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Principia Physica... or

Caveat Emptor?

An unusal paper appeared recently on the Los Alamos eprint server: "Principia Physica" by Mukul Patel (gr-qc/0002012) This is an unusual paper in the sense that it has not a single reference although it acknowledges Prof. Phil Parkers of TWSU. The abstract of the paper states: "A comprehensive physical theory explains all aspects of the physical universe, including quantum aspects, classical aspects, relativistic apects, their relationships and unifications." With such bold claims, the paper is strange indeed.

"Principia Physica" was brought to my attention by a colleague, a fellow member of an international physics discussion group that is very actively pursuing avenues of "New Physics" on the WWW. Being not only a theoretical physicist, but a mathematical physicist as well, I was asked to check out the claims of this paper including the mathematical terms, but it proved to be not an easy task, to say the least! The author and his "associates," such as they may be, were not only evasive, but aggressively so.

I approached this request in the same manner as any other contact between colleagues where certain courtesies are understood and extended in the never-ending pursuit of greater knowledge of the universe in which we live.

But, I must have stumbled onto something else unawares. At this point in time I can only conjecture that 1) the paper is a joke; or 2) the paper is NOT a joke. If it is not a joke, then I will leave other conjectures to the reader.

The following is a series of e-mails exchanged in the process of trying to determine the identity of the author, his academic affiliations/credentials, and obtaining some straightforward answers to technical questions.

Dear Colleagues, [This was addressed to the discussion group]

I normally don't say much except that which relates strictly to physics or math. But, I think it is time for an exception. Most physicists are aware of the "rivalries" that exist between mathematicians and physicists, so I don't need to explain it... But, because of my position as a mathematical physicist, I am often asked by theoretical physicists to "get down and dirty" with the math, which I am always happy to do.

The following exchange of e-mails is rather lengthy but, it is certainly worth looking at for some strange clues about the possible existence of a "secret group" out there deliberately spreading lies and disinformation. In the above case, Mukul Patel's "Principia Physica", I had a look at the paper...

First of all,there seems to be another version of this paper, sent by the author to a "group of physicists". This other version is, supposedly, written in a more serious way.

Second, I am in touch with the author, and he slowly withdraws from the formulations of the official version, and starts to rephrase all ....

It is possible that Mukul Patel indeed has something interesting to say, but makes "errors" on purpose. What purpose? I do not know.

To be more specific: I checked his "academic affiliations," by writing to the institution in question:

From: Arkadiusz Jadczyk
Subject: info/inquiry
Date sent: Tue, 29 Feb 2000 17:35:30 -0400

Two weeks ago a research paper appeared on Los Alamos e-print server No: gr-qc/0002012. The paper is signed by Mukul Patel from Department of Mathematics, Southwestern College. As an expert in differential geometry I was asked by a group of physicists to review and comment on this paper. After reading it I became curious about the background of the author.

Physicists publish their papers with references and their backgrounds are available for all. If a mathematician writes a paper, physicists normally assume the paper is mathematically correct. This is not the case with the aforementioned new paper, therefore I am wondering about institutional affiliation of the author.

Please, let me know if Mukul Patel is indeed associated with the Department of Mathematics of your College, and in what capacity.

Thank you,


Arkadiusz Jadczyk

I never received a reply from the uni (added on March 23: just got the answer), but did receive a reply from the e-mail listed on the paper, to which I had addressed a similar inquiry. The second inquiry and reply are as follows:

From: Arkadiusz Jadczyk
Subject: Principia Physica
Date sent: Tue, 29 Feb 2000 08:43:29 -0400
Dear Mehri Arfaei,

I am somewhat puzzled by the paper "Principia Physica" (gr-qc/0002012) apparently authored by Mukul Patel, Department of Mathematics, Sothwestern College. I noticed that Mukul Patel is not listed as a faculty memeber. What is his association with your Department?

