General Relativity and the universe

[Nereid]

In his post in the thread "Mass Exodus From Big Bang Begins" (in the ATM section of BAUT - here), ngeo writes "does GR contain a mathematical description for ‘empty’ space, like a kind of ‘inert field’, or is it that space-like characteristics arise out of the mathematics, or do these characteristics exist only as spatial measurements? If the addition of matter/mass-energy to ‘empty’ space leads to expansion, what is it that expands?".

I've seen good questions about GR, and what sort of universe follows from applying the equations, several times.

While there are good resources on the internet, at many different levels, which explain these sorts of things (and more), I don't think there's a single place in BAUT where our members can go to ask questions about this, at the level they're comfortable with, and get helpful answers.

This, then, is the purpose of this thread.

In particular, let's take a look at Einstein's GR equation (G = T), and apply it to the universe.

Let's also consider what GR does NOT get into (e.g. hadrons and leptons; galaxies and superclusters) ... i.e. what extra 'bits' (from other parts of physics) you need to make a 'real universe' (the kind we can see around us, with our eyes).

[Fortunate]

OK. I'll start. Please explain the nature of the terms in the equation G=T. I believe that it equates two symmetric tensors, with G being a metric for a four-dimensional space-time and T being the so-called "stress-energy tensor." Is this understanding correct? Why are the two sides equal to each other? Thank you.

[Tensor]

OK. I'll start. Please explain the nature of the terms in the equation G=T. I believe that it equates two symmetric tensors, with G being a metric for a four-dimensional space-time and T being the so-called "stress-energy tensor." Is this understanding correct?

Close, but no cigar. G=T is an extreme simplification, there are other terms (and indexes on the two tensor terms). G is the Einstein Tensor, T is the stress-energy tensor. The Einstein tensor includes the Ricci curvature tensor (and a few other terms). A metric is basically a solution of the equation. It describes the shape of spacetime. What you have to remember is that G=T is a symbolic form of the equation and cannot be directly used to solve for anything.
Try here for a more in depth explanation.

Why are the two sides equal to each other? Thank you.

Simply? The right side of the equation show how much energy is available, which directly relates to how much curvature is on the left side. Of course, if you know how much curvature is on the left side, you can find out how much energy is on the right. Hope this helped.

[Tensor]

In his post in the thread "Mass Exodus From Big Bang Begins" (in the ATM section of BAUT - here), ngeo writes "does GR contain a mathematical description for ‘empty’ space, like a kind of ‘inert field’, or is it that space-like characteristics arise out of the mathematics, or do these characteristics exist only as spatial measurements?

Not quite sure what he's looking for here. GR treats spacetime as a Reimann manifold, so I guess you could say it's in the math. Although most relativists will say spacetime is the actual physical component of the mathematical manifold.

If the addition of matter/mass-energy to ‘empty’ space leads to expansion, what is it that expands?

In terms of the entire universe, the addition of stress-energy to the manifold causes either an expansion or contraction. The static solution resembles a pencil balanced on it's point. As soon as something changes, the manifold either contracts or expands. Within the math, it's the manifold that expands (as currently viewed). So, since spacetime is the physical componet of the manifold, it is spacetime that is expanding.

I've seen good questions about GR, and what sort of universe follows from applying the equations, several times.

While there are good resources on the internet, at many different levels, which explain these sorts of things (and more), I don't think there's a single place in BAUT where our members can go to ask questions about this, at the level they're comfortable with, and get helpful answers.

This, then, is the purpose of this thread.

In particular, let's take a look at Einstein's GR equation (G = T), and apply it to the universe.

Not sure what you are looking for here Nereid, but here's an application of the Friedman equation. It doesn't start with the EFE, but the discussion starts with if EFE is correct, the first equation is correct. I'll see if I can scare up a page that shows how to get to the Friedman equation relates to the EFE.

EDIT Ok, I found this rather quickly.

Let's also consider what GR does NOT get into (e.g. hadrons and leptons; galaxies and superclusters) ... i.e. what extra 'bits' (from other parts of physics) you need to make a 'real universe' (the kind we can see around us, with our eyes).

