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Einstein in deriving the Lorentz transformations, used the principles of space-time homogeneity and Isotropy. Does space-time isotropy follow from space-time homogeneity or are they completely independent of one another?

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Possible duplicate: and links therein. – Qmechanic Dec 20 '12 at 9:15

Please look at Carroll's GR book in Chapter 8.1. or at the free available lecture notes Chapter 8.

Short version: They are independent.

Space homogenous + isotropic in 1 point --> Space is homogenous + isotropic everywhere

Space isotropic in every point --> Space is homogenous + isotropic everywhere

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If you can't have isotropy every where without homogeneity, then doesn't this imply they're not entirely independent of one another? – Physiks lover Dec 20 '12 at 21:44
But if you ask if "isotropic everywhere" and "homogenous" are related. Then of course the first implies the second. If you can not access the book look at Chapter 8. – ungerade Dec 21 '12 at 6:58

Isotropy is its own principle, corresponding to the generalized rotational/boosting symmetry inherent to Minkowski spacetime. One can build a mathematical structure on homogeneity without isotropy, but such a system is not a vector space like what we're accustomed to dealing with, so it's difficult (for me at least) to imagine.

Lorentz transformations are much more directly tied to isotropy--the freedom to change one's basis and still describe the same physical situation. You could recognize it as a generalization of rotational invariance.

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A magnetised piece of iron is homogeneous because it's the same everywhere, but it's not isotropic becase the magnetisation gives it a preferred direction. So you can have homogeneity without isotropy.

However I don't think you can have isotropy without homogeneity. Or, as Chris and ungerade have pointed out, you can't have isotropy everywhere without the system being homogeneous. You can certainly have a system isotropic about a single point and not homogeneous.

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You can have isotropy at a single point without homogeneity. Two points of isotropy, I believe, are sufficient to guarantee homogeneity. – Chris White Dec 20 '12 at 11:47
However I don't think you can have isotropy without homogeneity. <-- that is wrong. – ungerade Dec 20 '12 at 15:15
I had assumed isotropy meant isotropy everywhere and not just isotropy at a single point. However I concede that a system can be isotropic at a single point without being homogeneous. – John Rennie Dec 20 '12 at 15:20
I've edited my answer to take on board your and Chris' comments. Thanks :-) – John Rennie Dec 20 '12 at 15:46

The principles of space time homogeneity and isotropy are dependent of one another. The reason of dependency of one another refers to the region of space time of being homogenous. Homogeneity in space time results from being symmetric, and what causes space time to be symmetrical is simply the Laws of Nature. Hence our space time, or the shell we are living in, is homogeneous and isotropic.

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