Thank you,


To which I received the reply:

Date sent: Tue, 29 Feb 2000 13:03:23 -0600
From: Mehri Arfaei
Subject: Re: Principia Physica

He is a friend of this department, and has been working here for a while. I tried to send your e-mails to him, including this one, but for some reason it doesn't go through. sorry about that. Of course I don't understand why should it matter!

best regard

Mehri Arfaei

I then wrote to the mathematician who was referred to in the paper, (who I had to search to find as no e- mail address was given) and received the following reply:

Date sent: Wed, 1 Mar 00 01:15:46 CST
Subject: Re: Principia Physica
Date: Tue, 29 Feb 2000 09:13:06 -0400
From: "Arkadiusz Jadczyk"

I am reading, with some puzzlement, a paper "Principia Physica" submitted to Los Alamos e-print server by Mukul Patel (gr-qc/0002012)

In the acknowledgement section the Author is thanking you for "critical assesement". Did you indeed study this paper? Who is the author? I am puzzled, because the paper, on one hand, shows pretty good knowledge of certain aspects of differential geometry, but also it contains unsupported claims. Very strange....

He's a former student of mine (some 12 years or so ago). I did read through it and offered some advice. Some was taken and the rest wasn't. So it goes. I thought it needed more work and refinement, especially regarding the provision of at least reasonable support for all claims, even if not genuine mathematical proofs. I must admit I did not expect him to "publish" it at this (what I thought of as an) early stage, but I assume he has his own agenda.

If you wish to conttact Mukul directly, I can provide an email address if one was not included.
Phil Parker --------------------------------------------

As will be noticed by these two foregoing responses, there seemed to be some discrepancy between the "versions" of who and what Mukul Patel was, as well as the "connections" he was trying to establish by his references to Professor Parker, so I wanted to clear this up. I wrote the following:

From: Arkadiusz Jadczyk
To: Mehri Arfaei Subject:
Re: Principia Physica
Copies to:
Date sent: Wed, 1 Mar 2000 04:06:16 -0400

On 29 Feb 00, at 13:03, Mehri Arfaei wrote:
"He is a friend of this department, and has been working here for a while. I tried to send your e-mails to him, including this one, but for some reason it doesn't go through. sorry about that. Of course I don't understand why should it matter! best regard Mehri Arfaei "

Dear Collegue,
I am including below an e-mail from Prof. Parker. If Mukul Patel is indeed your friend, I would suggest that you ask him to withdraw the paper that involves your e-mail address and Department of Mathematics of Southwestern College - at least until errors and unsubstantiated claims are removed. That is a normal procedure and will save the time of many physicist who expect that a paper published in Los Alamos archive satisfy certain standard criteria. Unfortunately I am not able to contact Mukul Patel himself. E-mails are bounced.


Arkadiusz Jadczyk

This was followed by a standard inquiry:

Subject: Re: Principia Physica
Date sent: Wed, 1 Mar 2000 04:24:12 -0400
On 1 Mar 00, at 1:15, PPARKER@TWSUVM.UC.TWSU.EDU wrote: > If you wish to conttact Mukul directly, I can provide an email > address if one was not included. > > Phil Parker

Thank you for your kind information. I tried to contact him directly, but he has stopped responding. I think the paper is wrong, but perhaps you can convince me that it is not.... I have many problems with the paper, but the first one deals with affine nature of his maps omega_{xy}. I asked Mukul to explain, here is what he wrote:

Dear Colleague,

thanks for your kind interest in the paper. if we had required that f(x) = y, then the natural candidate would be the "ordinary derivative"....i.e. a LINEAR map from tangent space at x to that at y. But as things stand f(x) maybe a point differnt than y. so we have a linear map (the derivative) from tangent space at x to that at f(x) we apply our principle to the pair f(x) and y....again there is a derivative map form tangent space at f(x) to that at f(f(x)), and we appply the prinicple to the pair of paoints f(f(x)) and y....we repeat the process and get successive linear maps from f^n(x) to f^(n+1)(x)...for each n we have a compisiton from tangent space at x to that of f^n(x) can be show that the sequence of points f^n(x) converges to y and the wbove compistion converges to a linear map from tnagent space at x to that at y. But we have an extra pice of information...the sequence of points f^n(x)....this determines a vector at y. so what we have is a Linear map from x to y AND vector at y. In short, we have an AFFINE map form x to y (tangent spaces at these)....this map is what i calle "roughly speaking, an 'affine derivative of f'"....i hope this clarifies the vague allusion to the term affine. ************

to which I replied:

Patel wrote:
"But we have an extra pice of information...the sequence of points > f^n(x)....this determines a vector at y."

