On 9/13/2024 4:56 AM, Alan Grayson wrote:
It's already proven that ds/dt=ks => s=j*exp(kt) where k with dimensions of 1/time and j is an arbitrary constant of integration with the same dimensions as s. Going to second derivatives won't gain any more.On Friday, September 13, 2024 at 4:06:49 AM UTC-6 Alan Grayson wrote: On Thursday, September 12, 2024 at 11:07:49 PM UTC-6 Alan Grayson wrote: On Thursday, September 12, 2024 at 11:00:21 PM UTC-6 Brent Meeker wrote: On 9/12/2024 9:21 PM, Alan Grayson wrote:On Thursday, September 12, 2024 at 3:55:45 AM UTC-6 Quentin Anciaux wrote: Le jeu. 12 sept. 2024, 11:53, Alan Grayson <[email protected]> a écrit : On Thursday, September 12, 2024 at 2:40:56 AM UTC-6 Quentin Anciaux wrote: I just gave you a full proof that as long as the expansion is uniform and expansion rate > 0, then it follows objects will sooner or later recess from each other at speed > c. What was the justification for the geometric progression? I made no such assumption in my "proof". As explained multiple times and in the quote you made, expansion is uniform and happens at every point in space. What bothers me about your method is that you*assume* a geometric increase in the separation distance, when, IMO, that's the variable that must be calculated (which I did). So no matter how many times you affirm your proof as valid, I can't agree. AGYou didn't calculate the expansion parameter, which is the Hubble constant. It's an observed value. Brent Why must I do that, when I just want to show that eventually the recessional velocity exceeds c? Also, I don't see why theta is fixed, when the end of the arc defines the position of the receding galaxy. AG Now I am not sure I proved the recessional velocity is greater than c, after some time has passed. If the sphere is expanding, then the distance between any two fixed points on the sphere will increase as time passes. But that was obvious due to the expansion. What's wrong, if anything? AGNow I see the light. We've been struggling to prove that a receding galaxy will fall out of view if the universe is expanding, but all the so-called "proofs" fail, but for different reasons. What Quentin offers is not a proof. He's just repeating a result done by someone else,*using mathematics*, which he believes (and might be true). Brent is mistaken in his apparent belief that the proof of concept requires appeal to Hubble's law. This is also mistaken IMO since the result to be proven depends *exclusively* on the *geometry *of an expanding sphere. Finally, my proof also fails, since it's obvious that the arclength, s, between two galaxies on an expanding sphere, will obviously increase as the sphere expands. That is, ds/dt will obviously be positive since the arclength is increasing. IOW, a constantly increasing arclength s, assuming a uniformly expanding sphere, necessarily yields ds/dt > 0, but it does NOT demonstrate that the velocity of the receding galaxy eventually increases to be greater than c. When I have the energy, I will calculate the *second time derivative* of the arclength, s, hopefully to demonstrate, that for a uniformly expanding sphere, the *four* terms of the second derivative of s, imply a*positive acceleration*. This will establish that eventually the receding galaxy will pass out of view for the observer on the assumed stationary galaxy. Comments welcome. AG
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