On the order of indices of the Christoffel symbol of the 1st kind

[ric peregrino]

Homework Statement: The order of indices of the Christoffel symbol of the 1st kind seems to vary from source to source. Is there a preference, and if so why?
Relevant Equations: Christoffel symbol of the 1st kind.

The 1st definition of the Christoffel symbol of the 1st kind I came across was from Einstein's "The Meaning of Relativity", 1953, pg. 71, eq. 69:

$$[ij,k]=\frac{1}{2}\left(g_{ik,j}+g_{jk,i}-g_{ij,k}\right)$$

This is the same as I found in Barry Spain's "Tensor Calculus", 1953, but differs from other sources, for example wiki:

$$[ij,k]=\frac{1}{2}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)$$

I've taken the liberty to use i, j, and k to make the difference here obvious. I realize that with a symmetric metric, that perhaps the order of these 3 indices may not matter, but I wonder if the order may matter for an asymmetric metric? I've tried getting some answers on my own, and had limited success, lastly on stackexchange, where it seems the gatekeepers there are a tough crowd, and they closed my question on this with out any answer there, or any good answers to others' questions there concerning an asymmetric metric, mostly just responses that amount to "the metric HAS to be symmetric".

I had pointed out that Einstein and Kaufman published a paper in 1953 ( https://www.jstor.org/stable/1969690?origin=crossref) that included an asymmetric metric. If I read that correctly I may have learned a few things:

1. in 4d, there can be 2 unique components for the anti-symmetric part of such an asymmetric metric.
2. if the anti-symmetric part of the metric is non-singular at one point in space time, then it will be so for all points.
3. I didn't see a simple relation between the resulting metric compatible connection, and such a metric.
4. An asymmetric metric doesn't seem to immediately solve any issues. Nor does an asymmetric field, aka torsion.

I think another issue with my question at stackexchange was that I went big up front, and then when I tried to narrow it down to just this question here, I had proposed a preferred order that differs from the original, and from the wiki definitions, and made unsubstantiated claims about this proposed order. Any way, I've gone on long enough, and hope I got the root question across about the differing published orders of indices of the Christoffel symbol of the 1st kind, considering an asymmetric metric, is there a preference and if so why?

Cheers,
Ric

 

[anuttarasammyak]

The both are all right by symmetry of metric tensor, i.e.
$$g_{ij}=g_{ji}$$

 

[Orodruin]

The metric tensor is symmetric by definition. Einstein and others considered a general tensor formalism with the hope that it would lead to unification since the electromagnetic field tensor is rank 2 and antisymmetric. This however turned out to be fruitless. It is not something that is really considered much today as far as I am aware.

The both are all right by symmetry of metric tensor, i.e.
$$g_{ij}=g_{ji}$$

OP knows this, they have stated as much.

 

[ric peregrino]

My proposed order is: $$[ij,k]=\frac{1}{2}\left(g_{ik,j}+g_{kj,i}-g_{ij,k}\right)$$

The both are all right by symmetry of metric tensor, i.e.
$$g_{ij}=g_{ji}$$

Indeed, with a symmetric metric it doesn't matter. What I'm asking is does it matter with an asymmetric metric.

 

[anuttarasammyak]

As far as
$$ds^2=g_{ij}dx^i dx^j$$
holds, and
$$dx^j dx^i=dx^i dx^j$$
I have no idea how we can introduce non symmetric metric tensors.

 

[ric peregrino]

To be sure, I'm asking about asymmetric metrics, which are neither symmetric nor antisymmetric, but have both symmetric and antisymmetric components, and though the antsymmetric part does not contribute to ds^2 as anuttarasammyak points out, or affect geodesics, it does contribute to torsion.

 

[anuttarasammyak]

$$g_{ij}=s_{ij}+a_{ij}$$
$$ds^2=s_{ij}dx^idx^j+a_{ij}dx^idx^j$$
changing dummy indeces i and j
$$ds^2=s_{ji}dx^jdx^i+a_{ji}dx^jdx^i$$
Adding these two
$$2ds^2=(s_{ij}+s_{ji})dx^idx^j+(a_{ij}+a_{ji})dx^idx^j=2s_{ij}dx^idx^j$$
Antisymmetric components disappear.

