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310

Fuck geometry (lemm.ee)

submitted 1 month ago by [email protected] to c/[email protected]

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[–] [email protected] 4 points 1 month ago (1 children)

You can make something like this properly by defining a different metric. For example with metric dl^2^ = dx^2^ - dy^2^ the vector (1, 1) has length 0, so you can make a "triangle" with sides of lengths 1, -1 and 0.

[–] kogasa 2 points 1 month ago (2 children)

That's not a metric. In any metric, distances are positive between distinct points and 0 between equal points

[–] [email protected] 1 points 1 month ago* (last edited 1 month ago) (1 children)

It depends which metric definition are you using. The one I wrote is a pseudo-Riemannian metric that is not positive defined.

Normally physicists use that generalized metric definition because spacetime in most cases has a metric signature of (-1, 1, 1, 1). Points with zero distance are not necessarily the same point, they just are in the same null geodesic.

[–] kogasa 1 points 1 month ago

You're talking about a metric tensor on a pseudo-Riemannian manifold, I'm talking about a metric space. A metric in the sense of a metric space takes nonnegative real values. If you relax the condition that distinct points have nonzero distance, it's a pseudometric.

[–] [email protected] 0 points 1 month ago (1 children)

This is true for real-valued metrics but not complex-valued metrics.

https://en.wikipedia.org/wiki/Complex_measure

[–] kogasa 2 points 1 month ago

Metric, not measure. Metrics are real by definition.