Hi, I am still trying to understand your paper. I do not see how a sequence point y_n on a manifold, even converging to a point y, determines a tangent vector at y. Will you please provide a proof/construction?

Did you think about this problem when reading the paper? Although, I do not understand how he extends affine map omega from vectors to tensors. Is there a method that I am not aware of? I think you can't extend affine map to tensors, because extension depends of decomposition of the tensor....?


This was followed by a rather astonishing reply from Professor Parker who, apparently, objected to my inquiries:

Date sent: Wed, 1 Mar 00 12:42:09 CST
Subject: you erred

You quoted without permission: that's not only bad manners, rude, and thoughtless, its also a violation of federal US law. I suspect you are merely a frustrated jerk who is trying to cause trouble because you are so uncreative yourself. You will receive no further communications from me, either directly or indirectly. You owe me an apology.
Phil Parker --------------------------------------------

So, the project having taken an unusual turn, I responded:

Subject: Re: you erred
Date sent: Wed, 1 Mar 2000 14:08:12 -0400

When I am asked by members of the international physics community to examine a paper and pronounce on its usefulness, I act in the standard way which includes being open with all, since it saves time and energy for all concerned. For the record: the standard way is that ALL is quotable, as long as proper attribution is made, and no specific request for confidentiality is appended. Papers which are evidently WRONG, should not be published. As a referee for journals, you should know it. I don't apologize for doing my job.


I did not receive a reply from Parker, but DID receive info from Patel, indicating clearly that he WAS in communication with Parker and Arfaei:

Date sent: Wed, 01 Mar 2000 16:06:38 -0800
To: "Arkadiusz Jadczyk"(by way of Mehri Arfaei )
From: mukulp
Subject: Re: Principia Physica

Dear Colleague,
i recieved the following communication from you addressed to arfaei.. I am glad to learn that you are 'reviewing' the paper for a 'group of physicists'. If you had mentioned that earlier, i could have responded in more elaborate and precise fashion. I must comment though that some of your objections are rather hasty. For example, you kindly informed me that one needs a measure or a volume form to integrate. Now this point is already addressed to in a footnote on the same page. If you had been reading more carefully, i am sure you could not have missed it, as it is referenced exactly where that integral occurs.

I humbly suggest that you reserve your opinion until AFTER you have read the whole paper, and you direct these same to me personally, rather than running around bothering my aquaintances and colleagues. Another thing, since you have failed to tell me what your background is, i had to go look up on the net. While this 'group of physicists' might consider you an expert on differential geometry, judging from your papers, i am inclined not to quite concur with their opinion.

More importantly, i suggest that you also consider the physical significance of the IDEAS presented in the paper. I assure you that all your technical objections can be and will be answered in detail. Judging from your web-site, you seem to be of the opinion that understanding "what actually happens" should be the ultimate goal and the "ptolemian" quantum theory ought to be replaced. Given this conviction of yours, which i whole-heartedly agree with, i am sure that you will find it in your heart to consider the concepts and how they successfully do the job, rather than being foccussed completely on the technicalities...

I assure you again, that i will answer all the questions you have, in one single communication. (Indeed, i am in the process of writing down all the "there exists" "for all" and "if and only iffs" of the technical side, AS WELL AS, a detailed analysis of the whole conceptual framework. And that detailed paper promises to be of a length not quite appropriate for publication in a journal.)

Also, you obviously seem to have rather too exalted a view of the "standards of rigor" at los almos archive. There are hundreds of papers there which don;t even come close your idea of rigor there. I sincerely believe that all your comments are rather caused by "a mathematician writes a physics paper". I hope you will be able to put this obvious prejudice and the prejudice about "institutional affiliation" aside and review the worth of the ideas in answering the questions you yourself have deemed ncessary in your worthy project, "Quantum Future".

Sincerely, Mukul Patel

p.s. i also suggest that you review the paper as i submitted to "the group of physicists" rather than an older version at the archive.

So, essentially, he was saying that there was a version of the paper that was posted at Los Alamos that was DIFFERENT from the one he had submitted to a "group of physicists."
Now, apparently, he had the idea that I was affiliated with this "group of physicists" to whom he had submitted the "other" version of his paper...

