How does GR slow a homogeneous universe?

[mysearch]

While the title may suggest this thread belongs in cosmology, my questions are orientated towards a better understanding of general relativity. Basically, there seems to be an assumption that after the Big Bang, the initial expansion of the universe was slowed by gravity. Many sources appear to feel that this is so self-evident that no further explanation is usually given other than a possible passing reference to GR. Therefore, I was hoping that somebody might be able to outline the basic GR premise that supports this conclusion. However, I would like to initially try to link the discussion to 2 conceptual models:

• Model-1:
  Assumes a large spherical volume of homogeneous density, radius=R, exists within an infinite and absolute vacuum. This homogeneous volume has an effective mass and a centre of gravity. The gravitational effects are assumed to align to the logic of http://en.wikipedia.org/wiki/Shell_theorem" . As such:
  
  
  • The force on an object (m) at radius=r>R, i.e. outside this volume, is subject to the normal inverse square law [1/r²] based on its distance [r] from the centre of the homogeneous volume.
    
    
  • The force on an object (m) at radius=a<R, i.e. inside this volume, is now linearly proportional to [a], because the enclosed mass of the homogeneous volume itself is proportional to [a³]. The gravitational effect of the mass outside [a] is assumed to cancel out.
  
  
• Model-2:
  Also assumes a homogeneous density, but now its volume conceptually extends to infinity. The logic of Newton’s Shells now appears problematic because it is difficult to resolve whether the object [m] has a near-zero or near-infinite radius as the centre cannot be determined.

However, both these models undergo expansion.

• Model-1:
  Anything outside the homogeneous volume would not be affected by its expansion, i.e. Birkhoff’s theorem (?). Anything inside, co-moving with an expansion, would also see [a] expand, but as the mass contained within [a] doesn’t change, the gravitational force would fall during expansion. While this model may appear homogeneous at [a], it might not be isotropic, although if the expansion rate was greater than the gravitational collapse this might be difficult to measure?
  
  
• Model-2:
  Clearly, this is the one that appears to align to the standard cosmological model, at least, in its original form. While many of the details of this model are speculative, the original model assumed that nothing existed outside the universe; therefore any expansion of model-2 has to take place within the universe with no obvious centre of gravity.

However, the purpose of these models is not to debate the cosmological model, but to better understand how general relativity might modify the interpretations presented.

1. Model-1 seems to have a centre of gravity and, in the context of a weak gravitational field; it is assume that general relativity would converge towards the Newtonian interpretation?
   
   
2. Model-2 doesn’t seem to have a centre of gravity; therefore it appears difficult to quantify the effects of gravity, other than as a local perturbation of the density. So is there a general explanation of how gravity acts on this model, as a whole, and slows its expansion; hopefully in a form that does not require a prerequisite, and in-depth, understanding of GR maths?

Would appreciate any knowledgeable insights. Thanks

 

[John232]

I wouldn't be able to give you the calaculations but there are some problems at the start of the big bang. It had rapid periods of inflation close to the start of the big bang. No one knows what banged. These periods of inflation happened before the formation of matter. I think the big bang would have had to been made of mostly energy in order for it not to be trapped into a singularity. Then the reason why it would burst out several times is unkown. An accurate big bang theory would have to explain why these burst started and stopped for several periods without matter.

I don't think there was an abundance of anti-matter after the big bang since it is unstable. It would prob only exist for micro secounds if it did happen to form.

 

[mysearch]

Appreciate the feedback and agree that there are many speculative ideas surrounding the issue of cosmological expansion. However, within the scope of this forum, I was really trying to focus on the issue of how GR might be said to slow down any notion of expansion based on the 2 gravitational models outlined in post #1. Thanks

 

[pervect]

I'd recommend reading http://math.ucr.edu/home/baez/einstein/ to get some insight.

In the Newtonian model, you've noted that there would be a "center of attraction". In the GR model, you can consider a small spherical ball of particles, "coffee grounds", that don't interact with each other, but mark out some volume and are initially at rest. If you track the volume enclosed by said sphere, it will decrease at an accelerating rate. This acceleration will be proportional to the density of the matter enclosed by the sphere plus 3x its pressure (assuming the pressure is isotropic).

