Does homogeneity infer isotropism?
I believe homogeneity does imply isotropism. If the universe is the same everywhere than it should look the same in every direction. What am I missing?
See here and here. Or, just consider an infinite universe with a uniform, non-zero vector field. Every point can have the same value of the field (implying homogeneity), but it will have a preferred direction (implying non-isotropy).
If enforced at all points, however, isotropy does imply homogeneity.
I don' think so,
you can have homog /inhomog conditions and isotropic and uniso... cdtns. They do not imply the other. Consider the metric:
This is homogeneous and isotropic.
This is inhomogeneous but still isotropic,
This is inhomog. and unisotropic.
I am just learning GR but I think this is right. The general gist is right - there might be a couple of hiccups in the equations.
Maybe someone with more experience can set me right if I am wrong.
No. Homogeneity does not imply isotropy. The cylinder is clearly homogeneous but not isotropic.
How about for an infinite spacetime?
"Clearly, the universe cannot look isotropic to all obervers...Only an observer who is moving with the cosmological fluid can possibly see things as isotropic." --MTW, p. 714.
If the universe is homogeneous, then there can be one and only one local frame in which the universe looks isotropic. The CMB rest frame is fairly well defined to be approximately 368 km/sec (relative to the Sun) in the direction of the constellation Leo, and this is the local frame in which the cosmic microwave background radiation looks isotropic. Is there any other frame that is a more likely candidate to be our local comoving frame?
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