# What's the proper mathematical name for the Lorentz “inner product” on Minkowski space?

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).

• I've seen the PIP option used, though I can't find a source atm. – J.G. Nov 5 '19 at 22:20

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.
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.