Well, I don't think I have the math for this part of the question as I haven't studied QM, much less superstring theory, in mathematical depth. I think I could get concepts across. Although, since there are practicing particle phycisists on the board, they would be more qualified than I to describe these things.

[Fortunate]

Although most relativists will say spacetime is the actual physical component of the mathematical manifold.

What do you mean?

As soon as something changes, the manifold either contracts or expands. Within the math, it's the manifold that expands (as currently viewed).

Since the manifold already contains time as a dimension, how could it expand over the course of time? Maybe it is "successive" three-dimensional cross-sections that "expand." I don't pretend to understand. I'm asking. Thank you.

edit: 1. Thank you, Tensor, for the article linked in your your last post.
2. I'm still not clear what it is that is expanding. The putative 3-dimensional cross-sections mentioned above wouldn't even be well-defined locally. Cancel that idea. But, maybe, as we proceed along wordlines, the metric tends to change so that, in general, things are further away. Am I babbling yet? OK, maybe you could talk about lightlike cross-sections that would be well-defined. Then it would make sense to say that one of them represents a later time than another. One representing a later time would, according to the metric, be larger.

[Nereid]

Thanks Tensor.

Not quite sure what he's looking for here. GR treats spacetime as a Reimann manifold, so I guess you could say it's in the math. Although most relativists will say spacetime is the actual physical component of the mathematical manifold.

Let's see what ngeo has to say ... I was merely introducing the topic (and letting any other folk who might be interested know - they can join as well, of course).

In terms of the entire universe, the addition of stress-energy to the manifold causes either an expansion or contraction. The static solution resembles a pencil balanced on it's point. As soon as something changes, the manifold either contracts or expands. Within the math, it's the manifold that expands (as currently viewed). So, since spacetime is the physical componet of the manifold, it is spacetime that is expanding.

I suspect that this is super-mysterious to a lot of people - is this just the way it is, from the math? And what is this 'stress-energy'?

Not sure what you are looking for here Nereid, but here's an application of the Friedman equation. It doesn't start with the EFE, but the discussion starts with if EFE is correct, the first equation is correct. I'll see if I can scare up a page that shows how to get to the Friedman equation relates to the EFE.
EDIT Ok, I found this rather quickly.

Let's wait for ngeo.

Well, I don't think I have the math for this part of the question as I haven't studied QM, much less superstring theory, in mathematical depth. I think I could get concepts across. Although, since there are practicing particle phycisists on the board, they would be more qualified than I to describe these things.

I suspect the first step is a very simple one - what, if anything, does GR itself have to say about the nature of 'particles' or 'bodies'? Or, putting this another way, is GR itself completely 'blind' to the form that the things in the equations can take (mass, energy, ...)?

Not quite, because there's a distinction between massless entities (which must travel at c; but they don't have to be photons) and the rest (which cannot travel at c); and there's a special type of 'thing' - the black hole (whose mass can be anything, except zero!).

In other words, a purely GR universe is, potentially, a place with far more 'possibilities' than the real universe, if only because the interactions between real particles are constrained by the physics of electromagnetism, the weak force, the strong force, and so on.

[ngeo]

Thank you again Nereid. Tensor answered one question, in that "since spacetime is the physical component of the manifold, it is spacetime that is expanding."

This seems to mean that 'spacetime' has a physical existence, so would that be a kind of potential?

And (basing this question on complete ignorance of the maths after looking at the links) does the rate of expansion depend on the amount of mass?

[ngeo]

Re 'potential' - it seems that something underlies the 'reaction' of spacetime to mass. Is this in the 'time' component?

Also I wonder whether a 'potential' could be found in the way a volume expands. My elementary (elementary school?) understanding of the formula for the volume of a sphere (a 2-sphere, i.e. a common 'ball' shape?) leads me to wonder whether an expanding sphere (a 3-sphere?) ends up with potential, in that if you give the sphere the ability to expand by itself at a constant rate at its outer edge, you end up with less volume than you expect per expansion period.