Matter outside the sphere marked out by the coffee grounds will not affect the rate at which the volume changes (or rather, the acceleration of the volume) - though it will affect the shape of the grounds in a manner that doesn't affect the volume.

This is described in more detail in Baez's paper.

So, this is a bit like the Newtonian "shell" picture, but you don't have or need a "center of attraction" in the GR case.

The fact that pressure can be thought of as "causing" gravity is another important difference between GR and the Newtonian theory.

 

[Ich]

Pervect already an excellent explanation, but I'd like to point one thing out: Clearly, the infinite Newtonian model gets into conceptual problems, as you can't determine where all the things are actually falling to. But the math still works: according to the equivalence principle, you can't tell free fall from inertial motion, so there's no need to know how fast and in which direction you are "actually" falling. Instead, you may set the acceleration and velocity of any arbitrary point (the "observer" to zero, and get a well-defined and quite correct cosmology, with velocity and acceleration proportional to distance.

 

[mysearch]

Pervect,
Appreciate the feedback and the link to the John Baez’s site. I will take the time to try to work through some of the material in more detail, but would like to initially clarify some of the points you raised, if possible. Of course, now that this thread has been moved back to cosmology maybe the emphasis on GR will be lost.

You appear to be making some reference to the Friedmann equations of cosmology; therefore, I have included a form of these equations below for cross reference. The derivations of these equations are said to be rooted in GR, but can also be derived based on classical concepts: http://arxiv.org/abs/Astro-ph/0309756" . I have also normalised the equation back into energy-density $$[\rho]$$ via the relationship $$P=\omega \rho c^2$$ and applying the accepted equations of state; as this form allows the positive and negative effects of the various energy-density components to be seen more clearly. However, according to the reference above, the derivation of [1] is linked to the conservation of energy and, in isolation, it is difficult to understand how this explains the recessional velocity implicit in [H], i.e. as an escape velocity rather than free-fall velocity. Equation [2] can also be linked to the conservation of energy via the 1st law of thermodynamics, which also explains where pressure comes into these equations and why the negative sign appears. Broadly [3] seem to be derived by differentiating [1] and substituting in [2].

Does GR radically disagree with this simplistic assessment of the underlying physics?

In the Newtonian model, you've noted that there would be a "center of attraction". In the GR model, you can consider a small spherical ball of particles, "coffee grounds", that don't interact with each other, but mark out some volume and are initially at rest. If you track the volume enclosed by said sphere, it will decrease at an accelerating rate. This acceleration will be proportional to the density of the matter enclosed by the sphere plus 3x its pressure (assuming the pressure is isotropic).

Are you describing a process of expansion or contraction here?

Matter outside the sphere marked out by the coffee grounds will not affect the rate at which the volume changes (or rather, the acceleration of the volume) - though it will affect the shape of the grounds in a manner that doesn't affect the volume.

Presumably the accelerated expansion or contraction has to still depend on what assumptions you are making about the energy-density components?

So, this is a bit like the Newtonian "shell" picture, but you don't have or need a "center of attraction" in the GR case.

Are you saying that GR describes a curvature of space that acts in a way that is equivalent to a centre of gravity, even though in Model-2 no centre is said to exist?

The fact that pressure can be thought of as "causing" gravity is another important difference between GR and the Newtonian theory.

Agreed, but the equation of state also suggest that pressure can be both negative and positive, where the negative pressure assumptions of dark energy still seem to be quite speculative. Anyway, many thanks for the help.

Friedmann References:

[1] $$H^2 = \frac{8}{3} \pi G(\rho_M + \rho_R + \rho_D + \rho_K + \rho_\Lambda )$$

[2] $$\dot{\rho} = -H(3 \rho_M + 4 \rho_R + 3 \rho_D + 2 \rho_K + 0 \rho_\Lambda )$$

[3] $$\frac{\ddot{a} }{a} = -\frac{4}{3} \pi G(1 \rho_M + 2 \rho_R + 1 \rho_D - 0 \rho_K - 2 \rho_\Lambda )$$

 

[pervect]

Pervect,
Appreciate the feedback and the link to the John Baez’s site. I will take the time to try to work through some of the material in more detail, but would like to initially clarify some of the points you raised, if possible. Of course, now that this thread has been moved back to cosmology maybe the emphasis on GR will be lost.

