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This is related to a previous post. Assuming that the Cosmological Principle is correct, does this imply that the universe possess an empirically 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.

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  • $\begingroup$ I would point out that isotropy and homogeneity are properties of the stuff in the universe, not properties that can be ascribed to spacetime itself. Or actually, that may be misleading so let me put it like this: it is built into GR at a very fundamental level that spacetime itself is isotropic and homogeneous. It's a uniform "nothing" in all reference frames. Any anisotropy or nonhomogeneity comes from the matter and other stuff placed into the spacetime. I'm not sure if that helps clarify anything. $\endgroup$
    – David Z
    Nov 16, 2013 at 6:25
  • $\begingroup$ When You say "GM" do You mean "GR", "CM", or something else? $\endgroup$
    – xuinkrbin.
    Jan 8, 2014 at 17:16
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    $\begingroup$ @xuinkrbin. Its a typo, I mean GR..not General Mechanics or whatever. $\endgroup$
    – user31580
    Jan 8, 2014 at 17:39

3 Answers 3

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I think you're close to hitting on a key point about the currently accepted mainstream cosmological model -- it is in fact, quite special. In the above, I think you seem to be confusing a few things about the general case of a generic spacetime, and a few details of our current best model.

The key feature of our current best model, the Robertson-Walker cosmological model with a $\Lambda$CDM matter distribution is that it DOES, in fact, have a special time coordinate. This arises from the fact that we, in the coordinate system naturally picked out by our galaxy, observe that (given that we zoom out to a sufficiently large distance scale) the universe looks the same in all directions, and that it does not appear that any one point is "special".

We can leverage this fact to construct a cosmological model within the framework of general relativity. The end result that you get is that while general relativity itself does not give us a global inertial frame, this model does give us a special coordinate system -- the one that is at rest relative to the galaxies as they expand, and where time is measured "since the big bang" at any given point. You can, of course, still have any number of other coordinate systems, but if you are taking measurements in these coordinate systems, you will always be able to observe that you are moving relative to the galactic fluid, and you will always be able to come up with a number saying that "the dust at this place has existed X years since the big bang", so you can always convert back to this special system.

But this "niceness" is not a feature of general relativity -- it is merely a feature of our particular cosmological model. And we can only be confident that this model conforms to our universe due to the fact that our direct cosmological observations are consistent with isotropy and homogeneity, and it is, in fact, not valid in the "zoomed in" view, where the geometry is distorted by black holes, and galaxies and everything else that there is.

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    $\begingroup$ @John Rennie: I would share credit if I could. Thanks to both of you for your answers! I've summarized my take-aways as an answer of my own, but I've given credit where credit is due. Jerry, thanks for the thorough answer! $\endgroup$
    – user31580
    Nov 18, 2013 at 14:35
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I think Jerry's answer has covered most of the points. What follows is just a few comments of my own.

The first clarification is that the Einstein equation is local, that is it relates the curvature at a spacetime point to the stress-energy tensor at a spacetime point. However when we look for a solution we are generally interested in simple solutions and these tend to be global. So the Schwarzschild solution for a stationary black hole strictly speaking applies only when the black hole is the only thing in the universe, and also only when the black hole has existed for an infinite time and will exist for an infinite time to come.

So when you say whereas in GR, you must solve for an entire spacetime, this isn't true and there is no must about it. It's just that it's mathematically easier to solve for the entire spacetime and bear in mind that your solution is an approximation to a real black hole. So it's our (approximate) solutions that describe the entire spacetime.

The second point has been made by David in a comment. Einstein's equations relate the curvature to the distribution of matter/energy. Typically we start with the distribution of matter and solve the equations to find the curvature that matches, though you can start with the curvature and look for the corresponding stress-energy tensor (this is how the Alcubiere drive was discovered).

So when deriving the FLRW metric we start by assuming the distribution of matter is isotropic and homogeneous. The homogeneity isn't an outcome of GR - it's a starting point that we put in by hand. What GR tells us is that if we start with an isotropic and homogeneous distribution of matter we get a metric that is simply expressed using comoving coordinates. We could of course start with the metric and ask what distribution of matter would produce it and we'd end up concluding that the matter must be isotropic and homogeneous, but historically this isn't how it happened.

So it's the distribution of matter that's special and given a special distribution of matter it's not surprising the metric that corresponds to it looks a bit special too. But don't get this the wrong way round. It isn't the case that the universe has an FLRW metric and this forces matter to be evenly distributed. Instead it's the other way round. Whatever mechanism was responsible for the Big Bang produced evenly distributed matter and the universe obligingly adjusted it's curvature to match.