So I guess I am wondering whether expanding spacetime could be described in such a way that the spacetime operates on itself to produce an effect of some kind, without having a mass involved.

[Tim Thompson]

Easier I think, to write the equations like this, as ASCII allows, see a real picture on the Wikipedia page:

R(ab) - (1/2)Rg(ab) + Lg(ab) = 8*pi*T(ab)

Now, R(ab) is the Ricci tensor, and describes the curvature of space time. R is a Ricci scalar, which also describes curvature. g(ab) is a metric tensor, which describes the geometry of space time. L is the cosmological constant, inserted later by Einstein to prevent the universe from collapsing, since it was thought to be static at the time. This takes care of everything on the left side of the equation. Notice that it's all geometry & curvature, nothing else.

On the right side, we have only T(ab), the stress-energy tensor (which is also called the energy-momentum tensor). Everything having to do with mass, matter, or energy in any form goes here, inside T(ab).

The expansion of the universe can be more easily seen here as generally a result of stuff that happens on the left side, where matter & energy are irrelevant. That's where the cosmological constant is. The original expansion of the universe is a property of R(ab) (from which R is derived), or g(ab). This implies that the addition of matter or energy is not responsible. One hypothesis for explaining the accelerated expansion of the universe is to blame it on the cosmological constant, which becomes more effective, as galaxy clusters get farther apart.

However, since mater & energy & mass are all by themselves on the right side, they too can be responsible for an accelerating expansion (though I suspect not for the original expansion, when there was no matter). Quintessence is an idea based on a scalar field embedded inside T(ab). The cosmological constant is pretty well obliged to be "constant". However, any scalar field in T(ab) need not be constant, and so can create a far mode complicated expansion history. An accelerating universe could, conceivably come to a screeching halt, turn around, and contract, because of the form of the "quintessence field".

In any case, the "mathematical description of empty space" is the left side of the equation

[Tensor]

Easier I think, to write the equations like this, as ASCII allows, see a real picture on the Wikipedia page:

R(ab) - (1/2)Rg(ab) + Lg(ab) = 8*pi*T(ab)

snip....

In any case, the "mathematical description of empty space" is the left side of the equation

Thanks Tim.

[Fortunate]

Thank you, Nereid, Tim, and Tensor .

[ngeo]

Hoping that I understand Tim Thompson’s post correctly, following are some questions:

- What is the difference between the tensors describing ‘curvature’ and ‘geometry’ of spacetime. Would ‘geometry’ be a set of parameters within which the particular ‘curvature’ takes place? So is this an equation that can be used to describe a local area as well as to describe the entire universe? And do the two tensors and the scalar depend on each other? (Keeping in mind that after reading ‘tensor’ in wikipedia I am unclear just what it is.)

- If ‘spacetime’ incorporates 3 spatial measurements and 1 time measurement, does ‘curvature’ refer exclusively to the time measurement or can it refer to spatial or time measurements, interchangeably or together? In other words, can ‘curvature’ mean curvature in spatial measurements or in time measurement or in both? Apart from that, the possibility of expansion, contraction, or constancy implies an acceleration. Is the ‘original’ empty spacetime expansion an acceleration or a constant expansion? Is there any scenario taking both sides of the equation into account in which a spacetime expanding at a constant rate could be regarded as ‘flat’?

- If the universe is expanding as it currently appears to be (whether accelerated or not), would it be necessary to insert a cosmological constant/vacuum energy term into the equation if it were being written now, or could it be written to include the apparent expansion without vacuum energy?

- Does the left side of the equation (minus cosmological constant/vacuum energy) expand or is it, like the pencil stood on its point, equally likely to contract?

- If local spacetime curvature is an effect of an accelerated frame of reference due to a gravitational field or to other cause, and if the effect of spacetime curvature is in turn to create a variation in spatial or time measurement, and since all observers are in accelerated frames of various strengths, then it seems no two observers can make identical spatial or time measurements. Does this not also present a problem of measuring any object or event? For example a measurement taken to indicate a distant object in a frame of a certain strength could be misinterpreted as a closer or more distant object in a stronger or weaker frame. Is that right?