You appear to be making some reference to the Friedmann equations of cosmology; therefore, I have included a form of these equations below for cross reference. The derivations of these equations are said to be rooted in GR, but can also be derived based on classical concepts: "[URL link for details[/I]Does GR radically disagree with this simplistic assessment of the underlying physics?

I think you'll find it difficult to explain why radiation has a coefficient of two (the coefficient of $\rho_{r}$ in equation 3 in your version of the Friedmann equations) while matter has a coefficient of one, unless you include the pressure terms I mentioned (see Baez's paper, Baez does derive the Friedmann equations, though he may not have the same constant factors of proportionality).

Are you describing a process of expansion or contraction here?

This equation says that positive energy density and positive pressure curve spacetime in a way that makes a freely falling ball of point particles tend to shrink. Since E = mc^2 and we are working in units where c = 1, ordinary mass density counts as a form of energy density. Thus a massive object will make a swarm of freely falling particles at rest around it start to shrink. In short: gravity attracts.

Agreed, but the equation of state also suggest that pressure can be both negative and positive, where the negative pressure assumptions of dark energy still seem to be quite speculative. Anyway, many thanks for the help.

Negative pressure is the only way to account for an expansion that accelerates. Otherwise, "gravity attracts", so one expects the expansion to slow down.

More and more evidence seems to be turning up to support the existence of "dark energy". For instance the fluctuations of the cosmic microwave background radiation depend on the acoustic properties of the interstellar medium at the time when the universe became transparent. And these acoustic properties depend on the ratios of dark energy, dark matter, and baryonic matter. Plus there is other evidence that suggests the universal expansion is accelerating, which also would require dark energy.

 

[michelcolman]

In the Newtonian model, you've noted that there would be a "center of attraction". In the GR model, you can consider a small spherical ball of particles, "coffee grounds", that don't interact with each other, but mark out some volume and are initially at rest. If you track the volume enclosed by said sphere, it will decrease at an accelerating rate. This acceleration will be proportional to the density of the matter enclosed by the sphere plus 3x its pressure (assuming the pressure is isotropic).

Matter outside the sphere marked out by the coffee grounds will not affect the rate at which the volume changes (or rather, the acceleration of the volume) - though it will affect the shape of the grounds in a manner that doesn't affect the volume.

I've always had a problem with this idea. "Finite universes contract, and adding matter around that initial volume does not change anything inside, therefore an infinite universe will contract as well."

The problem is that, apart from having a center, there's another property that finite universes have and an infinite universe does not: an edge. Matter that is closer to the edge (in absolute distance) will have more matter on one side and will therefore tend to go towards the side where there is more matter. You can make the universe as big as you like, this property will always remain. But it is totally absent in an infinite universe! Therefore, I don't think you can consider the infinite universe as a limit model that you can approximate by just adding more and more mass around a finite starting bit, since you are losing this essential property in the transition to infinity.

It's a bit like saying that 3, 3.1, 3.14, 3.141, etcetera are all rational numbers, and you can add as many digits as you like without changing the rationality, therefore pi is rational. Replace rationality with some more important quality that all the rational numbers have in common but irrational numbers don't, and the argument starts becoming very similar.

Another argument against contraction is that gravity travels at the speed of light, so any point of mass will be affected by a sphere centered around it, with a radius equal to the speed of light times the age of the universe. OK, maybe the speed of light has not been constant since the big bang, and we don't know where all the matter came from, when gravity started working, etcetera etcetera, but I still have this rather compelling image of two different "spheres of influence" around the different points, instead of one single sphere containing both points. Considering a single sphere, you would tend to expect contraction. But with two different spheres, there's absolutely no reason for it.

I have yet to come across a really compelling argument why infinite universes should be expected to contract.

In fact, I might as well make the opposite argument: infinite universes should be expected to expand!

Imagine an infinite universe that is uniformly filled with a dense kind of matter or other energy. Accept for a moment that this universe would be stable. Now introduce some local "bubbles" of lower density. Those bubbles would tend to push everything around them away with a kind of repulsive anti-gravity, since there is a lack of matter inside of them, and therefore less attraction from that side.

