This is related to a previous post. Assuming that the Cosmological Principle is correct, does this imply that the universe possess an empircially privileged reference frame?
What I am trying to understand is related to the following: From what I understand of general relativity (GR) (which is NOT much), not only is it conceptually different from Classical mechanics (CM), but how it is applied is also different. In CM, you solve for a particular system within the universe, whereas in GR, you must solve for an entire spacetime. This seems to create challenges when physicists want to describe individual systems that behave relativistically, requiring the specification of an entire, ad hoc, and hopefully computationally benign universe surrounding the real object of interest. In other words, it appears that theoretically, there is no way to talk JUST about a black hole, or a neutron star, etc. It is always embedded in a completely specified 4-D spacetime (yes, redundant adjective, but I am emphasizing that time is not the independent variable here).
OK...I hope that was generally correct, because it pertains to my actual question. Since GM specifies an entire spacetime in an invariant way, is there a sense in which an entire spacetime is isotropic and homogenous even though different reference frames within the spacetime may see otherwise? Its hard for me to describe without the proper theory, but I am thinking of an abstract sort of homogeneity/isotropy in the tensor equations, where there is no "directionality" or "hereness" in the equations (not in a coordinate sense). In other words, I'm thinking more along the lines of abstract algebra, less differential geometry, if that makes sense to you theoretical types who actually can do this stuff (I'm merely a consulting engineer/applied math troglodyte).
To state this a bit more concretely, I offer this related question: Doesn't fact that we can describe our universe using comoving coordinates imply that the universe is fundamentally isotropic/homogenous in the algebraic sense above? I say this because I can imagine it's possible to have space-time specifications in GM where you cannot make corrections from your reference frame to get to an isotropic frame. In which case, every observer would agree that the universe is not isotropic. Therefore, to my amateur mind, it seems like a very special thing that we can make such simple "inertial" corrections, suggesting that there is something fundamentally correct about the isotropic frame, such that the most accurate way to look at our world is from a comoving frame since it reflects the underlying symmetry better than a frame with a peculiar velocity.
Hence, being at rest relative to this comoving frame seems to show you what the universe really looks like and defines what motion counts as "relativistic" vs. merely being a fixed point in the universal expansion. The only reason I think this relativistic vs hubble-flow part is relevant is because Brian Greene, in "The Fabric of the Cosmos" said that all commoving observers would have synchronized clocks, implying that even though they are in motion relative to each other they do not experience time dilation, since they are moving with spacetime, not through spacetime.
Sorry for the long post, but I am trying to convey in simple words what may be more succinctly expressed using theory. If my reasoning above appears correct, then why do we act as if all frames are epistemologically/experientially valid? The laws may work equally well in all frames, but it seems that frames with peculiar motion see a distorted now and where due to their motion in spacetime, somewhat like the distortion of a sound due to motion through a transmission medium.
Thanks again for any observations or thoughts and corrections to my thinking. If I'm not obviously wrong, then I don't know if there is a strict answer to resolving this...I just want to know what more informed minds think of this issue/confusion.