# Comoving Dynamics

1. Aug 5, 2014

### yogi

Conventional Cosmology would explain greater than c recessional velocities of the nebula as a non-problem regarding STR because the luminous matter is co-moving with respect to local space rather than with respect to local space (the latter generally referred to as peculiar motion).

In an accelerating universe, it is my understanding that the recessional velocity of luminous objects increases both with distance (the usual dv/dr from the velocity distance law) as well as time (dv/dt) and except for the peculiar motions, the recessional velocity of the luminous masses is always equal to the recessional velocity of space. But how is it that material objects can accelerate at the same rate as their local space (i.e, co-move with accelerating space) since these objects will be gravitationally slowed by the mass within their present radius from the earth - and if they are gravitationally slowed, they cannot be at once comoving with space. In other words, how does one rationalize zero net velocity between co-moving matter and an accelerating recessional velocity of space when presumably only material objects would feel the retarding influence of gravity?

2. Aug 6, 2014

### Ich

...cosmology uses GR, in which STR is (generally) only locally valid.
I don't think this is what you intended to say, but I can't reconstruct the meaning.

You have two possibilites:
1) Starting with the assumption of a homogenous space in which all matter/energy rests, you solve the Equations and find this space expanding in a certain way. As matter is per definitionem at rest in this space, it expands along. That is the complete picture, you can't add gravity to it. Gravity is already dealt with.
2) If you want to think in terms of gravitic attraction, that's locally also possible, where you do the calculations on a static background metric. Then you have not only the attraction due to the mass within their present radius from the earth, you have also the repulsion due to the amount of Dark Energy within their present radius from the earth. If the latter dominates, you get accelerated motion.

Both possibilities describe the dynamics exactly. You just can't mix them.

3. Aug 6, 2014

### Staff: Mentor

I think that "co-moving with respect to local space" may not be the correct way of looking at it. To my understanding you cannot assign a frame of reference to space, only to other objects. The resulting expansion and acceleration of distant objects is due to geometry, not due to "space expanding" as it is commonly said. In other words, you can't say what space is doing, only what objects within space are doing relative to one another. Someone correct me if I'm wrong.

4. Aug 6, 2014

### yogi

Response to Post #2: While density is presumably uniform on the large scale, it is definitely not so locally. Since the stars and other objects are not changing size due to global expansion - but spatial expansion is isotropic, there should be local pressure gradients and an overall gravitational retarding gradient that acts upon non-expanding matter. Dark energy, if its exists, would have to be precisely fine tuned to maintain the recessional velocity of lumpy matter equal to the recessional velocity of massless space.

By the way - in your last post in our previous conversation - could you supply the calculations (In simple Newtonian formalism please) that lead to zero energy expansion of a volume in hyperbolic space). I am reading your statement as inconsistent with the dark energy requirement of the LCDM model.

Thanks

5. Aug 6, 2014

### Ich

And because it isn't fine-tuned you see what is called structure formation, where overdense regions virtually decouple from the general expansion.

Right. In cosmology, space is defined by a congruence of "canonical observers", that is a bundle of worldlines to which "space" is orthogonal. The worldlines are supposed to represent particles that are at rest with space. So what "space" does and what these imagined particles do is one and the same.

6. Sep 7, 2014

### yogi

There still appears to be something missing in arriving at the summary dismissal. General Relativity specifies a conditioning of spacetime that causes masses to follow geodesics. The acceleration in the case of an orbiting mass does not involve a change in the magnitude of the velocity, but only a change in the direction of the velocity. No energy is added or subtracted - the same is true for a falling object - the total of the PE and KE is unchanged. In the case of expanding space, the nebula are presumably being accelerated - this means extra energy from somewhere (e.g. dark energy). I do not see how this can be accounted for by the General Theory.

7. Sep 7, 2014

### Staff: Mentor

More precisely, it causes *freely moving* masses--masses that feel zero acceleration--to follow geodesics.

No, the acceleration of an orbiting mass, like the acceleration of any freely moving mass, is zero. (There is a concept of "acceleration" that assigns a nonzero acceleration to an orbiting mass, but it's the wrong concept and you shouldn't be using it; it's a Newtonian concept of acceleration, not a relativistic one. The more precise way of distinguishing the two concepts is to call the correct concept, the one where acceleration means non-geodesic motion, "proper acceleration", and the incorrect concept, the Newtonian one, "coordinate acceleration". The key difference is that proper acceleration is an invariant, i.e., it is not frame-dependent; coordinate acceleration is frame-dependent. In relativity, physics can always be described using only invariants.)

