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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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    $\begingroup$ If there was a difference in the physics one of them would be wrong. It's just a convention. $\endgroup$ Jan 12, 2014 at 17:19
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    $\begingroup$ 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..." $\endgroup$
    – JohnnyMo1
    Jan 12, 2014 at 17:29
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    $\begingroup$ 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 $\endgroup$
    – Danu
    Jan 12, 2014 at 17:44
  • $\begingroup$ @Danu that's about the extent of it as far as I know. Perhaps you could post that as an answer. $\endgroup$
    – David Z
    Jan 12, 2014 at 18:47
  • $\begingroup$ The West Coast Metric is the Wrong One by Peter Woit (yes, the title is provocative but it's interesting to read and very well motivated). For conventions in the (-+++) signature, see M. Srednicki's QFT book or the "TASI 2011 lectures notes" doi.org/10.48550/arXiv.1205.4076 $\endgroup$
    – Quillo
    Apr 22 at 10:11

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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.

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    $\begingroup$ 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. $\endgroup$
    – Danu
    Jan 12, 2014 at 19:34

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