*Firstly, the metric tensor does NOT directly give the curvature. Rather, it's a bilinear function whose range is a real number, but which one? That depends on the two domain vectors in the vector space on the tangent plane at some point P where the tangent plane touches the spacetime manifold. The 4x4 matrix you refer to is a representation of its mapping, not the curvature at some point P on the spacetime manifold. To get the real number it maps to, you need 2 vectors, a column vector followed by a row vector, but which give the real number we're seeking? This is where the ambiguity that bothers me is manifested. In GR, to get the curvature, we need to solve for the metric tensor field, and then solve for the curvature scalar and the Riemann curvature tensor. Recall, that in Einstein's Field Equation, there are these 3 tensors on the LHS, if we place the Energy-Momentum tensor is on the RHS. As for the uncountable set of real numbers, since we can't write them down, they can't be put in list (which doesn't omit any), but which can be done for the rational numbers as proven by Cantor using his diagonal proof. AG* On Wednesday, September 4, 2024 at 6:30:56 AM UTC-6 John Clark wrote:
> On Wed, Sep 4, 2024 at 8:08 AM Alan Grayson <[email protected]> wrote: > > >> *> it seems that the metric tensor FIELD is NOT well defined. AG* > > > > *The metric tensor encodes spacetime curvature, and for every point in > spacetime that you can name I can give you a 4x4 matrix of unique > computable numbers that defines the curvature at that point. What's > ambiguous about that? It's true that I can't do that for points in > spacetime that you cannot name, but that is not a problem because you > cannot get experimental results from points that you cannot name. * > > John K Clark See what's on my new list at Extropolis > <https://groups.google.com/g/extropolis> > wai > > > > > >> >> On Wednesday, September 4, 2024 at 6:02:17 AM UTC-6 John Clark wrote: >> >>> On Tue, Sep 3, 2024 at 6:44 PM Alan Grayson <[email protected]> wrote: >>> >>> * > I fail to see how your comments relate to the possibly ambiguous >>>> concept of the latter. The metric tensor field seems ambiguously defined.* >>> >>> >>> *A N dimensional space is composed of an uncountable number of real >>> numbers but it can be unambiguously defined by just N countable rational >>> numbers, you can pair them up one to one. This is possible because there is >>> only a countably infinite number of COMPUTABLE real numbers, the same rank >>> of infinity as the rational numbers. So you can in effect give a rational >>> number name to every real number you are able to find on the number line. >>> You can do this even for a number such as π which is not only irrational, >>> it's transcendental, because it is also computable. You can use an infinite >>> series to get arbitrarily close to π. * >>> >>> *The vast majority of numbers on the number line are NOT computable (and >>> have no name) but that's not really a problem despite the fact that the >>> vast majority of numbers on the number line are NOT computable because, >>> except for Chaitin's Omega Number, every number that a mathematician has >>> ever heard of is a computable number. Computable numbers can have names, >>> uncomputable numbers can not.* >>> >> -- 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/97637e88-85d9-4076-94a9-da5691c6e86bn%40googlegroups.com.
