We're dealing with the tangent space defined by a vector space of velocities on the spacetime manifold, so there are at least a countable infinity of such vectors. And the issue is whether the metric tensor is well defined since it's a bilinear function of a pair of velocity vectors, which maps to the real numbers. But for the metric tensor FIELD to be well-defined, we need a *unique* real number as the result, but this seems impossible since there are many possible pairs which map to *different* real numbers. I don't understand Brent's suggestion. AG
On Tuesday, September 3, 2024 at 6:56:27 AM UTC-6 John Clark wrote: > On Sat, Aug 31, 2024 at 10:48 PM Alan Grayson <[email protected]> wrote: > > * > please explain how the metric tensor can be defined unambiguously at >> some point P on the underlying manifold, spacetime, if there is an >> uncountable set of pairs on a vector space on the tangent space at some >> point P on which the metric tensor is defined* > > > > If, as I suspect, your interest is physics and not pure mathematics then > it's a non-issue. The fact is nobody is even sure that 4D space-time > contains an infinite number of points, for all we know it may only contain > an astronomical number to an astronomical power number of points. That's > undoubtedly a very big number but it's no closer to being infinite than the > number one is. > > And even if 4D space-time does contain an uncountabley infinite number of > points, if you simplify your physical theory by assuming there is only a > countably infinite number of points it will have a negligible effect on > your theory; that is to say you could make the discrepancy between what > your theory predicts will happen and what you actually observed to happen > in experiments to be arbitrarily small. I am not aware of any physical > theory in which the difference between countable infinity and uncountable > infinity leads to different experimentally testable predictions. > > John K Clark See what's on my new list at Extropolis > <https://groups.google.com/g/extropolis> > n4x > > > > > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/926473ea-9795-4042-bd6a-99482a5c905fn%40googlegroups.com.
