On the Subject of gravity in general... split

[elfmotat]

I my understanding it is meant that for example the space between the Earth and the Sun does not expand, that the space between stars in the Milky Way does not expand, that the space between galaxies in a cluster does not expand, BUT that the space beween galaxy clusters does expand.

 

Your understanding is wrong.

[zapatos]

If Michel is wrong, does that mean that the expanding space between the sun and the earth is 'flowing' past us?

[michel123456]

 

Your understanding is wrong.

What else should I understand from Wiki's article?

[elfmotat]

If Michel is wrong, does that mean that the expanding space between the sun and the earth is 'flowing' past us?

 

Sort of. It's kind of like putting a coin on a sheet of rubber, and stretching the sheet. The coin isn't pulled apart by the stretching because the forces holding the coin together are far too strong. But if you put two coins on the sheet, they aren't bound together by any forces, so they will move apart from each other.

What else should I understand from Wiki's article?

 

Sorry, I don't understand your question.

[michel123456]

 

Sorry, I don't understand your question.

1-1

I don't understand your answer.

 

Wiki says:

"Metric expansion is a key feature of Big Bang cosmology, is modeled mathematically with the FLRW metric, and is a generic property of the universe we inhabit. However, the model is valid only on large scales (roughly the scale of galaxy clusters and above). At smaller scales matter has become bound together under the influence of gravitational attraction and such things do not expand at the metric expansion rate as the universe ages. As such, the only galaxies receding from one another as a result of metric expansion are those separated by cosmologically relevant scales larger than the length scales associated with the gravitational collapse that are possible in the age of the Universe given the matter density and average expansion rate."

 

I still interpret from Wiki's statement that space expansion is not making the Earth receding from the Sun because the Earth and the Sun are gravitationaly bound.

Where am I wrong?

[elfmotat]

1-1

I don't understand your answer.

 

Wiki says:

"Metric expansion is a key feature of Big Bang cosmology, is modeled mathematically with the FLRW metric, and is a generic property of the universe we inhabit. However, the model is valid only on large scales (roughly the scale of galaxy clusters and above). At smaller scales matter has become bound together under the influence of gravitational attraction and such things do not expand at the metric expansion rate as the universe ages. As such, the only galaxies receding from one another as a result of metric expansion are those separated by cosmologically relevant scales larger than the length scales associated with the gravitational collapse that are possible in the age of the Universe given the matter density and average expansion rate."

 

I still interpret from Wiki's statement that space expansion is not making the Earth receding from the Sun because the Earth and the Sun are gravitationaly bound.

Where am I wrong?

 

You're not wrong in this post, but you were wrong in your previous post. You said that the space between the Sun and Earth doesn't expand. But it does. Just because the distance between them isn't changing doesn't mean that metric expansion isn't going on. It's going on everywhere.

[Strange]

If Michel is wrong, does that mean that the expanding space between the sun and the earth is 'flowing' past us?

 

That would imply space is a "thing" or "stuff". It isn't. It is just a way of measuring the distance between things. It only expands on cosmological scales because it is only on those scales that the universe is (roughly) homogeneous.

[zapatos]

That seems to be at odds with what elfmotat is saying and in agreement with michel. Is that correct?

 

 

EDIT: Sorry, I edited while you were responding.

Edited September 27, 2014 by zapatos

[Strange]

That seems to be at odds with what elfmotat is saying.

 

It does, doesn't it.

 

This might help:

Expanding Space: the Root of all Evil?

Matthew J. Francis, Luke A. Barnes, J. Berian James, Geraint F. Lewis

http://arxiv.org/abs/0707.0380

[michel123456]

 

You're not wrong in this post, but you were wrong in your previous post. You said that the space between the Sun and Earth doesn't expand. But it does. Just because the distance between them isn't changing doesn't mean that metric expansion isn't going on. It's going on everywhere.

