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Physicists tend to call the spacetime interval $(\Delta s)^2$ an "inner product," where $(\Delta s)^2=(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2$ up to factors of $c$ and an overall minus sign. But it's not really an inner product, since it breaks positive-definiteness:

$\left< v,v \right> \geq 0$ with equality iff $v=0$.

For that matter, the "metric tensor" of general relativity describes a similar map for two vectors in any tangent space, but this is also not technically a metric (in the mathematical sense) since that would also require positive-definiteness.

So what is the correct mathematical term for this structure?

I would be tempted to call it a pseudo-metric, since spacetime is called a pseudo-Riemannian manifold, but in fact pseudo-metrics still require positivity. I might also expect "pseudo–inner product" to be used, but this phrase does not appear in any Wikipedia article, and the similar term "pseudo-scalar product" is defined here as something completely different.

I have heard the terms "pseudo-Riemannian metric" and "Lorentzian metric," but I doubt mathematicians endorse such terms, since they are of the form "(adjective) + (noun)" where the object is not a type of (noun).

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  • $\begingroup$ I've seen the PIP option used, though I can't find a source atm. $\endgroup$ – J.G. Nov 5 '19 at 22:20
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Some people use "scalar product" instead of "inner product", reserving the word "inner" for the positive-definite case. But I don't understand what the issue is with "pseudo-Riemannian metric" or "Lorentzian metric", that's what people use normally. The point is that you should not think of metric spaces here. A pseudo-Riemannian metric in a manifold only induces a distance function when the metric is Riemannian, and then you have relations between $(M,g)$ and $(M,d_g)$ given, for instance, by the Hopf-Rinow theorem.

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Certainly "non-degenerate $\mathbb R$-bilinear pairing" (or several similar things) would be correct, as an umbrella description. "(The) pairing" for short, once it's understood.

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