present things simply in an exciting and engaging manner. They are quite complete with over 1000 pages total, covering everything from Phys101 material to GR to quantum to the standard model, yet they are free online to students. I looked up about the book online, and want to defend it. First, the book is not "self-published" as the author is a PhD physicist and supported by the Klaus Tschira Foundation which funds projects explaining science to the public. It has been recommended by the American Association of Physics Teachers and the California Learning Resource Network. It has gotten recommendations to students from other physicists, including John Baez ( http://www.motionmountain.net/reviews.html ), and many physicists "have provided material for this text" including I think Prof. Jon Thaler at UIUC ( http://www.motionmountain.net/help.html ) and even famous physicists like Freeman Dyson. So please talk to Prof. Thaler, or better yet read the books yourself, before calling it "non-mainstream" or "gibberish".
- John (age 20)
Urbana, IL, USA

John- Thanks for these thoughts.
 
I'm not an expert in these things, but if quantum gravity sets a minimum distance scale, then there would be a maximum energy density and thus a maximum field. I'm not sure what Motion Mountain is referring to in claiming that there's been an observed field maximum already. I've never heard anyone cite evidence of that sort for any version of quantum gravity. There's an entirely different effect, not involving gravity, which actually limits field strengths. Very high fields cause sparking of the vacuum, i.e. they cause the virtual particle-antiparticle pairs of the vacuum to become real, sort of the way a big field in the atmosphere causes a breakdown of the atmosphere into charged particles. It's hard to imagine how you could get past that process to observe the gravitational effects.

It's great that the Motion Mountain book is free.  Despite the acknowledgment, Jon Thaler doesn't recollect any involvement with it. I wrote Baez to ask about his current views on the book, but he says he has not had the time to keep up with the various parts added since he made those initial generally positive comments. Someone submitted this book as a possible "Baloney" candidate, but we didn't put it in that category because we reserve that for cases where there's no ambiguity.

Parts of the book are fun and explanatory. The beginning of the speculations on entropy (equation 112) looks like gibberish because the author seems to be conflating the information content of a message transmission process with the minimum entropy of a physical state, without providing sufficient justification. As I wrote, you shouldn't take my view as gospel.

Following the entropy discussion further, Eq. 113 is presented in a rather elaborate way as a sort of "uncertainty relation". It is in fact just Gibbs' old formula for the mean-square fluctuations of the energy of an object in thermal equilibrium with a heat bath, with one factor in the squared energy fluctuations re-expressed as if it were fluctuations in inverse temperature.* It gives no information at all on a "minimum entropy".

This happens to be a topic I'm familiar with, since as a grad student I measured precisely those fluctuations. ( M. B. Weissman and G. Feher.  Observation of energy (thermal) fluctuations in an electrolytic solution.   J. Chem. Phys. 63, 586-587  (1975).  also  M. B. Weissman and G. D. Dollinger.  Noise from equilibrium enthalpy fluctuations in resistors.   J. Appl. Phys. 52, 3095-3098  (1981).) The current we used to make the measurement drove the system slightly out of equilibrium in a way that reduced the fluctuations below the equilibrium value. In other words, we violated the inequality of Eq. 113.

It's quite possible for a book to be a mixture of good stuff and nonsense. Consider an extreme case- Galileo's Dialogue Concerning the Two Chief World Systems. The question is whether any of the new ideas in Motion Mountain are worthwhile, or just some of its energetic presentation of old ideas.

Mike W.


*Technical point. If one were really to define "1/T" by finding the  T for which <U> would equal the actual value of U, one would obtain an infinite <Δ(1/T)>, since there's always a non-zero probability of being in the ground state, for which this "1/T" would be infinite. So apparently the author really has done nothing more than re-express ΔU in terms of Δ(1/T) using a linear approximation. 

(published on 03/04/2011)

The worst kind of baloney. The author receives funding from a physics institution to write a free set of physics textbooks, but instead he used the money to promote his non-mainstream baloney and peddle it as free educational resources. He misunderstands action, makes really bizarre claims on quantum mechanics, completely misunderstands general relativity, and the coup de grace -- promotes his own pet 'theory of everything' in the last chapter, which is unmistakable crackpot baloney. Unfortunately according to the site, the California Learning Resource Network is promoting it to physics teachers since 2010, and the EU-funded European Gateway to Science Education is distributing the text to science teachers, science communicators and pupils since 2006. Just because the author has funding and a PhD doesn't mean it isn't baloney! Its clear they never looked at it, since even a cursory skim of his 'pet theory' chapter would convince any sane physicist that this guy is a crackpot!
- Martin (age Grenzo)
Cincinnati, OH, USA

Hi Martin- I've marked this as a follow-up to some of our old discussions of certain parts of MotionMountain which could legitimately provoke your reaction. On the other hand, it also contains enthusiastic vivid presentations of real physics.

Is there any book which avoids individual crackpot digressions without settling for the  homogenized blather of the standard huge texts?  I kind of like the series of "Units" by Thomas Moore, which get to much of the core of modern physics in a direct way.

Mike W.

(published on 09/24/2011)

How could someone with a PhD in physics "completely" misunderstand General Relativity?? He may only be stating that in the quantum limit of volume there also exists an upper field limit. After all GR does seem to break down on Planck scales.
- Larry P (age 65)
Carrollton TX USA

Yes, it's possible that these obscure passages have been misinterpreted.

On the other hand, you're way overestimating the reliability of PhD physicists. We can all misunderstand a great deal, in part because it's a broad area and most of us have big holes in our knowledge. I'm planning to sit in on a series of undergrad lectures on GR in the next few weeks, to try to get a grasp of it. I'm helping colleagues write parts of a general textbook for freshmen because they don't understand thermodynamics at that level. I talked with a recent PhD who genuinely didn't believe that the distinction between bosons and fermions could have much implication for the behavior of helium 3 and 4. We all have gaps.

Mike W.

(published on 03/16/2012)