I understand that it is more common in GR for the metric to be given a $(-,+,+,+)$ signature and more common in particle physics (or field theory, as Peskin & Schroeder tells me) to use the $(+,-,-,-)$ metric signature. I'm wondering why. Is there any advantage in these disciplines in terms of actual physics to using one over the other, or it simply an entrenched arbitrary convention?
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5$\begingroup$ If there was a difference in the physics one of them would be wrong. It's just a convention. $\endgroup$– dmckee --- ex-moderator kittenJan 12, 2014 at 17:19
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1$\begingroup$ Well, obviously. The character of my question was more, "Do they make calculations simpler somehow?" I guess I should have said, "...in terms of DOING physics..." $\endgroup$– JohnnyMo1Jan 12, 2014 at 17:29
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5$\begingroup$ One reason that I could find for the $+,-,-,-$ is that $p_\mu p^\mu=m^2$ instead of $-m^2$... I guess it all depends on which signs you prefer in certain equations $\endgroup$– DanuJan 12, 2014 at 17:44
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$\begingroup$ @Danu that's about the extent of it as far as I know. Perhaps you could post that as an answer. $\endgroup$– David ZJan 12, 2014 at 18:47
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In the theory of geometry of space-time, it makes much more sense to use $-+++$ because geometric view comes from intuition about 3D space, where we have metric $+++$. Time requires opposite sign, so we end up naturally with $-+++$ (or $+++-$ in some books).
In other areas where geometry and space distances are not that important but particles and their proper time is, and four-momenta appear a lot, using $+---$ makes the expression $p^\mu p_\mu$ proportional to rest mass squared, and $dx^\mu dx_\mu$ on the trajectory of particle turns out positive, which is more convenient if we want it to be differential of proper time squared. I hear also that in the theory of spinors, $+---$ is much more convenient.
answered Jan 12, 2014 at 18:58
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1$\begingroup$ Note that, in 3D Euclidean space, one might as well say that we have metric signature $-,-,-$, so it does eventually depend only on our preference for signs. $\endgroup$– DanuJan 12, 2014 at 19:34