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So the minkowski metric is -,+,+,+. I was wondering if you can have more combinations of + and -. How would these spaces behave and is there anything interesting to say about them? I cannot really imagine what happens with these kind of metrics. I first wanted to ask in the physics stackexchange but I think this is more a mathematics question.

$ ds^2 = -dx_0^2 - dx_1^2 + dx_2^2+dx_0^2 $

Or more in general:

$ ds^2 = \sum_{i=0}^{n} dx_i^2 -\sum_{j=0}^{m} dx_j^2 $

Does this make any sense? How is this called and are there any interesting things happening at certain 'n'and 'm'

Thanks in advance!

bananenheld
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    In your first example the $dx_0^2$ would just vanish and your quadratic form would not be non-degenerate anymore. – C_M Nov 10 '21 at 14:46
  • I suggest you take a look at a math textbook on semi-Riemannian geometry, say, O'Neill, https://www.amazon.com/Semi-Riemannian-Geometry-Applications-Relativity-Mathematics/dp/0125267401. Also, proofread your questions before posting. Incidentally, the standard form of a flat semi-Riemannian metric is not what you wrote, but $\sum_{i=1}^p dx_i^2 - \sum_{j=1}^q dy_j^2$. – Moishe Kohan Nov 10 '21 at 20:23
  • thanks for commenting. I proofread my question but I have no experience with how you should write it because I dont know this area. – bananenheld Nov 10 '21 at 21:20
  • Whatever your experience, you should expect that $X-X+Y=Y$. – Moishe Kohan Nov 11 '21 at 00:23
  • I don't understand what you mean, because $ x_i \neq x_j $. Thanks for the suggestion, that is a nice book and it contains what I am looking for. – bananenheld Nov 11 '21 at 06:32

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