But if the metric expansion is at work between the earth and the Sun, we should observe it through light. Because light is the "thing" that makes us know about expansion.

[elfmotat]

But if the metric expansion is at work between the earth and the Sun, we should observe it through light. Because light is the "thing" that makes us know about expansion.

 

It's too small to be measurable at small scales. We "know" it must be happening because the cosmological constant is nonzero.

[Strange]

 

It's too small to be measurable at small scales.

 

Actually, it isn't that small. I worked it out when this came up on another forum. By a weird coincidence it is about the same as the rate at which the moon is moving away from the Earth due to to tidal forces. So it would be quite easily measurable.

That seems to be at odds with what elfmotat is saying and in agreement with michel. Is that correct?.

 

Elfmotat appears to know more about this than I do, so draw your own conclusions ...

[elfmotat]

 

Actually, it isn't that small. I worked it out when this came up on another forum. By a weird coincidence it is about the same as the rate at which the moon is moving away from the Earth due to to tidal forces. So it would be quite easily measurable.

 

I'm not sure what you mean by this. What exactly did you calculate? Let's assume the Earth and the Moon are the only two bodies in the entire universe. With a linear approximation of GR with non-zero cosmological constant, we can derive the following force law:

 

[math]\frac{\mathbf{F}}{m}=-\frac{GM}{|\mathbf{r}|^3}\mathbf{r}+\frac{c^2\Lambda}{3}\mathbf{r}[/math]

 

(For more information about where this equation came from, see my post here: http://www.scienceforums.net/topic/72550-antigravity-is-the-source-of-dark-energy-accelerating-expansion/?p=728909 .)

 

The value of the cosmological constant is of the order 10^-52 m^-2. Plug that into the second term, along with average Earth-Moon distance, and you get a correction to the gravitational acceleration between them of the order of 10^-27 m/s², which is about four orders of magnitude smaller than the secular pole motion acceleration. Additionally, the recession of the Moon from the Earth was calculated based on centuries worth of data, and it cannot be measured over short time scales.

Edited September 28, 2014 by elfmotat

[Strange]

 

I'm not sure what you mean by this. What exactly did you calculate?

 

A simple approximation using the Hubble constant and the Earth-Moon distance. It may not be accurate...

[elfmotat]

 

A simple approximation using the Hubble constant and the Earth-Moon distance. It may not be accurate...

 

Hubble's Law only works on bodies that are not gravitationally bound to us. So it's not surprising that it spits out strange values when you attempt to apply it where it doesn't belong.

[Strange]

 

Hubble's Law only works on bodies that are not gravitationally bound to us. So it's not surprising that it spits out strange values when you attempt to apply it where it doesn't belong.

 

Fair enough. (Although I was still surprised how large the [incorrect] value was. )

[elfmotat]

 

Fair enough. (Although I was still surprised how large the [incorrect] value was. )

 

The error was so massive because, referring again to that force law I posted, when two massive bodies are near each other the GM/r² term completely dominates cosmological constant term. Hubble's Law is actually the special case where Λc²r/3 >> GM/r² ≈ 0. To see this you can let the GM/r² term in the force law go to zero, and solve the corresponding differential equation:

 

[math]\frac{d^2 r}{dt^2} = \frac{\Lambda c^2}{3} r = H^2 r[/math]

 

where I've defined H² = Λc²/3. The solution to this equation is:

 

[math]r=k e^{Ht}[/math]

 

for some constant k. The velocity is therefore:

 

[math]v= Hk e^{Ht}=Hr[/math].

Edited September 28, 2014 by elfmotat

[hoola]

yes, there seems a bit of a confusion between space expansion and gravitationally bound object orbital expansions...

Edited September 28, 2014 by hoola

[swansont]

Additionally, the recession of the Moon from the Earth was calculated based on centuries worth of data, and it cannot be measured over short time scales.

