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In David Tong's lecture notes on general relativity (page 12) he denotes Minkowski spacetime as $\mathbb{R}^{1,3}$. Also, on Wikipedia, I found that both $\mathbb{R}^{1,3}$ and $\mathbb{R}^{3,1}$ are used and the choice depends on the metric signature used, $(-,+,+,+)$ or $(+,-,-,-)$. I think $\mathbb{R}^{1,3}$ is associated with $(-,+,+,+)$ since this is the signature Tong uses in his GR lectures, which would leave $\mathbb{R}^{3,1}$ to be associated with $(+,-,-,-)$.

What is the link between these? That is, what is the significance of the order of the coefficients? Does the first coefficient denote the number of negatives in the signature and the second number denote the number of positives? Or, is there some other, possibly deeper, meaning? Is this notation used in pure math too, or is it simply a notation introduced by physicists for this purpose?

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  • $\begingroup$ There is no standard convention. You should figure out the convention that Tong is using in his notes. $\endgroup$
    – Prahar
    Commented Aug 20 at 16:05
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    $\begingroup$ Related Math.SE question: Equivalence of Two Lorentz Groups $O(3,1)$ and $O(1,3)$ $\endgroup$
    – Qmechanic ♦
    Commented Aug 20 at 16:11
  • $\begingroup$ Wikipedia: “A real vector space with an indefinite nondegenerate quadratic form of index $(p, q)$ (denoting $p$ $1$s and $q$ $−1$s) is often denoted as $\mathbf R^{p,q}$ particularly in the physical theory of spacetime.” $\endgroup$
    – Ghoster
    Commented Aug 20 at 17:48

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Sadly, there is no real convention about which number denotes the number of $+$ and which one denotes the number of $-$. As it turns out however, the two metric signatures are entirely equivalent (they are the same up to a constant factor of $-1$), so which one is being used really comes down to a matter of taste. Traditionally, the "mostly plus" metric is being used by relativists, while the "mostly minus" metric is used by particle physicists since it simplifies some calculations. Since they are equivalent, pure mathematicians don't really care about whether the metric has more plus or more minus, and there is no deeper significance about the notation. As with all things notational, it doesn't really matter what you do, as long as it remains comprehensible and consistent.

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    $\begingroup$ Since they are equivalent, pure mathematicians don't really care They’re not equivalent for real Clifford algebras. $\endgroup$
    – Ghoster
    Commented Aug 20 at 17:55

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