Now imagine a single finite-sized bubble that contains mass, but at a much lower density than the surrounding, infinitely large, extremely dense universe. This bubble would tend to expand, right? (Assuming the rest of the universe is stable). Now make the bubble larger. If you add some more low-density matter around the intial bubble in a uniform way, this will not affect the rate at which the inner bubble will expand. Keep doing this up to infinity. It follows that any infinite universe must undergo an accellerating expansion.

This is just to show that you can "prove" pretty much anything by improperly extrapolating to infinity.

 

[marcus]

...
More and more evidence seems to be turning up to support the existence of "dark energy". For instance the fluctuations of the cosmic microwave background radiation depend on the acoustic properties of the interstellar medium at the time when the universe became transparent. And these acoustic properties depend on the ratios of dark energy, dark matter, and baryonic matter. Plus there is other evidence that suggests the universal expansion is accelerating, which also would require dark energy.

Pervect, could you not also phrase this slightly differently and say as follows?
"More and more evidence seems to be turning up to support the belief that the law of gravity contains a certain constant, called the cosmological constant, with a small positive value."

That is, there could be no "dark energy" in any sense analogous to microwave or heat or any other familiar conventional (more material) forms of energy. So calling the cosmo const. by the name "dark energy" is in some ways a bad analogy.

It leads people to have bad mental pictures and bad logical expectations. If it is really just a constant appearing in the law of gravity (the Einstein field equation, EFE) which has always been there and is a natural constant to put in when you write the EFE down, but for many years everybody thought was zero.

IOW it may not actually be necessary to force upon this constant the inappropriate and phony interpretation of an "energy".

Although an imaginary "energy" with certain unintuitive properties does do a good job of imitating some effects caused by that EFE constant.

That is one possible philosophical take on the cosmo constant.

It is presented in this essay by Bianchi and Rovelli
http://arxiv.org/abs/1002.3966
Why All These Prejudices Against a Constant?

It is short (9 pages) and written for a broad non-specialist audience. It argues that it is not necessary (or even especially reasonable) to think of the cosmo constant as arising from a vacuum energy density. It is just how it is: laws of nature have constants in them. This is our law of gravity, it has G, c, and Lambda in it. It describes how we see nature working. Get over it. (Or words to that effect )

 

[pervect]

A number of published papers use the term "dark energy", for instance google finds:http://prd.aps.org/abstract/PRD/v72/i10/e103503
http://www.springerlink.com/content/w541407713838m25/
http://onlinelibrary.wiley.com/doi/10.1111/j.1745-3933.2005.08577.x/abstract
http://prl.aps.org/abstract/PRL/v82/i5/p896_1

Many of them discuss other origins for dark energy other than the cosmological constant as a possibility, though the cosmological constant seems to be the leading contender. Therefore my current thinking is that it is more prudent to continue to use "dark energy" rather than "cosmological constant" in order to avoid being over-specific, though I do have to agree that the "big mystery" aspect associated with dark energy is sometimes a bit over-done in the popular media.

Having heard my concerns, do you still think that "cosmological constant" should be used rather than "dark energy"? Do you think that theories such as "quintessence" with a different equation of state than one would get from a cosmological constant can be rejected with the current state of observational evidence?

My main focus is on relativity rather than cosmology (I saw the thread before it was moved), though of course both are related, I'm trying to do the best I can for our readers here.

 

[Chronos]

I'm still trapped by observables. This particular universe arose a finite time ago in the past. Does that mean there were no prior universes? - No, so time may be eternal. Does it mean this universe is spatially infinite - No, we are forever incapable of observing anything older than this universe. That which is unobservable is irrelevant in science.

 

[bcrowell]

In the Newtonian model, you've noted that there would be a "center of attraction". In the GR model, you can consider a small spherical ball of particles, "coffee grounds", that don't interact with each other, but mark out some volume and are initially at rest. If you track the volume enclosed by said sphere, it will decrease at an accelerating rate. This acceleration will be proportional to the density of the matter enclosed by the sphere plus 3x its pressure (assuming the pressure is isotropic).

Are you describing a process of expansion or contraction here?

GR basically predicts the second derivative of the volume with respect to time. Pervect was describing the case where dV/dt is initially zero. In the cosmological case, dV/dt is initially positive, but d2V/dt2 is negative.

