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What's the difference between $\Bbb R^{4}$ and $\Bbb R^{1,3}$?

I know that the first one has metric Kronecker delta $\delta_{ij}$. Does the second one have Minkowski metric $g_{\mu \nu}$?

Xoque55
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nightmarish
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  • Could you be so kind as to define them? The $R^4$ I had in mind had the standard Euclidean metric... – pjs36 Dec 05 '15 at 17:10
  • What would you like the definition of? The spaces or Minkowski metric? – nightmarish Dec 05 '15 at 17:15
  • I just thought it would be good to say exactly what $R^4$ and $R^{1, 3}$ are. I assumed $R^4$ means $\Bbb R^4$ with the usual Euclidean metric, but the Kronecker comment made me doubt that. But this is not my area, so maybe it's already clear to others what you are talking about. – pjs36 Dec 05 '15 at 17:20
  • I've included your LaTeX edits for the notation of the spaces. The Euclidean metric is the Kronecker delta $\delta_{ij}$. – nightmarish Dec 05 '15 at 17:34
  • The second one seems to be relatively non-standard notation; I would understand it as "one dimension in time, three dimensions in space", or in other words "relativistic spacetime". Physically it is natural to put the Minkowski metric on such a thing. Whereas $\mathbb{R}^4$ is just 4-vectors, with no distinction between the components. – Ian Dec 05 '15 at 17:36
  • I would like the definition of $\mathbb R^{1,3}$. The Kronecker metric is the standard Euclidean metric which would apply to $\mathbb R^4$. Ian's comment sure makes sense to me and I would presume it is the correct answer but I actually don't know what $\mathbb R^{1,3}$ is. – fleablood Dec 05 '15 at 17:44

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$\newcommand{\Brak}[1]{\langle#1\rangle}\newcommand{\Reals}{\mathbf{R}}$The meaning of $\Reals^{p,q}$ isn't universal, but typically $\Reals^{1,3}$ refers to the vector space of ordered real $4$-tuples equipped with the indefinite quadratic form $$ \Brak{v, v} = v_{0}^{2} - (v_{1}^{2} + v_{2}^{2} + v_{3}^{2}) $$ or its negative $$ \Brak{v, v} = -v_{0}^{2} + v_{1}^{2} + v_{2}^{2} + v_{3}^{2}. $$

In either case, one fundamental technical difference between $\Reals^{4}$ and $\Reals^{1,3}$ is that the orthogonal group $O(4)$ (the group of linear isometries of $\Reals^{4}$) is compact, and the orthogonal group $O(1, 3)$ (the group of linear isometries of $\Reals^{1,3}$) is not.

Less commonly, $\Reals^{p,q}$ may also refer to a super vector space, with even (or parity $0$, or Bosonic) part isomorphic to $\Reals^{p}$, and with odd (or parity $1$, or Fermionic) part isomorphic to $\Reals^{q}$. It's possible that $\Reals^{p|q}$ is becoming the "standard notation" for a super vector space, however.

Andrew D. Hwang
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