To answer his inquiry about my background in differential geometry, I forwarded the following:

From: Arkadiusz Jadczyk
To: mukulp
Subject: Re: Principia Physica
Date sent: Wed, 1 Mar 2000 17:56:52 -0400

On 1 Mar 00, at 16:06, mukulp wrote:
While this 'group of physicists' might consider you an expert on differential geometry, judging from your papers, i am inclined not to quite concur with their opinion.

From: Dr. Arkadiusz Jadczyk (yup, he has a Ph.D. in physics!) is a curious case indeed. On the one hand, he seems believe in lots of weird, weird stuff. (He complains that his colleagues at the Institute of Theoretical Physics, University of Wroclaw asked him to move this page off their website; it is easy to see why they didn't want their institute associated with this stuff.) On the other hand, he has coauthored a serious book on Riemannian geometry with R. Coqueraux, who is known to me for his perfectly sane papers on Cayley-Klein geometries,, and he has coauthored numerous papers with Phillipe Blanchard, who is known to me for his excellent review paper on the classical theories of Julia and Fatou (as in the Mandelbrot-Hubbard set). So I figure Jadczyk must be sane on even numbered days and channel with the Cassiopeians on odd numbered days.

Well, today is an odd day....


Apparently Mr. Patel was quite impressed by SOMETHING in the above quote. Whether it was dependent on the odd or even days, I am not quite sure. Nevertheless, he responded:

Date sent: Wed, 01 Mar 2000 19:04:00 -0800
To: "Arkadiusz Jadczyk"
From: mukulp
Subject: Re: Principia Physica

:):) now we are speaking the same language. (not the mathemtically obscure text on geometry, but the communicating with casseopians.)

Look, looks like we got off on the wrong foot with one another. let us put this thing behind us and concentrate on the problems at hand. You and i both agree that physics has SEVERE problems, AND most physicsts are shy of admitting those. I congratulate on your vision to see this, and the courage to speak out.

Before i attempt to answer your objections, with all due respect, i want to record my objections to your tactics of reviewing: i find it extremely offensive that you misguided me into believing that you are merely "reading" my paper. On top of that you go and bother my ex-professor, and my current academic abode, and then you cross-refer them to each other...and send private communications to third parties without express permission of their respective writers. You must agree that this is way outside of your regular duty as judging the content of the paper.

That being so, i am not holding it agaist you, as long as you don;t insist on lecturing me on how one needs a measure to integrate.

About your latest objection, this is what i have to say: not every converging seuqence determine a vector. But this one does beuase of the way it arises. In fact, even this "sequence" is just a first approximation that i thought of on the spur of the moment, to answer a question by someone who i thought must be a casual amateur reading my paper. In the actual formulation of the principle, i don;t even need that sequence. And of course it IS a technical matter (even if what you call an error IS an error indeed.) The crux of the matter is that what we have is a (nonlocal) *affine* connection. The upshot is that its infinitesimal aspect is a usual affine connection. BUT this affine connection has two components, the linear componet and a "rotational" component (corresponding to the semidirect product of the affine group in to the linear gorup and R^4). The interesting thig is that this rotational aspect turns out to be the "canonical form" which is equivalent to a tensor field on the base space, which is given by the identity transformation on each tangent space.

Now you must be familiar with the fact that canonical form in itself does not represent any extra sturcture on a manifold. So essentially what we have is a linear connection. (plus the canonical form). Basically, the "field" consistes of the exterior derivative of the linear connection with respoect to itself AND the exterior covariant derivative of the canonical form with respect to the former. So all our information IS ineed contained in the linear part of the, even if your hasty assertion is correct and certain sequence does not determine a vector as advertised, it is not a serious fundamental problem of my theory. I implore you again, to stipulate temporarily that we do have what i call a "nonlocal [affine] connection".

Based on that, see if other ideas have any significance at all in your opinion. Believe me you will answers to many of the questions you have outlined in your "Quantum Fuuture" project....and more. meanwhile i will try to write up a precise formulation which would spin the heads of even the most pedantic mathemticians. One professional courtsey you can do to me is tell me exactly what "group of physicsists" has asked you to review this paper? Because if this group is what i think it is, you are reading the wrong version of the paper.