 

[Orodruin]

$$g_{ij}=s_{ij}+a_{ij}$$
$$ds^2=s_{ij}dx^idx^j+a_{ij}dx^idx^j$$
changing dummy indeces i and j
$$ds^2=s_{ji}dx^jdx^i+a{ji}dx^jdx^i$$
Adding these two
$$2ds^2={s_{ij}+s_{ji})dx^idx^j+(a_{ij}+a_{ji})dx^idx^j=2s_{ji}dxjidxij$$
Antisymmetric components disappear.

Again, this is not what the OP is asking about. They have said so already in the OP and clarified again.

There are connections with non-zero torsion that are metric compatible and formalisms where the connection is a priori separate from the metric. It then turns out to be Levi-Civita from the equations of motion in the basic scenario.

How this connection would be affected by the introduction of an antisymmetric part to the metric I will leave unsaid as I have not studied this in detail.

I believe it is however fair to assume that Christoffel symbols of the first kind assume a symmetric metric, ie, the metric that goes into the definitions of the line element etc. To those, the antisymmetric part would not contribute, but they would generally not be equal to the connection coefficients.

 

[ric peregrino]

How this connection would be affected by the introduction of an antisymmetric part to the metric I will leave unsaid as I have not studied this in detail.

Thank you, this is the best answer I've received so far. I've self studied this in detail long ago. At the time, I did not have anyone I could discuss my findings with. This is why I'm here, and I'm trying to slowly ask about my findings in the hopes of finding others who may already know. I've also seen new methods, namely differential geometry, which may provide an easier approach. I recall the tensor calculus involved to have been many hours and many pages of tedious machinations. I've went over them many times, and at first did not get a favorable result, but after years of occasionally working on this, I came to a reasonable result in 1993.

A connection compatible with an asymmetric metric, where the symmetric portion of the connection is related only to the symmetric part of the metric, same as the symmetric Levi-Civita connection, and the antisymmetric part of the connection is related only to the antisymmetric part of the metric, and has the same form as the Levi-Civita connection, except only using the antisymmetric part of the metric with the proposed new order of indices.

I would like to post further equations on this, but I'm not sure if or where I should. Perhaps I could edit the OP? I await guidance.

Cheers,
Ric

 

[Orodruin]

I did not have anyone I could discuss my findings with.

This may prove difficult here due to forum rules. This is not a forum to discuss original research - it is simply outside the scope of the forum. You are expected to be able to refer to published material in reputable sources.

 

[ric peregrino]

Thanks for that reply. In the Einstein and Kaufman publication I referenced, I suggest they arrived at a relation between an asymmetric metric and the ensuing connection. However, their result seems overly complicated in my estimation, and I'm offering a simple alternative. Could you suggest another appropriate forum that might be a good place to discuss this? Seems stackexchange won't have it, and I'm too old to go back to university.

 

[ric peregrino]

Thanks for the participation. The answer I'm also receiving here in physicsforums is that there is currently no known preference, and all 3 of the order of indices noted above are equivalent even for the case of an asymmetric metric.

I'll note one fundamental difference with the order I proposed: the antisymmetric part of the [ij,k] in the indices i and j, depends only on the antisymmetric part of g, and the symmetric part of the [ij,k] depends only on the symmetric part of g. Using the original definition this is not the case, and two terms from the antisymmetric part of g enter into the symmetric part of [ij,k]. I hope this one easy to prove statement of fact is not seen as controversial and can be agreed.

The next steps in my homework are not easy to see, perhaps controversial, though straightforward but tedious to prove. However, in attempt to abide by rules here, I'll keep these to myself, but may be willing to share with any interested parties outside of this forum. I hope I'm not overstepping.

I understand the reason to avoid crackpots. Who remembers newsgroups and the "status quo" guy on sci.physics?

 

[strangerep]

@ric peregrino : It sounds like you're not yet aware of Einstein's "Non-symmetric Unified Field Theory". (Google will yield plenty of references.) I did a bit of work on it in the late 1970's, but remain angry to this day that my supervisor guided me to participate in this deadend research that he had worked on for quite a while.

