Expanding Universe here?

Naty1, take a look at Newton's law for the gravitational force, while taking into account the effect of a cosmological constant: $$F = {GMm \over r^2} - {\Lambda m c^2 \over 3} r$$ You can see that the cosmological constant reduces that gravitational force between two objects, expanding orbits.

That's because dark energy is a constant repulsive gravitational force (whether it be from a negative pressure, or a constant curvature.).

Take a look at the paper you posted - exponential expansion affects bound objects, however slightly.

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 Quote by Mark M It depends on the nature of dark energy. Firstly, keep in mind that even without dark energy, the orbit of the earth will grow. That's because the Sun and the earth emit gravitational waves over time, and exhibit gravitational recession, a consequence of general relativity. But let's ignore that.
Yes, I agree that it exists and I agree that it should be ignored because it's not what we are discussing, which is the effects of dark energy / the cosmological constant.

 Next, let's assume dark energy has a constant strength, so that it doesn't vary with time. With that in mind, I would have to say two is correct. Let's say we had two objects moving through a region in which dark energy was extremely strong (just a thought experiment). Since the force from DE is constant, the two objects will accelerate away, diverging to infinity. So, we should be able to conclude that two is correct. Naty1, regular expansion does not affect gravitationally bound objects. Dark energy does.
Yes, all of this is now what I believe to be true

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 Quote by Naty1 So it still seems to me we instead say something like 'gravitationally bound systems and things inside them are not thought to expand [or are generally not considered to expand] but we have no exact solution, no good model, for such conditions.
I LIKE that, with a trailer caveat saying "so, they MAY expand, but if they do so, the result is so small as to be totally negligible"

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"In an expanding universe, what doesn't expand?" by Price and Romano,

http://arxiv.org/abs/gr-qc/0508052,

I skimmed the article and the conclusion seems to be :

 We have presented a simple definitive question about the influence of the expansion of the universe on a very particular system: a classical “atom.” ...... atoms are in no danger of being disrupted by cosmological expansion.

 That Price & Romano paper ("What doesn't expand") uses an awfully simple model of an atom, specifically, of the electron's orbit. We know that an electron doesn't have a planetary-type orbit around the nucleus (in fact, it passes through the nucleus). I assume that this still fits the Price & Romano argument, however, because the momentum of the electron increases as r decreases and counteracts any anticentric forces, including expansion. Still, the authors use Newtonian and relativistic analyses but ignore quantum mechanics. According to quantum mechanics, there is a non-zero probability of finding any given electron anywhere in space. So... is it possible that expansion gets lucky now and again, capturing an electron that has strayed so far from its atom's nucleus that the electromagnetic force is too weak? Is it possible, in fact, that expansion is what is responsible for the inconsistency of electron orbits in the first place? Or is that just stringy chaos? Incidentally, Price & Romano cite Bonnor (1999, Class. Quantum Grav. 16 1313) and claim that their analysis is consistent with his. But Bonnor additionally considered an Einstein-de Sitter model and concluded that under that system "the atom expands, but at a rate which is negligible compared with the general cosmic expansion."

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Mark M: Is that equation in your post within the quoted article? I am not familiar with that equation.....is that available in say Wikipedia?.... I have no idea about the assumptions from which it is built.

 Take a look at the paper you posted - exponential expansion affects bound objects, however slightly.
Perhaps, but that is not how I read the article.

from the conclusions:
 ....And we have found a simple definitive answer: Expansion forces increase with increasing atomic radius, while atomic forces decrease. This amounts to an instability with respect to the disruption of an atom. If the atomic accelerations are initially stronger than the cosmological, then the subsequent expansion will become less and less important. The atom will not “partially” take part in the expansion. If, on the other hand, the cosmological effect is initially stronger, the atomic radius will increase and the atomic forces will become less and less important. The atom will fully take part in the expansion.......atoms are in no danger of being disrupted by cosmological expansion
Even if my interpretation is accurate, I would not necessarily take this as definitive, either, as

" We will put this classical atom in a homogeneous universe
in which expansion is described by an expansion factor a(t), where t is time..."
How realistic IS that? I do not know. And the author points to a different paper [#6] for cosmological expansion effects on galaxy clusters...I don't know what that one sez.