Again, he talks about "two versions" of the paper. The fact that he likes my wife's Cassiopaeans has nothing to do with the scientific approach that gives structure and form to my work as a physicist.
I replied:

To: mukulp Subject:
Re: Principia Physica
Date sent: Wed, 1 Mar 2000 22:40:12 -0400

Dear Mukul,

As you certainly know certain requests for a "review" are confidential. We are dealing with such a case. Perhaps you are right, perhaps I am reading the wrong, that is the "official" version. If you like to send me the "right" version - please do it. Otherwise I will have to follow blindly my duty - which will consume my time and yours.


He responded:

Date sent: Thu, 02 Mar 2000 12:25:25 -0800
To: "Arkadiusz Jadczyk"
From: mukulp
Subject: Re: Principia Physica

Dear Ark,

i am certianly not aware of a protocol where the body requesting a review is "confidential". Referees are often "confidential", of course. At any rate, this is the case. I have submitted a version of the paper to a "group of physicists". This version slightly differs from the one you seem to be looking at. I am assuming your "group of physicists" asked you to review the version on the xxx archive and not the one i submitted to my "group of physicists". If that is the case, it is possible that your "group" and mine are different. One question before i send you the other version: did this "group of physicists" say or imply that I have formally submitted my paper to them for publication in a print journal? I am sure you can tell me that much. I think you owe me that information before i waste any more of my time and yours---i mean what is the point in convincing a person of the verity of my paper if I (or my journal publisher) have not asked him to 'review' my paper?


So, we seem to be getting into the realm of DELIBERATE DISINFORMATION. So, having some personal objections to folks who post lies and obfuscations in the public arena which seem to be designed to lead the majority of physicists who don't have a good handle on the more complex math (not because they are not GOOD physicists! Believe me, I KNOW it!) I wrote back to Patel:

From: Arkadiusz Jadczyk
To: mukulp Subject: Re: Principia Physica
Date sent: Thu, 2 Mar 2000 15:14:57 -0400

Hi Mukul,
Yes, probably our "groups of physicists" are nonoverlapping. So, let me proceed with the only version that is official. I am back to my question: how you define "affine derivative". Affine derivative of WHAT? And how it defines affine map between tangent spaces. ark P.S. I am ready to accept: "I do not really need all these constructions, let us start with an arbitrary affine connection, and we will construct all our f's and omegas from this connection. Then, if we want to reverse the precess, we certainly, after sufficiently hard and, in principle, useless work, can do it and find axioms that f-s must satisfy ..." Or something to thi s extent. Otherwise I will press you to fix all wrong or unsuported claims (and you know you are making such claims, believing that they will be undetected, like with "sequence defining a tangent vector". Believe me, I will detect all such little sweet swindles.



Well, needless to say, Patel must have decided that physicists/mathematicians as a group, those who are searching for answers, do not deserve to read the "right" version of his paper... and he wrote back:

Date sent: Thu, 02 Mar 2000 20:26:42 -0800
To: "Arkadiusz Jadczyk"
From: mukulp Subject:
Re: Principia Physica

hi ark,

get lost.

How dare you badger me as if you are doing me a favor by reading my paper??? i didn't ask you to "process" me. So you can do whatever you want. I will take my "sweet swindles" directly to intelligent physicists. Actually, i already have. It is exactly your type of bitter old bastions of mediocrity that has helped keep physics in the quagmire that it has been for over a centurey now. Look at you, instead of thinking about physics, you are boasting about co-authoring obscure books on geometry that no respectable geometer reads. I am not much impressed by anyone lesser than Gauss, Riemann, Cartan, Ehressman etc. So it is hard to intimidate me using your pathetic little book that nobody reads. And your work speaks for itself---which isn't much i must add. I consider it beneath by dignity to even hold a conversation with you. Do not attempt to write to this address any more. Because if you do write any more unsolicited mail, i am going public with all your infringements of protocols, personal slanders, and other such strong-arm tactics.

Note added October 26, 2001
The statement above about the book "that no respectable geometer reads" somewhat contradicts a recent post to the newsgroup sci.physics.research in the "Re: D vs. E in vacuum" thread (post no 25 in this thread". Quoting from this post:

From: Charles Torre (
Message 25 in thread
Subject: Re: D vs. E in vacuum
Newsgroups: sci.physics.research
View this article only
Date: 2001-10-21 21:58:44 PST
ark <> writes:
> Also earlier: "Electromagnetic permeability of the vacuum
> and the light cone structure"
> A. Jadczyk, Bull. Acad. Pol. Sci. 27 (1979) 91-94.
> Online at > with extra info.

