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
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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