You wrote, "...*the second term must be positive based on the physical
assumption that d(theta)/dt must be positive (since the arclength s must
be increasing as the universe expands)" The arclength is increasing but
the arcangle, theta, is not.
I argued that the recession is not due to a force but simply due to the
expansion of space.
Brent
*
On 9/12/2024 9:10 PM, Alan Grayson wrote:
On Thursday, September 12, 2024 at 5:47:27 PM UTC-6 Brent Meeker wrote:
That's backwards. theta represents a fixed point on the expanding
balloon universe so dtheta/dt=0 and all the change is in r the
scale of the universe. Assuming expansion is constant means
dr/dt=Hr where H is Hubble's */observed/* constant.
Brent
IMO, theta represents the angular displacement along the arc
connecting two galaxies, where one is assumed as fixed, the other
receding. I don't see what backwards about this. When we discussed
this ages ago, I recall your argument that the increasing recessional
velocity was purely a geometric consequence. This continues to be my
view. Consequently, there is no need to appeal to Hubble's law. AG
On 9/12/2024 4:12 AM, Alan Grayson wrote:
*I prefer this method. s = r * theta, where s is the
arclength or separation distance of two galaxies residing on
a circle of radius r, where theta is the angle subtended by
s. Differentiating, ds/dt = dr/dt * theta + r * d(theta)/dt.
Even if the expansion rate, dr/dt, is constant, the RHS is
positive since the second term must be positive based on the
physical assumption that d(theta)/dt must be positive (since
the arclength s must be increasing as the universe expands).
So, eventually, ds/dt will exceed the velocity of light, the
condition that the galaxies will lose contact. Any flaws in
this logic? AG*
*If the parameter r captures size of the universe, and if we
assume it's expansion is constant, then dr/dt = 0 and the first
term when differentiating s is zero. So the increase of ds/dt is
completely captured in the second term. AG*
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