No, they aren't; they are freely moving, just like the orbiting mass.

Dark energy changes the geometry of spacetime, compared to a spacetime without dark energy but with all other matter-energy content the same. The change in the spacetime geometry is what changes the paths that freely moving masses follow. These paths look "accelerated" relative to a particular coordinate system, but this is coordinate acceleration, not proper acceleration. As I noted above, the proper acceleration of the nebulae is zero, so nothing is expending energy to "push" on them; they are just following the geodesics of spacetime.

8. Sep 7, 2014

### yogi

Thank you for the comments Peter. Yes, I have switched back and forth between acceleration and the GR concept of geodesics. So given some ambiguity in the description, I do not understand your rationale for:

Please clarify what part of General Relativity applies to radial expansion (of space) and of co-moving matter.

If possible, can you start with Einstein's original formulation - a static universe where empty space gets curved by inert matter. Then explain how this translates to flat space on the large scale, does this not imply the world lines of the galaxies are perpendicular to a flat space at the distance of Hubble sphere. And if they are perpendicular to a flat space, are they not properly described by Newtonian acceleration?

I will separately comment on your paragraph re dark energy

Thanks, Yogi

9. Sep 7, 2014

### Staff: Mentor

I think you may be confusing General Relativity, as a theory, with particular solutions within GR. See below.

This is one particular solution in GR: the Einstein static universe, which has a nonzero cosmological constant that creates gravitational repulsion, which exactly cancels the gravitational attraction of the matter. This solution is valid, mathematically, but it is unstable, like a pencil balanced on its point; any small perturbation will cause the universe to either collapse or expand.

Also, of course, galaxies in this universe are at rest relative to each other, on average, which they are not in our actual universe; so this solution does not describe our actual universe.

It doesn't. To describe our actual universe, we use a different solution, one in which the universe is expanding--more precisely, in which galaxies, on average, are receding from each other. There are actually many solutions of this type, which differ in things like the exact rate of recession of galaxies relative to one another, and how it changes with time; the solutions also differ in the curvature of space--more precisely, in the spatial curvature of surfaces of constant cosmological time. We do not know exactly which one of these solutions describes our universe, but we can pin down the set of solutions fairly precisely in terms of the key parameters.

No. First of all, we don't know for sure that the spatial curvature of surfaces of constant cosmological time is zero--i.e., that they are flat. (Note that in any spacetime, you can choose different slicings into space and time, which can lead to different spatial curvatures; space curvature, unlike spacetime curvature, is frame-dependent. The spatial curvature I'll talk about here is the spatial curvature in the standard coordinates used in cosmology.) That is our best current estimate, but the "error bars" are still large enough to leave open the possibility of the surfaces being urved.

Second, while the worldlines of galaxies are orthogonal to surfaces of constant cosmological time, assuming those surfaces are indeed flat, they are *not* "flat spaces at the distance of the Hubble sphere". They are flat spaces that cover the entire universe--more precisely, each such surface is the entire universe at an instant of cosmological time.

No; the opposite. The worldlines being orthogonal to surfaces of constant cosmological time means each worldline has constant spatial coordinates; i.e., that each worldline is at rest in cosmological coordinates, meaning it has zero coordinate (i.e., Newtonian) acceleration.

As I said before, these worldlines also have zero proper acceleration, so in these particular coordinates, coordinate acceleration happens to exactly match proper acceleration. But that's a happy feature of those particular coordinates.

10. Sep 7, 2014

### yogi

Ok - if your are saying that with reference to the expanding coordinate system, all the nebula are fixed, I understand. In my original post, I was attempting to draw a distinction between the rate of expansion of the galaxies and the rate of expansion of the coordinate system which is linked to space based expansion. To analogize, think of a coordinate system expanding at uniform velocity c at the Hubble length and an additional acceleration induced by some dark agent distributed throughout the cosmos that causes the galaxies to accelerate away from each other at a greater rate than the expansion of the coordinate system linked to ordinary c expansion of space at the Hubble distance. I suppose you would say that by virtue of GR one would have to embrace a new coordinate system that takes into account the dark energy - and based upon this fixture, the galaxies are still fixed since it is necessary to add the new rate of expansion which creates an accelerated expansion system and so on. But what if the mechanisms are different - for example the peculiar motions of the galaxies evolve from different sources, we do not attempt to claim these are not legitimate accelerations wrt space.

