On Saturday, January 18, 2025 at 4:28:06 PM UTC-7 Brent Meeker wrote:
On 1/18/2025 5:42 AM, Alan Grayson wrote: On Saturday, January 18, 2025 at 6:13:27 AM UTC-7 John Clark wrote: On Fri, Jan 17, 2025 at 1:41 PM Alan Grayson <agrays...@gmail.com> wrote: *> IMO SR can handle curved spacetime. All one has to do is make the partitions very fine, so we're approximating inertial motion along very short paths. AG * *All one has to do? Well yes but that's easier said than done, it took Einstein 10 years of grueling work to figure out exactly how to do it, and the effort nearly killed him, he got sick, lost 50 pounds and figured he would die soon. Fortunately he did not. One of the most difficult things he had to figure out was how to measure distance in 4D non-Euclidean spacetime that was curved in any given way that was useful and never produced self-contradictory results. Mathematicians insist that distance must have the following properties:* *1) Non-negativity: d(x,y) ≥ 0 2) Identity of indiscernibles: d(x,y) = 0 if and only if x = y 3) Symmetry: d(x,y) = d(y,x) 4) Triangle inequality: d(x,z) ≤ d(x,y) + d(y,z)* *After years of false starts and dead ends Einstein eventually found a measuring stick that worked, it's called the Metric Tensor. * *John K Clark See what's on my new list at Extropolis <https://groups.google.com/g/extropolis>* Sorry to consume so much bandwidth here. I have one question about the LT and one about the Metric Tensor (MT) which will hopefully resolve most of my confusions. I'll pose the LT question in the context of the TP. If the stationary twin at rest on the Earth uses the LT to calculate the clock reading at some time on the traveling twin's clock, or its clock rate using two or more time readings, what relationship, if any, does this have on what the traveler twin's clock actually reads, Actually reads when? and what he observes as his clock rate? He observes his clock rate to be one second per second. *IOW, using the LT, the stationary twin knows precisely what the traveling twin will measure for his clock rate, but the traveling twin detects nothing. You gotta luv it. AG* IOW, to what extent does the LT transform the parameters of one frame, to actual observations on another frame It transforms spatial and temporal intervals in one frame, say Earth's frame, into intervals in a frame moving relative to Earth. (previously referred to as the source and target frames respectively)? Concerning the MT; it's defined as bilinear map to the real numbers on vectors in the tangent vector space at each point on a manifold. So, when trying to solve Einstein's field equation, the experts claim one must start by calculating the MT, presumably for some given distribution of matter and energy. That's for general relativity in which spacetime is not approximately flat, but it is for the twin paradox. Since, in general, there exists an infinite set of pairs of vectors in the tangent vector space at each point on the manifold, and assuming the MT has a unique real value at each point on the manifold, at each point on the manifold how do we choose which pair of vectors to do the calculation? If you have a pair of vectors you just get the product of projected lengths. So for a path you're just interested in two copies of the vector along the path. Then multiplying them with the metric tensor gives you the squared length of the vector. Brent -- 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 everything-list+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list/ac017916-2896-4370-9528-c0a3a81999f8n%40googlegroups.com.