Nice. I will keep your result showing how Hodge * on 2-forms determines conformal structure in mind next time the issue comes up, which it does from time to time.

As long as I am writing, I might mention (changing the subject entirely) that your series of papers with Coquereaux and the corresponding book (all from long ago) on Kaluza-Klein reductions have been very useful to us here at USU since we are working on various aspects of symmetry reduction in gravitational theories. I would highly recommend them to anyone who is interested in such things.


It is always useful to see that different people have different opinions, but it is even more useful to try to understand a basis for each opinion.

So, what to make of it all? I don't know. Throughout the exchange, whenever a math question was asked, which was the main focus, (and in the exchange quoted above I skipped much of the technical math e-mail exchange between us), answers between mathematicians are given as a courtesy - and this is standard - the answer always seemed to twist around to "why do you want to know?" or "who ARE you that you want to know?" etc. And then, when called on these antics, both Patel and Parker jump into the highly defensive insult mode. I just simply could not get a straight answer about the math questions, nor could I get a reason for why the "wrong" paper was posted for the public, and the "right" one was reserved for "a group of physicists."

His attitude toward hard-working physicists who haven't sold their scientific principles is that they are "bitter old bastions of mediocrity that has helped keep physics in the quagmire that it has been for over a centurey now."

Everyone knows that there are many, MANY physicists who are working day and night, wading through mountains of papers, theories, and the quagmire of disinformation, to which Mr. Mukul Patel has now contributed so willingly by posting a version of a paper that he also freely admits is full of errors, but "so what?" In point of fact, based on all of the above, how can we know that the "right" version of the paper is even "right?"

Well, I have long thought about the idea that there IS a group of physicists who DO have some REAL answers... of course, that is mere speculation. Well, I guess you can draw your own conclusions.

Some of the technical exchange can be found here.

At my last attempt I tried to find out about a missing term in an important formula. Here what I got:

Date sent: Thu, 02 Mar 2000 20:56:30 -0800
To: "Arkadiusz Jadczyk"

From: "Arkadiusz Jadczyk"
(by way of mukulp )
Subject: Re: Principia Physica

The following message is undelieverable due to a filter set on this server.
-------------------------------------------------------------------------- -- ---
Hi Mukulp, I got a problem with the following: On p. 10 you write "Here, G is the Riemann curvature tensor field of the metric connection \gamma" But what happened to \delta\wedge\delta term which is also in Gothic g? ark

Mukul Patel must have set up a "filter", and does not want to discuss missing terms.... Perhaps someone else will also find this missing term?


Date sent: Thu, 23 Mar 2000 17:30:34 -0600
From: Reza Sarhangi
Subject: info/inquiry

Sirs, Two weeks ago a research paper appeared on Los Alamos e-print server No: gr-qc/0002012..................

Dr. Jadczyk, Mr. Mukul Patel was an adjunct faculty in our department. Currently he does not have any contract with our department or with the college. Regards, Reza Sarhangi, Ph.D. Chair, Mathematics and Computer Science Southwestern College 100 College Street Winfield, KS 67156

For the Expert:

At this place I am making some comments that address some the technicalities of the paper.

Recently there have been many discussion of torsion theories (Einstein-Cartan,Shipov, ....), and we are dealing here with torsion, so some readers may like to have some help in assessing some of the geometrical ideas presented in the paper.

In the e-mail exchange quoted above Mukul states " this affine connection has two components, the linear componet and a "rotational" component (corresponding to the semidirect product of the affine group in to the linear gorup and R^4). The interesting thing is that this rotational aspect turns out to be the "canonical form" which is equivalent to a tensor field on the base space, which is given by the identity transformation on each tangent space."