It involves both a nonsymmetric metric and a nonsymmetric connection. He also invented 2 different covariant derivatives associated with the 2 different conventions for the ordering of the connection indices. After learning more differential geometry I realized that this distinction is nonsense (imho). It seems plausible only if one has not clearly thought about the meaning the connection indices. It's just mindless playing with indices, without a rigorous underlying geometric foundation.

Nevertheless, various people tried to develop this theory further (with Einstein, and long after his death). There is even work (by Tonnelat, iirc) that derives a specific form of this generalized connection in terms of the nonsymmetric metric (a horrendous computation -- trust me).

This theory never went anywhere useful. You are wasting your time (like Einstein wasted several decades on it towards the end of his life ).

Bottom line: yes, this can be discussed on PF -- because many articles on it have been published in reputable journals. You'll need to obtain some of these papers and study them before trying to discuss here.

But, I suggest you try and sort out the underlying meaning of the connection indices (as mentioned above) as the quickest road to realizing that it's all nonsense. The previous work on this "theory" doesn't properly address this point at all, afaik.

 

[PeterDonis]

Moderator's note: Thread moved to relativity forum.

 

[Dale]

I don’t have any direct answer to the question, but by way of motivation aren’t these types of metrics and connections important for Einstein Cartan gravity? If so then it should be a niche subject, but in the professional scientific literature

 

[renormalize]

I don’t have any direct answer to the question, but by way of motivation aren’t these types of metrics and connections important for Einstein Cartan gravity? If so then it should be a niche subject, but in the professional scientific literature

To my understanding, the metric (but not the connection) remains symmetric in Einstein-Cartan gravity.

 

[strangerep]

To my understanding, the metric (but not the connection) remains symmetric in Einstein-Cartan gravity.

Yes, and the torsion is only relevant inside matter.

 

[Orodruin]

Bottom line: yes, this can be discussed on PF -- because many articles on it have been published in reputable journals. You'll need to obtain some of these papers and study them before trying to discuss here.

To clarify what I was saying above: I did not mean that the subject itself would be off limits. However, trying to further develop the subject (as the OP seemed to indicate wanting) would be beyond the scope of Physics Forums.

 

[ric peregrino]

After learning more differential geometry I realized that this distinction is nonsense (imho).

Fair comment, and I find current topics are utilizing differential geometry and I plan to get a good textbook on the subject to hopefully be able to "see" better. Any suggestions?

However, trying to further develop the subject (as the OP seemed to indicate wanting) would be beyond the scope of Physics Forums.

I'm trying to respect this. An issue at stackexchange was the "check my work" issue. There I had approached this subject as surely this has been hashed out already, here's what I got in 1993, is this right and is there a good reference on this subject? Even though my posts on this narrow subject were closed, some answers were helpful to me and I learned a lot there, namely about contorsion, and that I should study differential geometry.

I see also the now closed related thread here by member 11137, which had some nice replies posted by strangerep and vanhees71. Thanks for that.

I am aware of Einstein's and Kaufman's work on this subject, as I noted in the OP, and also that nothing useful has or may ever come of this. Perhaps I'm a bit of a mathematical weirdo though, and tensor calculus machinations somehow sooth my brain. I hope differential geometry doesn't ruin that for me, but I expect not.

 

[strangerep]

[...] I find current topics are utilizing differential geometry and I plan to get a good textbook on the subject to hopefully be able to "see" better. Any suggestions?

I've only been referring to DG at a rather basic level. The first few chapters of Carroll's "Spacetime and Geometry..." should be enough -- if studied properly. If you can master that, a more advanced presentation can be found in Matthias Blau's Lecture Notes on General Relativity. (Personally, I find the more abstract pure-math treatments of DG too opaque to be useful for easy practical application, but opinions on this may differ.)

And let's not forget "Mathematical Methods for Physics and Engineering", by Matthias Blennow (a.k.a Orodruin ) which has an extensive chapter about "Calculus on Manifolds", and a lot of other good stuff.