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The metric induced by the p-norm:

$d((x_1,\dotsc,x_n),(y_1,\dotsc,y_n)) = \left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}$

is often called the Minkowski distance.

There is also Minkowski space, which as I understand is a bit like Euclidean 4-space. And there is the Minkowski metric tensor defined for it.

Is there a relationship between Minkowski distance and the Minkowski metric tensor? If not, why is the metric induced by the p-norm called Minkowski distance? Does anybody have a reference for this name?

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    @Borbus, why did you said you think is like the euclidean 4-space? isn't it more like euclidean n-space?2017-01-01

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No. The Minkowski metric tensor is actually based on a very different metric: $d((x_1,...x_n),(y_1,...,y_n))=\sum_{i=1}^{n-1}{(x_i-y_i)^2}- (x_n-y_n)^2$ Which in fact is not really a metric, since $d$ can be smaller than zero (if you want to be formal - this is a pseudo-Riemannian manifold)

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    Not quite. The correct formula is $d((x_1,...x_n),(y_1,...,y_n))=\sqrt{ |\sum_{i=1}^{n-1}{(x_i-y_i)^2}- (x_n-y_n)^2|}$ and it is a pseudo-distance. By abuse, though, it is still called "metric".2017-05-10