There is nothing unusual in the above. It is a standard knowledge from textbooks on differential geometry (Kobayashi-Nomizu for example) that a theory with torsion can be equivalently presented as "gauge theory of the affine group". It is also known that the translational part of so obtained affine connection is then nothing else than the canonical form (Kobayashi-Nomizu, Vol 1, Ch III.3, Proposition 3.1), and that translational part of the curvature coincides with torsion Proposition 3.4). So presenting the theory with torsion as as a gauge theory of he affine group living on the bundle of affine frames adds nothing new. I believe that Mukul is unaware of the vast literature on this subject (It is true that many of the relevant papers were written by physicists, and Mukul, as he frankly admits, is reading only Cartan, Riemann, Ehresmann and Gauss. If that is true he does not even know Kobayashi-Nomizu standard monograph). In fact, Kobayashi-Nomizu (Vol 1, Ch. III.3, p. 127) discusses also "generalized affine connections", when the translational part of the connection form does not necessarily comes from the canonical form. WE do not know why Mukul neglects this possibility....

This being said let me add also the following: one CAN go beyond the above scheme by "unsoldering" the principal bundle from the bundle of affine frames. In such a case one gets indeed interesting new possibilities. In particular the famous Einstein-Rosen bridge is a nonsingular configuartion in such a theory. Part of this program has been discussed in a classical paper by K. Pilch ( "Geometrical meaning of the Poincare group gauge theory", Lett. Math. Phys. 4 (1980) p. 49-51), another part in my own paper "Vanishing Vierbein in gauge theories of gravitation", available on this web site. This generalization allows for "phase transitions" (changes of signature) of the metric. This interesting idea belongs probably to A. Sakharov, and gets recently more and more attention (Prof. R. Jackiv brought recently to my attention his own paper "Gauge theories for gravity on a line", while F. W. Hehl kindly informed me about his own ideas about "Avoiding degenerate coframes in an affine gauge approach to quantum gravity"). Links to my own papers dealing with multidimensional gravitation can be found on my Kaluza-Klein page.

Physicists usually assume that there is a metric tensor. Mukul claims that metric tensor is, by itself, non-measurable, that only curvature is measurable. The standard wisdom of general relativity is that nine of ten components of the metric tensor are measurable. Whether this is indeed the case is disputable, but Mukul does not address this standard wisdom at all. His theory is a theory with metric tensor, or with "auxiliary metric tensor." Again, a pure affine theory is not a new invention. J. Kijowski has published a couple of papers on pure affine unified field theory. The main idea is to construct the Lagrangian density from the determinant of the (non-symmetric) Ricci tensor, and DEFINE metric as canonically conjugate object to the Ricci tensor (therefore metric, in this approach, is also non-symmetric).

Mukul's paper starts with assuming that, for every two points (x,y) and a sufficiently small neighborhood U of x there is a map f that maps U onto a neighborhood V of y. Unfortunately Mukul is not using a precise mathematical terminology, so we can not really know what are his assumptions. But, I guess, he is resoning as follows: suppose we have an affine connection, then if things are not too weird we can connect x and y by a unique geodesic gamma. Then a neighbourhood of x is mapped, via exponential map (see K-N, Ch. III.8, Proposition 8.2), onto a neighborhood of zero vectors in the tangent space Tx at x. We can also use the exponential map to identify V with a neighborgood of zero vector of the tangent space Ty at y. But now, by parallel transport (K-N, Ch.II.4, p. 130) we have affine map between Tx and Ty. If our connection is complete (K-N, Ch. III.6), then composing: inverse exponentila map at x with parallel transport affine map with exponential map at y, we get map f from U to V. This is, I guess, what Mukul has in mind.

Now, what I wrote below (in italics) is wrong .... So, do not read it. I have to think for a little while how to change it, so that will be correct....

The map such constructed maps geodesics through x into geodesics through y. If our connection is more general than admitted by Mukul,
(that is when the translational part of the connection is a general 1-1 tensor, then this last property (that f maps geodesics through x into geodesics through y) does not have to hold. This follows from the fact exponential map maps straight lines through zero into geodesics if we have affine connction, but not necessarily so if have generalized affine connection.

I keep it here, so that readers can see that it is a natural process in creating math that one has a clear idea of how to make things right, but making them really right takes some time, some thinking, some computation, and sometimes a complete turn .... It is only natural that when working with math one makes mistakes. What is important is to be willing to DISCUSS all possible errors, and all possible improvements. My error in the reasoning above is caused by the fact that for a while I have forgotten that affine map sends zero vector into non-zero vector. Therefore geodesics through x is mapped into straight line in Ty, but not through zero. Therefore exponential map will not, in general, send it into a geodesic through y. By the way, instead of using the term "geodesic line" I should use the term "autoparallel", because the term "geodesics" is usually reserved for SHORTEST lines. But we do not have metric to measure the distance!