 

Decade(s). The recession has beens measured with the retroreflectors placed on the moon, mainly those by Apollo and Luna missions, 1969-1973; individual measurements yield a precision of a few cm. Over a decade or so one can get enough statistics to deduce the recession rate.

 

http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment

http://eclipse.gsfc.nasa.gov/SEhelp/ApolloLaser.html

http://ilrs.gsfc.nasa.gov/missions/satellite_missions/current_missions/ap11_general.html

 

Whether a decade is short is left for discussion.

[michel123456]

Quoting from the other thread mentioned by elfmotat: (enhancing)

The Newtonian approximation of gravity with nonzero Cosmological Constant is well-known, and falls directly out of linearized General Relativity:

[math]S=\int \left [\frac{1}{8\pi G}|\nabla \phi|^2+ \left (\rho-\frac{c^2\Lambda}{4\pi G} \right ) \phi \right ]d^4x[/math]

This action yields the field equations for Newtonian gravity:

[math]\nabla^2 \phi +c^2\Lambda =4\pi G\rho[/math]

If you assume that all of the mass in the universe is just a single point particle at the origin of our coordinate system, i.e. [math]\rho =M\delta (\mathbf{r})[/math], then by applying Gauss' Law to the field equations with the point particle density (and assuming the equations of motion for a test particle are [math]\frac{d^2 \mathbf{r}}{dt^2}=-\nabla \phi[/math]) you arrive at the following force law between the gravitating mass [math]M[/math] and a test particle of mass [math]m[/math]:

[math]\frac{\mathbf{F}}{m}=-\frac{GM}{|\mathbf{r}|^3}\mathbf{r}+\frac{c^2\Lambda}{3}\mathbf{r}[/math]

Since our field equations are linear, the superposition principle applies and the force contribution from multiple point particles may be simply added with vector addition. From this, the shell theorem follows, etc., and we see that in a universe filled with complicated arrangements of matter ([math]\rho(\mathbf{r})[/math] can be any complicated function) all matter acts on all other all other matter according to the above force law. So clumps of matter which are close enough together will be attracted to each other, while clumps spread out far enough will actually repel each other.

This means that in the context of Newtonian gravitation, the presence of a positive nonzero Cosmological Constant is interpreted as a repulsive term in the force law. This means that, as far as Newton is concerned, gravity is indeed repulsive once you're far enough away from sources of gravitation (masses). So in this context "Dark Energy" is simply the result of a modification of Newtonian gravity. The observation that far-away galaxies are accelerating away from us is all explained by our force law.



In full-fledged General Relativity, the action is:

[math]S=\int \left [ \frac{c^4}{16\pi G}(R-2\Lambda )+\mathcal{L}_{matter} \right ]\sqrt{-g}d^4x[/math]

This yields the following (nonlinear) field equations:

[math]G_{\mu \nu}+\Lambda g_{\mu \nu }=\frac{8\pi G}{c^4}T_{\mu \nu}[/math]

From these field equations you can get exact solutions for the universe (using assumptions like isotropy and homogeneity) such as the FLRW metric. In this solution the spacial part of the metric (i.e. how spacial distances between points are determined) is dependent on a function of time [math]a(t)[/math] called the "scale factor." Plugging this solution into the field equations yields a few useful equations (called the Fiedmann equations), one of which is:

[math]\frac{\ddot a}{a} = - \frac{4\pi G}{3}\left(\rho + \frac{3p}{c^{2}}\right) + \frac{\Lambda c^{2}}{3}[/math]

where [math]\ddot{a}=\frac{d^2}{dt^2}[a(t)][/math]. Here, we can see that ordinary energy density and pressure are acting to "contract" the distances between points (i.e. trying to make [math]\ddot{a}[/math] negative). On the other hand, the term involving the Cosmological constant is working to increase the distances between points.

We can also see from the Friedmann equations that the Cosmological Constant term acts as some sort of energy with negative pressure (which is dubbed "Dark Energy"). So in the context of GR, the Cosmological Constant is something which works to expand the universe itself.