Naty1,

I've seen the equation from a post by Chalnoth, see post #27 here:

The reason behind the equation is simple - the cosmological constant has an effect that is opposite to gravity. It accelerates objects away from each other.

I was citing the caption under Figure 2 in the paper you posted, namely the line saying:

 Due to the exponential increase in a(t), the physical radius grows without bound.
Also, I never said that expansion affects bound systems. I said that accelerated expansion does, the cosmological constant. It's because it takes a constant value everywhere. So, it has a small effect on all systems, however negligible.

 See this article by John Baez about how normal metric expansion affects objects within a gravitationally bound system: http://math.ucr.edu/home/baez/physic..._universe.html Obviously, as he explains, objects in bound systems are NOT affected by metric expansion. Dark energy is what I've been speaking about - since it's a uniform negative pressure (or a constant negative curvature), it affects everything. However small these effects are (small enough that they won't even affect atoms), they can increase orbits, by an extremely small margin.

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 Quote by Naty1 Mark M: Is that equation in your post within the quoted article? I am not familiar with that equation.....is that available in say Wikipedia?.... I have no idea about the assumptions from which it is built.
Skimpy explanation:
 Quote by George Jones The weak-field limit of Einstein's equation without cosmological constant/dark energy leads to Poisson's equation, $$\nabla^2 \Phi = - \vec{\nabla} \cdot \vec{g}= 4 \pi G \rho,$$ where $\Phi$ is gravitational potential and $\vec{g}= - \vec{\nabla} \Phi$ is the acceleration of a small test mass. The weak-field limit of Einstein's equation with cosmological constant/dark energy $\Lambda$ leads to a modified "Poisson" equation, $$\nabla^2 \Phi = 4 \pi G \rho - \Lambda c^2.$$ For a spherical mass $M$, the divergence theorem applied to the above gives $$\vec{g} = \left(-\frac{GM}{r^2} + \frac{c^2 \Lambda}{3} r \right) \hat{r}.$$ The second term is a "springy" repulsive term for positive $\Lambda$.
 Quote by George Jones Yes. I also posted something similar in http://www.physicsforums.com/showthr...41#post2799641. I don't know of any online sources, but it is given in the books: General Relativity: An Introdution for Physicsists by Hobson, Efstathiou, and Lasenby; Gravitation: Foundations and Frontiers by Padmanabhan.

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 Quote by Mark M Dark energy is what I've been speaking about - since it's a uniform negative pressure (or a constant negative curvature), it affects everything. However small these effects are (small enough that they won't even affect atoms), they can increase orbits, by an extremely small margin.
Here's how I'm interpreting this. Expansion can be seen as a slight reduction in the "force" holding things together. Meaning that the Earth is very very slightly further out than it would be if there were no expansion. The effect of dark energy, or the acceleration of this expansion, results in an increase in the rate of this expansion, further reducing the attractive force between objects.

How's that sound?

 Quote by Drakkith Here's how I'm interpreting this. Expansion can be seen as a slight reduction in the "force" holding things together. Meaning that the Earth is very very slightly further out than it would be if there were no expansion. The effect of dark energy, or the acceleration of this expansion, results in an increase in the rate of this expansion, further reducing the attractive force between objects. How's that sound?
Well, normal expansion has absolutely no effect inside of galaxies. See the website I posted above by John Baez.

Think of the force from dark energy as repulsive gravity. But, the key difference is that it exerts this repulsive gravitational force at every point (or, in the language of a cosmological constant, there is a constant curvature at every point.). So, it continually expands the orbit of the earth at a constant rate.

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 Quote by Mark M Well, normal expansion has absolutely no effect inside of galaxies. See the website I posted above by John Baez. Think of the force from dark energy as repulsive gravity. But, the key difference is that it exerts this repulsive gravitational force at every point (or, in the language of a cosmological constant, there is a constant curvature at every point.). So, it continually expands the orbit of the earth at a constant rate.
Ah I see. The FRW spacetime simply doesn't apply at the local scale. So how does one reconcile the two different spacetimes from the article? If the universe as a whole is expanding, but locally it has zero effect, where's the middle ground? How weak does gravity need to be between two objects for expansion to occur?

