In physics, a sign convention is a choice of the signs (plus or minus) of a set of quantities, in a case where the choice of sign is arbitrary. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A negative number is a Number that is less than zero, such as −2 "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 different 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. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point
Sometimes, sign convention is used more broadly to include factors of i and 2π, rather than just choices of sign. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems
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Relativity
Metric signature
In relativity, the metric signature could either be + − − − or − + + +. The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive The latter form is often called the Landau-Lifshitz (spacelike) sign convention. Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968 Evgeny Mikhailovich Lifshitz (Евгений Михайлович Лифшиц February 21 1915 &ndash October 29 1985) was a leading Soviet A similar dual convention is used in higher-dimensional relativistic theories.
Regarding the choice of − + + + versus + − − −, a survey of some classic textbooks reveals that Misner, Thorne, & Wheeler chose − + + + while Weinberg chose + − − −. Books Popular Leisurely pace provides superb intuition for Schwarzschild geometry Subsequent authors writing in particle physics have generally followed Weinberg, while authors of papers in classical gravitation have generally followed MTW (as do most WP articles related to relativistic physics). Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Nevertheless, the Weinberg form is consistent with Hyperbolic quaternions, a forerunner of Minkowski space. In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity
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. The University of California Berkeley (also referred to as Cal, Berkeley and UC Berkeley) is a major research university located in Berkeley James Wallace Givens Jr ( 1910 December 14 – 1993 March 5) was a mathematician and a pioneer in computer science Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Princeton University is a private Coeducational research university located in Princeton, New Jersey. Year 1955 ( MCMLV) was a Common year starting on Saturday (link displays the 1955 Gregorian calendar) Oswald Veblen ( 24 June 1880 in Decorah Iowa &ndash 10 August, 1960) was an American Mathematician, 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. Papal infallibility is the Dogma in Catholic theology that by action of the Holy Spirit, the Pope is preserved from even the possibility of " 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. In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined In the Mathematical field of Differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express Some authors use the contraction 
, whereas others use the alternative 
. Due to the symmetries of the Riemann tensor, these two definitions differ by a minus sign. In the Mathematical field of Differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express
In fact the second definition of the Ricci tensor is 
. 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).
Thermodynamics
The sign of work in the first law of thermodynamics. In Thermodynamics, the first law of thermodynamics is an expression of the more universal physical law of the Conservation of energy.
Other conventions
- The choice of
in the Dirac equation. In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides - The sign of the field strength tensor
in gauge theories and classical electrodynamics. The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric - Time dependence of a positive-frequency wave:
(mainly used by physicists)
(mainly used by engineers)
- The signs of distances and radii of curvature of optical surfaces in optics
It is often considered good form to state explicitly which sign convention is to be used at the beginning of each book or article. Radius of curvature has specific meaning and Sign convention in Optical design.
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