Are the principles of space-time homogeneity and Isotropy independent of one another? 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?
 A: 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
A: 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.
A: 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.
A: 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.
