This is true for real-valued metrics but not complex-valued metrics.
Comment on Fuck geometry
kogasa@programming.dev 1 month agoThat’s not a metric. In any metric, distances are positive between distinct points and 0 between equal points
technocrit@lemmy.dbzer0.com 1 month ago
kogasa@programming.dev 1 month ago
Metric, not measure. Metrics are real by definition.
OrganicMustard@lemmy.world 1 month ago
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@programming.dev 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.