The point of this post was to show you that your idea that Dark Energy is actually a form of "antigravity" actually does work, to a certain approximation. But what's "really" going on is that the distances between points in the universe is increasing. GR is a much more accurate model of gravity than Newtonian gravitation, especially when things are moving fast compared to light and when gravitational fields are very strong. But the effects of a nonlinear gravity theory are also apparent within our own solar system, for example with the precession of Mercury. Applying Newtonian gravity on the cosmological scale is bound to give lots of inaccurate predictions. Your idea works to develop an intuition about the expansion of space, but it's bound to fail if you try to use it to make predictions about the behavior of the universe.

That comes out also from the present discussion: if I understand clearly it looks just as if gravity became repulsive as a function of distance.
But not as a function of distance squared, because Hubble law is rectilinear.

Edited September 28, 2014 by michel123456

[elfmotat]

 

Decade(s). The recession has beens measured with the retroreflectors placed on the moon, mainly those by Apollo and Luna missions, 1969-1973; individual measurements yield a precision of a few cm. Over a decade or so one can get enough statistics to deduce the recession rate.

 

http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment

http://eclipse.gsfc.nasa.gov/SEhelp/ApolloLaser.html

http://ilrs.gsfc.nasa.gov/missions/satellite_missions/current_missions/ap11_general.html

 

Whether a decade is short is left for discussion.

 

I stand corrected. My point was that we haven't been looking for DE for very long, only about ~15 years, and that the direct effects of DE on our solar system would be extremely tiny.

That comes out also from the present discussion: if I understand clearly it looks just as if gravity became repulsive as a function of distance.

But not as a function of distance squared, because Hubble law is rectilinear.

 

The "reason" is because of the Einstein field equations, not because of Hubble's law. If you take the Einstein field equations with nonzero cosmological constant in the limit that everything moves slowly compared to light and gravitational fields are weak, then you get the modified Poisson equation as the field equation. You might find this post to be helpful: http://www.scienceforums.net/topic/72570-how-can-galaxies-exist-with-the-expansion-of-space/?p=728733

Edited September 28, 2014 by elfmotat

[MigL]

Elfmotat's point remains, that the expansion is modeled on the Cosmological Constant, and is present everywhere. This constant is just one term in an equation, and only becomes the dominant term when 'gravity' is extremely weak, such as areas of nearly flat space-time, i.e. between galaxies/clusters. At shorter distances/densities the 'gravity' term is much larger than the Cosmological Constant, and dominates.

[michel123456]

Elfmotat's point remains, that the expansion is modeled on the Cosmological Constant, and is present everywhere. This constant is just one term in an equation, and only becomes the dominant term when 'gravity' is extremely weak, such as areas of nearly flat space-time, i.e. between galaxies/clusters. At shorter distances/densities the 'gravity' term is much larger than the Cosmological Constant, and dominates.

 

Speculating a bit:

 

It can be seen as follows:

In diagram 1, there is the gravity curve, that diminishes as distance increases, and the red cosmological constant wich is separate from gravity and acts linearly.

 

 

In diagram 2, there is only gravity. The only "thing" is that the axes are tilted. Which means that for extremely large distances gravity becomes negative, and for extremely small distances, gravity never reaches infinity.

 

But I may be wrong of course.

[elfmotat]

 

Speculating a bit:

 

It can be seen as follows:

In diagram 1, there is the gravity curve, that diminishes as distance increases, and the red cosmological constant wich is separate from gravity and acts linearly.

 

diagrams-29-29-14 at 08.09 PM.JPG

 

In diagram 2, there is only gravity. The only "thing" is that the axes are tilted. Which means that for extremely large distances gravity becomes negative, and for extremely small distances, gravity never reaches infinity.

 

But I may be wrong of course.

 

I'm not sure what you mean by this. Rotating a graph doesn't physically change anything.