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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: physics.stackexchange.com/q/24881/2451 –  Qmechanic Dec 20 '12 at 9:15

3 Answers 3

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

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 preposterousuniverse.com/grnotes Chapter 8. –  ungerade Dec 21 '12 at 6:58

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