- Are ‘mass’, ‘matter’ and ‘energy’ treated differently on the right side, or are they given some kind of equivalence? For example, if all the ‘mass’ or ‘matter’ of the universe were treated as ‘energy’, or vice versa, presumably that would change the left side. Is that right? What would the effect be in those cases?

- Could the equation be used to describe a universe expanding at a constant rate at its event horizon (or is the event horizon already observed to be receding at a constant rate)? Specifically at light speed?

- If the universe is not homogenous or isotropic, can the equation still describe it?

- If the original expansion of ‘empty space’ takes place without vacuum energy or matter, where in R(ab) or g(ab) could the (mathematical) cause of this expansion be found?

- Could the idea behind ‘tensor’ be described more simply than it is in wikipedia?

[TheBlackCat]

Someone on another forum I go to was talking about faster-than-light travel. According to him, takyons (which I understand most likely do not exist) would have the effect of "smoothing" the curvature of space-time. This would have the affect of speeding up travel. This seems a bit strange to me, and is not something I have heard anywhere else. I know takyons have imaginary mass based on the special relativity equations, but I do not have the mathematical background to apply the general relativity equations so there is no way for me to confirm his statements. A google search revealed nothing useful. I was wondering if someone could tell me whether his assertions regarding takyons are valid based on general relativity.

[Fortunate]

ngeo, I don't know as much physics as Tim and Tensor, but I do understand some of the math. Maybe my input can complemement theirs. If I say something that is wrong, I hope someone corrects me.

The quantities in Tim's post with the "ab" written in parentheses are tensors. Tensors transform in a predictable way whenever the local coordinate system is changed. Thus, although the quantities in a tensor are described in terms of local coordinates, the tensors themselves are independent of the choice of coordinate system. Hence, we know, in a sense, not only how they look from a particular point of view, but also how they would look from any other point of view.

The key quantities on the left are the functions g(ab), where a and b are indices each of which runs from 1 to 4 (because spacetime has 4 dimensions). Hence, there are really 16 of them, although some of them are equal to each other. The g(ab) determine the Riemannian (actually, pseudo-Riemannian) metric at each nonsingular point in spacetime. The metric tells how to measure distances and angles at each point. Thus, for instance, it can be used to calculate the length of each path and to determine which paths are geodesics, a geodesic being the shortest path between two points.

The metric determines the curvatures, so the R(ab) and R on the left are not independent of the g(ab) and, in fact, can be calculated from the g(ab). Since L is a constant, the entire left side of the equation can be expressed in terms of the g(ab) and their derivatives, but that would look like a huge, complicated mess.

So far, no physics has been introduced. All the terms on the left are geometric. Depending on the values of the g(ab), they could describe any pseudo-Riemannian manifold, whether or not that manifold had any relation to physics or physical reality.

The left side alone is like a subject without a predicate. It is not constrained. Setting it equal to the right side creates a statement which relates that subject to the universe. The equating of the two sides incorporates the g(ab) into a set of equations, which constrain the g(ab). Now they must conform to the laws of physics. These constraints (Einstein equations) determine the g(ab), which , in turn, determine the curvatures and, hence, the shape of spacetime.

[ngeo]

Thank you Fortunate that helps a lot. There seem to be so many mathematical tools available, it is a wonder how people figure out which tools to use to do what, and how they invent new ones.