 Quote by Drakkith Ah I see. The FRW spacetime simply doesn't apply at the local scale. So how does one reconcile the two different spacetimes from the article? If the universe as a whole is expanding, but locally it has zero effect, where's the middle ground? How weak does gravity need to be between two objects for expansion to occur?
It needs to be extremely weak, and there needs to be a sufficiently large distance in between the objects. Unfortunately, it's a grey line. It's like asking 'When do we start using global coordinates instead of local coordinates?'. There isn't a well defined answer.

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 Quote by Mark M It needs to be extremely weak, and there needs to be a sufficiently large distance in between the objects. Unfortunately, it's a grey line. It's like asking 'When do we start using global coordinates instead of local coordinates?'. There isn't a well defined answer.
Got it.

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MarkM:
 Well, normal expansion has absolutely no effect inside of galaxies. See the website I posted above by John Baez.
from Baez:
 Neither Brooklyn, nor its atoms, nor the solar system, nor even the galaxy, is expanding.
yes, regarding EXPANSION; but Baez makes no statement about whether an atomic orbital might be ever so slightly larger in theory due to the change in force Chalnoth's formula indicates. I do not know if the orbit is changed a smidgen or not. I do not know if Chalnoth's forumla applies.

Chalnoth posts in the other thread regarding his posted formula:

 ...You can see that the cosmological constant reduces the attractive force of gravity by some small amount. For atoms, this would have the effect of making atoms ever so slightly larger than they otherwise would be (the difference really is utterly negligible, however).
yes the formula says that, but what are the underlying assumptions...are they based on
an indealized model?

Drakkith:
 The FRW spacetime simply doesn't apply at the local scale.
This remains my perspective.
I still am of the opinion, based on arxiv references utilized in another thread [can't find THAT discussion] that the cosmological constant arises from the assumption of a homogeneous and isotropic universe and that those assumptions do not apply on lumpy galactic nor atomic scales. I do not understand the realm of applicability of the formulas from Chalnoth and George Jones although I trust their knowledge.

Examples: Do 'weak field limits' mentioned by George apply in the real world? With Price and Romano, "In an expanding universe, what doesn't expand?", I do not know if their assumptions, their simple model, applies in the real world...call be 'skeptical' I would not draw any absolute and far reaching conclusions from their result.

and I do not understand statements like:

MarkM:
 ...regular expansion does not affect gravitationally bound objects. Dark energy does.
I tend to disagree and it seems to conflict with Chalnoth's formula. What's the distinction between the negative pressure of dark energy, if that is what is meant, and the cosmological constant.

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 So how does one reconcile the two different spacetimes from the article? If the universe as a whole is expanding, but locally it has zero effect, where's the middle ground? How weak does gravity need to be between two objects for expansion to occur? It needs to be extremely weak, and there needs to be a sufficiently large distance in between the objects. Unfortunately, it's a grey line. It's like asking 'When do we start using global coordinates instead of local coordinates?'. There isn't a well defined answer.
This is exactly the 'undefined' circumstance that was a conclusion of another thread. The cosmological constant applies to the 'whole' but nobody really knows which 'part'. Where do the assumptions lose their validity...

 Naty1, a negative pressure dark energy is the same thing as a cosmological constant. The difference is that the curvature from the cosmological constant is intrinsic, it's just there. With a dark energy case, the negative pressure creates the curvature. For a cosmological constant $\Lambda$, the corresponding vacuum energy is $$\rho_{vacuum} = \frac {\Lambda c^{2}} {8 \pi G}$$ I'm using them interchangeably. Do you agree that regular metric expansion (as in FRW) does not have an effect within gravitationally bound systems? Dark energy (or the cosmological constant) is fundamentally different from the normal metric expansion. You can consider it to be a force exerted everywhere (Because it's a constant). Because of this, it has a slight effect within galaxies. That's why, as George Jones showed, it's factored in for gravitational interactions, although it is far to weak to have a meaningful effect.

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