After sleeping over the problem, I give up. I do not know how to make this "affine business" to work. It causes other problems too. Mukul's formula (1.1) makes no
sense in affine category. I do not know how he extends his omega to tensors. Integral involves addition, and addition is not defined in affine category, only subtraction. So, I am able to make some sense only when we take the linear part of the connection (the translational part is anyhow given by the canonical form, so there is no additional information there ....)

Before going further on, let me explain what is this "affine" extra translation that causes problems: suppose we have a connection (some will call linear connection, some will call it affine connection). Suppose we have two points, x and y and a path C(t) connecting x and and y: C(t0)=x, C(t1)=y. Then, by parallel transport we have linear isomorphism, say Fc: Tx -> Ty. Take now a frame e at x, and transport it paralelly along C. Let e(t) be paralelly transported frame. Let v(t) be the tangent vector to C at t. Let v[i](t) be components of v(t) with respect to e(t). Let a[i] (t) be the integral of a[i](s)ds from s=t0 to s=t1. Then, using frame e(t1), a[i](t1) defines tangent vector in Ty. This is the extra affine shift. Notice that if C is a geodesic then v[i](t) is a constant function, and so the affine shift is equal to a multiple of the tangent vector. For what Kobayshi-Nomizu calls generalized affine connection, calculation of the affine shift would involve application of the 1-1 tensor (of the translational part of the connection) to the tangent vector first.

Let me move now to Mukul's formula (1.1):

I will assume his omega is linear rather than affine. Question is: which volume form to take? If the formula is valid for one volume form, it will not be valid for another. He mentions "class of permissible fields", but we do not know what he means. His condition (1.1) is a strong condition, it is an integral equation. It imposes some kind of "harmonicity" on "physical fields". One possibility is that his connection admits a parallel volume form, then this form is unique up to a constant scalar, but even then we would have a dependence on this scalar. There is one possibility to make Mukul's equation invariant, namely to assume that the connection defines parallel transport not of vectors, but of vector densities.... But even then we have a problem with (1.1) ... So I do not know.... It would take me a couple of hours to think of a possible way of giving a meaning to Mukul's formula (1,1).

After having a one hour stroll, the following two ideas came to my mind:

1) Perhaps in his formula (1.1) Mukul wants to take "average" rather than "sum". That is, he would extend his integral over some open neighborhood V, divide it by the volume of V, and then take the limit of V going to infinity, assuming the limit exists. In this way the formula would be independent of a constant scalar factor, and it can be considered as "balance formula" or "equilibrium formula", when "external" sources are not present.

Or, possibly, he assumes that space-time is compact (R4 plus light-cone at infinity?), and then he assumes that the volume form is normalized. But, even thenm this will work only if the volume form is unique up to a constant. So that we do not have to divide by the volume of V. Do we need to assume that our affine connection has a parallel volume form? We will come to this question later on when discussing curvature 2-form. Is it assumed to be traceless? Notice that once we have a connection with torsion, we have a whole 1-parameter family of connections (as we can add a scalar multiple of the torsion) - thus we have some freedom of manipulation....

2) If we have a map f: U->V, f(x)=y, then the first jet of f defines a map from Tx to Ty. This is how Mukul proposes to get a connection. But we can continue. We can take second jet of f. This would map second order frames at x to second order frames at y. In particular this would map connection at x to connection at y. If so defined connection at y happens to be the same as the connection obtained from first order jets, then it would impose strong selfconsistency condition on the family of maps f. Whether Mukul takes it into account or not - I do not know....

You may ask why I am so preoccupied with this paper, which is so "USER UNFRIENDLY"? Well, first, it fits my field of professional interest and it touches many points that I was thinking about through the years. On the other hand it is a fun to be "filtered out" - so that I am forbidden to talk directly to Mukul or Phil Parker. Strange, strange are these mathematicians.... Fortunately we have Internet ... so I can talk to everybody who is interested and willing to use search engines.


To be continued... maybe.....




Last modified on: June 27, 2005.