[Ken G]

I am not a GR expert, so I cannot give answers with any authority, and I won't try to match the level of rigor of the preceding entries. However, I hear one question in what ngeo is saying that is easier to answer than to use the full GR formalism. I think I'm hearing "what is causing the universe to expand", in hopes that it is contained in the G=T equation. However (and please correct me if I'm wrong), the G=T equation has rather the flavor of Newton's equation for acceleration, i.e., acceleration equals force divided by mass. If you apply that equation to a cannon ball, and ask, why is the cannon ball flying through the air, you see that the answer is not contained in the equation. This is a very important point about the Big Bang, I think, that is not at all widely appreciated. The cannon ball is flying through the air because it was shot from a cannon, not because of Newton's equation. It is just an initial condition, whose subsequent evolution is governed by Newton. You could imagine a very different governing equation, yet cannons could still shoot balls. When applied to the Big Bang, this means that the most important thing about the universe is the expansion, which arises not from GR but from a presently ad hoc initial condition. You could have other theories of gravity, hypothetically, and you should expect that a universe that was originally expanding might still be, given the right conditions. Worse, since we now have to postulate both dark matter and dark energy, both unknown, to make things work out, I would argue that the only element of GR that has anything directly to say about whether or not the Big Bang is an accurate phenomonelogical description, is the fundamental instability in the structure of GR. That is, an unstable force principle should lead to a dynamical result, and the Big Bang is indeed dynamical. Beyond that, it's all details.
In short, GR works because of its independent verification in our own backyard, not because of anything that has to do with cosmology. Since whenever we see a glitch in the joint application of GR and the observed expansion, we merely postulate new physics, I would claim that
GR and the Big Bang are completely independent theories (both on very solid footing, in terms of their usefulness as organizing principles).

[Fortunate]

Riemann was a mathematician who lived in the 1800's. As far as I know, he had no interest in physics. Before his monumental contributions to the field of geometry, we didn't really know how to visualize a curved space from within the space itself. For instance, we visualized a two dimensional sphere by picturing that sphere embedded (sitting within) in a larger three-dimensional space. Curvature, for instance, of the two-dimensional sphere was merely curvature with respect to the larger three-dimensional embedding space.

Riemann showed how to consider the geometry of a manifold "intrinsically," (from within) in terms of a tensor, called the "metric tensor." It was no longer necessary to think in terms of an embedding space. What a tool he had devised. I doubt that he ever, in his wildest imagination, thought that physicists would eventually picture the universe as a four-dimensional spacetime incorporating his metric tensor. It is a good example of progress building upon a foundation of previous contributions.

[Ken G]

Wonderful summary, Fortunate. It is also an example of the amazing tendency for mathematical ideas to precede physical application, a phenomenon which must be more mysterious than all the so-called paranormal phenomena put together. That's probably worth its own thread!

[Halcyon Dayz]

It is also an example of the amazing tendency for mathematical ideas to precede physical application

It generally takes 50 years or so to happen.
Mathematicians usually become celebrated when they're long dead.

[Ken G]

yeah, that is a bummer, especially since they never get to appreciate the physical importance of their own ideas. Then again, they're mathematicians-- maybe they don't care!

[Tensor]

Shouldn't the above two posts be in ATM?

[Fortunate]

So far, I've learned a lot from this thread. We have addressed the meaning of the left side of the equation, but not so much the right side. Stress-energy? Momentum? Is gravitational energy treated differently from other types of energy?

[Nereid]

ChaByu's posts are now in a new Unified Field Theory thread, in Against the Mainstream.

[Ken G]

If I have this right, the 'cause' you are looking for is not in the equations. It is the initial condition of expansion/curvature, call it what you want. The point is, it's very hard to set up a universe so that it works, if you start from the Planck scale. The problem is, the GR equations are highly unstable, and the Planck time is very short. Thus you expect one of two things to happen, generically, and neither does. Either there is no cosmological constant, in which case if you don't start out with a huge amount of expansion, your universe caves in on itself on the Planck time. If there is such a constant
whose value is of importance at the Planck scale, then your universe inflates spectacularly and never stops inflating, so you never get the chance to form anything of interest, it all freezes out so fast. You really have to set things up in an exquisitely balanced way to get your Planck-length photon to expand into something that looks like our universe. No current theory can
explain why this happened, other than perhaps the anthropic principle, or intelligent design, neither of which are theories they are just ways of ameliorating the cognitive dissonance of not having a theory.
Note that the details of GR are hardly important in all this, all you need is quantum mechanics and a gravitationally unstable universe, both of which we would appear to have (though really I'm not sure if we know much of anything at the Planck scale).

