Principia Physica?

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.....