Finally, though it's probably obvious, the universe is not isotropic and homogeneous. It only looks that way at very large scales. That means the FLRW metric is an approximation that is good on the very large scale but poor on the small scale - for example the Earth and Moon aren't moving apart due to expansion of spacetime. Interestingly, it seems that the universe isn't as homogenous as we once thought. The record for the largest structures keeps being broken and is now at 10 billion light years, and a 13.7 billion light year universe with 10 billion light year sized objects in it really isn't that homogeneous!

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  • $\begingroup$ +1. Reading your response reminds me of when I read Sean Carrol's blog entry "Energy is NOT conserved", which argued that its more natural to explain physics dynamically as opposed through global conservation laws when dealing with GR. Mind blowing...at least for me. In fact, I was not aware of the large scale inhomogeneity of our universe. Us amateurs are told that while the universe is locally nonhomogenous, the universe if asymptotically homogenous. In other words, there is a natural scale to the perturbations from uniformity. $\endgroup$
    – user31580
    Nov 18, 2013 at 14:42
  • $\begingroup$ (cont'd). What you seem to be suggesting instead, based on those observations, is that far from there being a natural scale to the universe, the universe is scale-invariant! In math terms, the perturbations are heavy-tailed, not gaussian...this is totally new to me, ans substantially alters my views. If such is the case, then cosmic time ceases to be anthing but a convenient, approximate, local calibration of our sense of time. By local, I mean that we can conclude that we are approximately at rest relative to the observable universe $\endgroup$
    – user31580
    Nov 18, 2013 at 14:43
  • $\begingroup$ @Eupraxis1981 I wouldn't go so far as to say the universe isn't homogeneous on large scales. It was the confirmation of such large scales with early high-redshift galaxy surveys that solidified the FRLW model in the first place. It's true that the universe could have been lumpy up to very large scales, and that would have hindered cosmology. But these claims of inhomogeneity and anisotropy are probably more the product of a poor statistical treatment by astronomers who want to make a name for themselves. If you analyze noise enough different ways, you'll eventually see patterns. $\endgroup$
    – user10851
    Nov 18, 2013 at 15:46
  • $\begingroup$ @Eupraxis1981: yes, I agree with Chris, and it's probably a bit mischievous of me to be suggesting otherwise. Whether the 10 billion light year structure really exists remains to be seen. $\endgroup$
    – John Rennie
    Nov 18, 2013 at 16:01
  • $\begingroup$ @JohnRennie: Ok, in that case, we have a strong case for an absolute ordering of events in the universer, per my analogy to "finding rabbits in the precambrian"...particular motion can distort preceived ordering of spacelike connected events, yet we know via isotropy/homogeneity that certain densities and states occurred earlier than others. Seems like cosmic time is something quite fundamental for a universe like ours, where the mass distribution re-establishes an epsitelogically privileged frame. I have edited my own "answer" to reflect this. $\endgroup$
    – user31580
    Nov 18, 2013 at 17:16
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Summary of presented views

I started writing comments to each of you, but I figured it would be easier just to summarize what I've leared from you three in an answer. It's too bad I cannot share credit to each of you, so the answer goes to Jerry for his complete answer, but David and John also gave excellent supplementary answers. Overall, this was a very informative exchange for me...looks like I came to the right place.

First of all, I better understand what GR is about, and it appears to have a very metaphysical (not in the disparaging sense!) flavor to it in that it can be used to describe worlds vastly different from the universe we inhabit. Of course, in such hypotheical worlds it is not clear that GR does in fact hold, but it appears to be accepted to use GR for counter-factual theorizing, since we really don't have anything else. I find this similar to string theory, in that it offers a plethora of answers, not just for our particular world, but for many types of worlds..so GR almost a meta-theory. Therefore, I understand why I've recieved the answers I have in the past when I've brought up this question..I was focusing on GR, at-large, instead of GR constrained by empirically sound boundary conditions (i.e, homogenity, isotropy)...GR w/o boundary conditions is a bit too, well...general.

Therefore, it seems that insofar as the Cosmological principle appears true, given the specific distribution of matter in our universe, then there exists a privileged frame, not in the sense that physical laws take the proper form there, but in the sense that you will place objects in their proper epoch. That is, if an observer with a very pecular velocity calculated that brown dwarfs and free-floating quark-gluon plasma were equidistant from her, based on relativistic corrections, then others would have solids grounds to say that that is nonsesnse, as we know quark-gluon plasma was a precursor to star formation and, vis-a-vis the Cosmological Principle, the two would not occur in the same epoch anywhere in the universe. However, sans the Cosmological Principle, the prior objection to the validity of the observer's conclusion would be groundless -- this is the sense in which I think the Cosmological Principle can establish an epsitemologically privileged frame.

Thanks again for all your help!!

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