[ngeo]

That is a very interesting outcome in the case of no cosmological constant Ken G, a collapsing universe with no ‘mass’. I was not including a cosmological constant in my query, I imagined the answer would be that the massless universe would expand into infinity without it. This raises the question of what happens to the massless ‘expanding particle‘, which must have a calculable amount of energy. For example if the energy is now doubled but the expansion rate remains the same, how would that affect the equation? Also I wonder what would happen if the actual Planck dimensions were replaced with the number ‘1’. Could the relevant equation(s) be posted?

[Tensor]

Could the relevant equation(s) be posted?

The Friedman equation was posted here. It's the link in the phrase ..."here's an application of the Friedman equation"....

[Ken G]

I imagined the answer would be that the massless universe would expand into infinity without it.

I suspect this is because you are imagining that the expansion is somehow caused by the energy (or pressure), which is a common misconception. Mass-energy (even the energy of a photon) wants to contract space, and space is a very unstable place, so the universe must be *dynamical*. This is the real message of GR, in regard to the Big Bang. The rest is details. You need an initial expansion, or a cosmological constant, to get an expanding universe, and either way it has to be excruciatingly finely tuned. This is the big mystery, not whether or not the Big Bang occurred or whether GR is correct.

[ngeo]

Ken G you said: “Mass-energy (even the energy of a photon) wants to contract space”, so that even a hugely energetic region (of whatever dimension?) without gravitational mass, would contract without a prior or independent action (or both), acting to expand space. Is this right? So there appear to be three separate entities involved: the energy represented by the photon; the ‘expansive’ energy represented by the prior/independent action ; and space - which seems to take on a physical existence since it is acted upon both by mass-energy and by the expansive energy.

You also said the expansive energy “has to be excruciatingly finely tuned”, and that this is the big mystery (is this the ‘flatness‘ problem?). It seems so unlikely that I sense there is an explanation for it other than prior/independent action. Perhaps this would be an ‘initial condition’ that is not so much a condition as a force that is both ‘initial’ (as in existing at the earliest measurable time) and constant, acting to stabilize what is currently viewed as ‘unstable’ space. So I come back to the idea of ’space’ itself, as a physical entity, being the source of the expansive energy. An energetic field, acting not only to expand but also within the existing field, would create mass by its own action in a self-regulating process. There would be no need for ‘fine tuning’ because mass would arise naturally.

It appears that would require a different set of equations than those found in the posts above, so I will leave it there. I am sorry that after making an initial effort, I cannot follow to any great depth either the tips or the vast icebergs identified by Tensor and Tim Thompson - the field is just too deep for me - although I wonder whether the gravitational constant G is the source of a contracting universe, even in the absence of gravitational mass, in the Friedman equation. Also (and related to the above ‘initial condition‘) it troubles me that rho appears to be based on a finite ‘amount’ of mass-energy. Is this the case (say, in electron-volts)? I have read on the NASA website that ‘all the mass-energy of the universe was released within a year’ (or some similar terms) of the Big Bang. This seems absurd on its face, although I hesitate to question the judgement of many people with much more reasoning capacity than me.

Which leads me to a related question, which is being discussed in another thread (dealing with gravitational redshift). I have trouble staying logged on when I try to post a reply so I will include it here. In that thread Tensor says: “Except for our local group, almost all of the redshift is due to expansion (the closer the galaxy, the less the cosmological redshift).”

