In article <0c7a7059-b37b-4024-aad4-93536e301d1d@googlegroups.com>,
Savin Beniwal <darshanbeniwal11@gmail.com> writes:
I have questions regarding the Cosmological Principle that usually we's Law
study universe is SPATIALLY homogeneous and isotropic(around every
point) at large scale (>150MPC). Here homogenous means--> No special location and Isotropic means-->No special point. Also, this was
confirmed by Hubble in 1929 that if distances are expanding (or contracting), the speed must be proportional to distance =E2=80=93 Hubble=
is inevitable.
But my questions is that if there were a proportionality relation
between velocity and square of distance rather than a linear relation between r and v. Even then can we understand the homogenous and
isotropic concept from Hubble's law under this nonlinear relation?
No.
Say you are at the origin, at distance 1 velocity is 1, at distance 2 velocity is 4, at distance 3 velocity is 9, and so on. For an observer
at distance 1, your distance 2 is just 1 unit of distance away, but its
speed relative to the observer at 1 is 3 (4-1), whereas it should be 1
if the distance is 1.
In short, homogeneity and isotropy demand a linear velocity--distance
law, since otherwise homogeneity and isotropy couldn't persist. (Note
that this is purely kinematics, no dynamics, hence this does not depend
on general relativity in any way.)
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