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In his milestone paper on general relativity, Einstein not only introduces the Einstein summation convention, but also (formula (45) in [1]) abbreviates a minus at the Christoffel symbols away by introducing the Gamma notation for the connection coefficients of his variant of the covariant derivative, constructed on cotangent space first from given geodesics.

In later standard literature the minus is gone. (E.g. Schrödinger p.66 [3])

Why, and who dropped it?


Addendum/scholium: Thanks for linking from a related question on https://hsm.stackexchange.com/questions/7974/notation-for-christoffel-symbols !

The physical relevance of that sign might explain why it is gone, and I first thought about not asking on mathoverflow. But Einstein's introduction of the covariant derivative strikes me as masterful abstract differential geometry (of his time). Plus, it can be simplified and abstracted to general symmetric connections with given geodesics without Christoffel symbol stuff (if I'm not mistaken). I have not yet found it in other literature. (Already the standard geodesic equation with Christoffel symbols looks like an ugly case of superfluous zero to me. Like Einstein I prefer an own side for the 2nd derivative.)


[1] A.Einstein: Die Grundlage der allgemeinen Relativitätstheorie (1916) p.802 https://web.archive.org/web/20060830030952/http://www.alberteinstein.info/gallery/pdf/CP6Doc30_pp284-339.pdf

[2] Manuscript of [1], translation and more: H.Gutfreund, J.Renn: The Road to Relativity (2015) p.92 (German manuscript facsimile) p.209 (English translation). https://press.princeton.edu/books/hardcover/9780691162539/the-road-to-relativity

[3] E.Schrödinger: Space-Time Structure (1950/4) https://www.cambridge.org/core/books/spacetime-structure/554B50728DF38139E42E60BBED654D85

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  • $\begingroup$ I guess the culprit is Hermann Weyl. In Space Time Matter he introduces "The Conception of Affine Relationship" with a superfluous minus in the very first formula (33) p.112. I haven't yet comprehended his next page ... I guess his motive is to get the "right" sign for the curvature tensor... $\endgroup$ Mar 26 at 2:35
  • $\begingroup$ it might be helpful if you could indicate what kind of answer you are looking for; a paper that says "we choose a different sign from Einstein for this or that reason" and which was so influential that later users followed it? Since Einstein himself was not consistent in the choice of the sign, I doubt that such a paper exists (although there are many sources that noted that the identification of minus the second Christoffel symbol with the gravitational field is mistaken, obviating the need for the minus sign) $\endgroup$ Apr 20 at 8:25
  • $\begingroup$ Carlo Beenakker: Yes, that's what I'm looking for. Something after Einstein's 1916 paper. Before that he (and Grossmann) were inconsistent, naturally, as they were figuring out the maths and notation. The 1916 paper is not just a milestone in physics, but also in tensor calculus (mixed tensors). So there should be some very good reason for dropping Einstein's minus and making things (a little) messier. $\endgroup$ Apr 20 at 10:37
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Since the question has now narrowed down to "who lost the minus sign" in the Christoffel symbol, let me start a new thread: The OP asks for a reputable source, later than Einstein's 1916 paper, in which the minus sign is abandoned. I propose that it was Einstein himself who dropped it. Below I copy from his 1921 lectures at Princeton University, page 46:

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  • $\begingroup$ Thanks! I haven't yet sufficiently checked Hermann Weyl's Space Time Matter (plus his comment on Riemmanns Probevorlesung 1854. See my comment to my question. First eds. are 1918/9. In the later editions I got the minus gone. (Yes, one of my bibliophile treasures is an original 1923 Springer Verlag booklet. - But I don't have the internets line & equipment to dig out snippets like yours. Much appreciated! Alas for the next days I wont have nerves & time to check more.) $\endgroup$ Apr 23 at 19:56
  • $\begingroup$ This is sweet reading compared to the messy Weyl book! In formula (67) he introduces a minus out of no reason, like Hermann Weyl loc.cit. Thus an ugly geodesic equation. But Einstein's parallel transport equation (no number, right after the snippet) looks nicer without the "Einstein minus". :-) $\endgroup$ Apr 23 at 20:15
  • $\begingroup$ Methinks I can click "answered". Still I'm puzzled. Why then not just turn the {} symbols upside down? $\endgroup$ Apr 23 at 20:32
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Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

Why was the minus sign dropped?
It was understood that the identification of the Christoffel symbol with the gravitational field is mistaken: you can have a nonzero $\Gamma^\sigma_{\mu\nu}$ and zero gravitational field, all you have to do is to introduce curved coordinate systems in flat space. And conversely, the Christoffel symbol can vanish along a geodesic even if the gravitational field is nonzero.


Addendum: It seems Einstein was also not quite consistent with respect to the minus sign; in a 1914 paper he defined the $\Gamma$ without the minus sign:

In an interesting discussion on HSM it is suggested Einstein chose the symbol $\Gamma$ to refer to the first letter of "Gravitation".

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  • $\begingroup$ Yes. And he didn't like to carry the minus around at his canonical covariant derivative. My question in history of mathematics is: Why isn't this an Einstein convention, just like the summation convention? $\endgroup$ Mar 17 at 23:59
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    $\begingroup$ Added a paragraph on the "why". On a separate note: the summation convention is a convention of mathematical notation, its use is not limited to GR; the minus sign in the Christoffel symbol is a matter of physical interpretation limited to the gravitational context (and the interpretation turned out to be mistaken). $\endgroup$ Mar 18 at 7:25

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