This leads me to wonder how much expansion there really is, and whether it really leads to the extreme recessional velocities attributed to distant sources when ‘summed’ over the intervening distances from Earth. What part of the redshift (if any) of sources in the ‘local group’ is attributed to expansion? If less of the redshift of closer galaxies is attributed to expansion and (I presume) more to gravitation than more distant galaxies, and the light from more distant sources can be affected by intervening galaxies whose own redshift has a larger gravitational component, is it not possible that the redshift of more distant sources that is now attributed to expansion could instead be incrementally added to the gravitational redshift of the nearer sources? So that an incremental ’gravitational redshift’ could replace an incremental ‘expansion redshift’? Then the expansion would not necessarily be so extreme over extreme distances and times. I relate this to the discrepancy in rotational rates (attributed to ‘dark matter’) which gives entire galaxies (not just the regions near the galactic center) the appearance of rigid bodies. This, by some kind of non-rational osmosis I guess, tells me that galaxies - and clusters - are isolated regions of ‘flat‘ space - neither expanding nor contracting - surrounded by voids that appear to be expanding. If the local void expansion rate is inferred from redshift, when that redshift is actually caused by galactic or cluster gravitation, the local void expansion rate may be proportionately less than supposed. Would that affect not only the calculation of the local galactic or cluster spacetime curvature but also the calculation of the size of the universe and of rho?

[Ken G]

You have made a lot of speculations that I am not competent to comment on (especially since some of it amounts to as-yet unknown physics!), but I'll pick out some things I think I do know something about:

Ken G you said: “Mass-energy (even the energy of a photon) wants to contract space”, so that even a hugely energetic region (of whatever dimension?) without gravitational mass, would contract without a prior or independent action (or both), acting to expand space. Is this right?

As I understand the Friedman equation, yes. I am not a GR expert. But you mention "gravitational mass"-- all energy has gravitational mass. Photons have no "rest mass", but they have gravitational mass. Mass and energy are essentially the same thing.

So there appear to be three separate entities involved: the energy represented by the photon; the ‘expansive’ energy represented by the prior/independent action ; and space - which seems to take on a physical existence since it is acted upon both by mass-energy and by the expansive energy.

Yes, that sounds like the three terms in the Friedman equation (and a fourth term, the cosmological constant, can perhaps be subsumed into what you are calling space). Space itself is a dynamical player, not just the stuff in space. That's the main moral of GR.

You also said the expansive energy “has to be excruciatingly finely tuned”, and that this is the big mystery (is this the ‘flatness‘ problem?).

No, it's worse than the flatness problem. The fine tuning comes in when you start at the Planck scale with a universe whose only natural timescale is something like 10^-42 seconds and end up with one that's been doing interesting things for billions of years, or 10^17 seconds. This is a little like having a doctor tell you that you have a year to live, and ending up still feeling fine after 10^59 years. The flatness problem is fairly easy, if you accept that there was a process of very rapid expansion called "inflation". The fine tuning comes in because you need to inflate enough to get flat, as is observed, but you don't want to inflate too much or everything will freeze out and you'll have a very empty and uninteresting universe.

What part of the redshift (if any) of sources in the ‘local group’ is attributed to expansion?

Essentially none, since the local group is gravitationally bound to itself and so does not participate in the expansion. But you are right that incremental gravitational redshifts can happen, and can be calculated in GR. As I see it, there are two ways these can appear, one is the Sachs-Wolfe effect, which is due to isolated gravity "wells" like galaxy clusters, but it's very small. The other is the incremental effect of the gravity of the homogenous universe, which is changing as the entire universe climbs out of its own huge well. But this latter effect is viewed, as I understand it, as an equivalent way of describing the expansion redshift.

I relate this to the discrepancy in rotational rates (attributed to ‘dark matter’) which gives entire galaxies (not just the regions near the galactic center) the appearance of rigid bodies.

Actually, they tend to rotate with constant speed, like runners in lanes on a circular track. That's not solid-body.

This, by some kind of non-rational osmosis I guess, tells me that galaxies - and clusters - are isolated regions of ‘flat‘ space - neither expanding nor contracting - surrounded by voids that appear to be expanding.

This is the right picture, except for the business about the space being flat, and the implication that flat space can't expand-- you can still have expansion if only the time axis is curved, as appears to be the actual case on large scales in our universe. Basically, the people who do these calculations have included everything that is in GR, so to alter the result you have to invent new physics or at least include new terms, a la dark matter and dark energy. But it sounds like your intuition is grappling with a lot of the right issues. I think we all get very quickly to the edge of what we can understand without actually doing this stuff for a living!