This brings me back to my OP. Is it not possible that the expansion velocity of the nebula are different than the expansion velocity of coordinate system linked to expanding space? And if they are one in the same how is it that the massive objects are not slowed by gravity. The peculiar motions that involve accelerations are due to local momentum change. Perhaps dark energy does not act upon space to accelerate the expansion of the coordinate system, but only upon mass.

11. Sep 8, 2014

### Staff: Mentor

No, it isn't. The "rate of expansion of the coordinate system" depends on the coordinate system, and in and of itself it doesn't correspond to anything physical, certainly not "space based expansion". There are coordinates that make Minkowski spacetime (the flat spacetime of SR) look like "space is expanding". "Space" is not something that can expand, at least not in any invariant sense; in relativity "space" is coordinate-dependent, and only spacetime is a coordinate-free geometric object. The only physical "expansion" that is going on in the universe is the expansion of the galaxies.

GR does not say you have to choose any particular coordinates. A particular system of coordinates may be more convenient for a certain problem, but that's just convenience, not physics.

There is no such thing as "legitimate accelerations wrt space". Coordinate acceleration is not something physical; it's just an artifact of a particular choice of coordinates.

You can certainly make galaxies move, on average, in your chosen coordinates, if you choose coordinates that way; this would mean the expansion of the galaxies would be different from the change in the scale factor of the coordinates. In cosmology, we normally choose coordinates in which, on average, the galaxies are at rest, so the observed expansion of the galaxies matches up exactly with the change in the scale factor of the coordinates. But that's just a convenient choice of coordinates; it doesn't mean that "space is expanding".

They are. Up until a few billion years ago (according to our best current estimate), the rate at which the galaxies, on average, were receding from each other was slowing down, because of their mutual gravitational attraction. A few billion years ago, the universe shifted from being matter dominated (which means the biggest effect on the dynamics is the mutual gravitational attraction of the matter) to being dark energy dominated (which means the biggest effect on the dynamics is the gravitational repulsion produced by dark energy). When that happened, the rate at which the galaxies, on average, were receding from each other started speeding up, because the effect of dark energy was greater than the effect of their mutual gravity. (This is popularly referred to as the expansion of the universe "accelerating", but that's a very bad term, IMO, because it invites the sort of confusion you seem to have fallen into.) But the mutual gravity of all the galaxies is still there; it is just no longer the biggest effect on the overall dynamics.

Also note that, through all of this, two things remain true: (1) all of the galaxies are in free motion, with zero proper acceleration; (2) all of the galaxies, on average, remain "at rest" in the standard cosmological coordinates; all of the dynamics, in these coordinates, is contained in the scale factor, which relates coordinate "distances" to actual, physical distances.

12. Sep 8, 2014

### yogi

I am sorry I do not agree with your limited mathematical view of space and the non functional
interpretations you have insisted upon for expanding space.

Here are a couple of quotes from Ed Harrison:

"In summary we may say that motion in an expanding universe is compounded from recession and peculiar velocities; recessional velocities are due to the expansion of space ...." (p 283). On page 285 in connection with a discussion of Machian theory: "Even the ubiquitous explorer is tempted to believe that empty space is meaningless and the physical properties of space are dependent upon the presence of matter. Then we learn the trick of scattering around a few tiny commoving particles or (what amounts to the same think) of chalking on the sheet a network of commoving coordinates. This illustrates what happened originally with the empty universe proposed by William De Sitter: nobody knew that it was expanding until it was sprinkled with test particles and equipped with a network of commoving coordinates. It was then found to have kinematic properties even though it contained no matter."

13. Sep 8, 2014

### Staff: Mentor

Which book?

He left out the key qualifier: "in the standard cosmological coordinates". The fact that lots of references leave out this qualifier does not mean it's not there.

He forgot to mention that, while de Sitter spacetime does indeed have a space/time slicing in which it is expanding, it also has a slicing in which it is static, i.e., not expanding. So, once again, "expanding space" is not an invariant: it depends on the coordinate choice.

It is true that test particles which are at rest with respect to the expanding space/time slicing are traveling on geodesics, whereas test particles which are at rest with respect to the static space/time slicing are not (they have nonzero proper acceleration). But that's also true in other spacetimes where we do *not* say that "space is expanding": for example, in Schwarzschild spacetime. So you can't use that to define "expansion of space" in a coordinate-independent way either.

14. Sep 8, 2014

### yogi

2000 edition reprint with 2003 corrections

15. Sep 8, 2014

### Staff: Mentor

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