Sign convention


Sign convention

In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. The choices made may differ between authors. Disagreement about sign conventions is a frequent source of confusion, frustration, misunderstandings, and even outright errors. In general, a sign convention is a special case of a choice of coordinate system for the case of one dimension.

Sometimes, the term "sign convention" is used more broadly to include factors of i and 2π, rather than just choices of sign.

Contents

Relativity

Metric signature

In relativity, the metric signature could either be + − − − or − + + +. A similar dual convention is used in higher-dimensional relativistic theories. The choice of signature is given a variety of names:

+ − − −:

− + + +:

Regarding the choice of − + + + versus + − − −, a survey of some classic textbooks reveals that Misner, Thorne and Wheeler (MTW) chose − + + + while Weinberg chose + + + − (with the understanding that the last sign corresponds to "time"). Subsequent authors writing in particle physics have generally followed Weinberg, while authors of papers in classical gravitation and string theory have generally followed MTW (as do most Wikipedia articles related to relativistic physics). Nevertheless, the Weinberg form is consistent with Hyperbolic quaternions, a forerunner of Minkowski space.

The signature + − − − would correspond to the following metric tensor:

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}

whereas the signature − + + + would correspond to this one:

\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}

Einstein's "ex cathedra" pronouncement

While in some sense this is a mere notational convention, the choice of the signature has always engendered considerable passion and even some degree of "controversy" (not entirely serious).

In an interview given on the campus of University of California, Berkeley, Wallace Givens (an applied mathematician who was active in the early development of computer science) recalled an incident from his experiences as a graduate student at Princeton University, circa 1955:

Anyway, (Veblen) had been trying to persuade me that in the metric for general relativity the signature of the quadratic form was quite clearly three minuses and a plus rather than three pluses and a minus, just a change in sign because it's the foundation of the concept of causality and no other signature will do for that. It really should be called a causality metric rather than a gravitational metric, but after all it was done by a physicist instead of a logician or a mathematician. Anyhow, Veblen had been trying to persuade me that it made a difference which you used, three minuses and a plus, or its negative, three pluses and a minus. Well, he was much too good a mathematician in every respect to tell me authoritatively. That was not the nature of the relationship. Veblen wasn't that kind of a person. He didn't do that to graduate students, and he didn't do it to me. But he was not without guile.

The occasion was that I was in my office waiting for the usual morning call to go into Veblen's office and talk. No one came. Veblen didn't knock, and I guess it was getting along towards lunch, so I thought I had better see what was going on. I stepped out my door and knocked on Veblen's door, and Veblen said come in and I went in. I saw what the difficulty was. He had been having a conversation with Einstein. Well, I'd met Einstein—his office was two or three doors down the hall—but I never knocked on Einstein's office because I had too much respect for his privacy and his time.

Anyway, on this occasion Veblen took the opportunity to fire a big gun on this little question of the signature. Well, both of us knew perfectly well what was going on. I don't know what the subject of the conversation with Einstein had been about. They both agreed that they were concluding it, and Einstein was about to leave. So Veblen said, "Professor Einstein, perhaps you'll decide ex cathedra a little question for us in regard to the signature of the metric." Well, Einstein laughed, quite a hearty laugh; he rumbled in laughter I think would be an appropriate way to describe it. He was flattered a little; he enjoyed it. He understood the question (and its phrasing!) and remarked quietly with some answer. This was more or less the end of the conversation and Einstein left, and I had a quiet, brief conversation with Veblen.

Now the story doesn't quite end there. Someone is supposed to ask which signature Einstein chose. Well, as a matter of fact, I don't remember, but the nature of the work at that time was of the following character. Einstein didn't give his reasons, so why did it matter which he said. That was the way things were done at Princeton in those days. Actually of course the question is easily answered by looking in Einstein's little book called Relativity, and I think it's three minuses and a plus. I think that's what he said, but I can't even be absolutely sure of that. But as I point out, I don't really think it matters very much. At least I wasn't convinced, even as a graduate student that it mattered very much.

Curvature

The Ricci tensor is defined as the contraction of the Riemann tensor. Some authors use the contraction R_{ab} \, = R^c{}_{acb}, whereas others use the alternative R_{ab} \, = R^c{}_{abc}. Due to the symmetries of the Riemann tensor, these two definitions differ by a minus sign.

In fact the second definition of the Ricci tensor is R_{ab} \, = {R_{acb}}^c. The sign of the Ricci tensor does not change, because the two sign conventions concern the sign of the Riemann tensor. The second definition just compensates the sign and it works together with the second definition of the Riemann tensor (see e.g. Barrett O'Neill's Semi-riemannian geometry).

Other sign conventions

It is often considered good form to state explicitly which sign convention is to be used at the beginning of each book or article.

See also

References


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