I think the reason you're having so much trouble here is that you're trying to use the concepts of force and energy. In GR, gravity is not described as a force. The way you're attempting to apply conservation of energy to cosmological models also doesn't really work. GR has local conservation of energy, and it also has global conservation of energy in certain special cases. Those special cases include stationary spacetimes and asymptotically flat spacetimes. Cosmological models aren't stationary and aren't asymptotically flat.

 

[mysearch]

I think the reason you're having so much trouble here is that you're trying to use the concepts of force and energy. In GR, gravity is not described as a force. The way you're attempting to apply conservation of energy to cosmological models also doesn't really work. GR has local conservation of energy, and it also has global conservation of energy in certain special cases. Those special cases include stationary spacetimes and asymptotically flat spacetimes. Cosmological models aren't stationary and aren't asymptotically flat.

I think you are probably right. However, let me say in advance of comments below that I am not in denial about general relativity, just in the process of really trying to understand the premise on which it is built, when applied to various cosmological models. However, GR can 'appear' a bit like a mathematician’s version of programming, i.e. a set of formulaic processes, which the programming adage of ‘garbage in, garbage out’ might also apply. Therefore, I was also interested in trying to understand the assumptions being made.

That which is unobservable is irrelevant in science.

Sorry, I disagree. Speculation has always been a part of science; we just tend to refer to this process as theories or hypotheses. IMO, there is nothing wrong with speculation as long as we don’t forget that such ideas are not verified science and that cosmology is an area of science that 'appears' to have more than its fair share of speculation. Therefore, for me, the issue concerning dark energy and/or the cosmological constant, as raised by https://www.physicsforums.com/showpost.php?p=2990754&postcount=9", seems to fit into this category. In the context of the standard energy-density model, dark energy has virtually no active presence in the universe till about 7 billion years ago, when its density rises to a sufficient proportion to be a ‘possible’ explanation to the ‘apparent’ observation of an accelerated expansion. It is unclear to me how dark energy maintains a constant value under expansion, as this ‘seems’ to be in contradiction of the conservation of energy, at least, within an isolated thermodynamic system. Of course, ‘maybe’ the universe is not all there is, ‘maybe’ there is a quantum source, which may all be valid speculations, but not ‘necessarily’ verified fact. However, dark energy doesn’t ‘seem’ to explain how the universe initially expanded or why it kept expanding in the absence of any obvious energy-density for the 1st 7 billion years. At this point, we might ‘speculate’ further regarding scalar fields and introduce the idea of inertia of expanding space or alternatively we might ‘speculate’ that space is infinite and that expansion of our local universe took place within existing space. I am not sure whether any of these ‘speculations’ can be ruled out by the mathematics of GR or whether it is just as susceptible to ‘garbage in, garbage out’ as any other branch of science.

…..Therefore, I don't think you can consider the infinite universe as a limit model that you can approximate by just adding more and more mass around a finite starting bit, since you are losing this essential property in the transition to infinity.

…..OK, maybe the speed of light has not been constant since the big bang, and we don't know where all the matter came from, when gravity started working, etcetera etcetera,
……I have yet to come across a really compelling argument why infinite universes should be expected to contract. In fact, I might as well make the opposite argument: infinite universes should be expected to expand!

…..This is just to show that you can "prove" pretty much anything by improperly extrapolating to infinity.

I must admit to also being puzzled by such issues. However, in many ways, the reason I originally submitted this thread into the relativity forum was an attempt (failed) to limit the discussion to GR and the specific models outlined in post #1. Still I will follow pervect advice and follow the http://math.ucr.edu/home/baez/einstein/" in the hope that GR does hold the answers to my original questions. Thanks

 

[marcus]

My main focus is on relativity rather than cosmology (I saw the thread before it was moved), though of course both are related, I'm trying to do the best I can for our readers here.

You are doing great! I was intending to say that in my post but got rushed to finish and neglected. Mysearch is getting an ideal response. I don't want to interfere.

My comment was mainly for you---it's a possible different take on the subject (but doesn't really fit in this thread)---just a footnote.

Having heard my concerns, do you still think that "cosmological constant" should be used rather than "dark energy"? Do you think that theories such as "quintessence" with a different equation of state than one would get from a cosmological constant can be rejected with the current state of observational evidence?

If you have to choose one or the other then you have to say "dark energy" because the equation of state has not been nailed down. And the parameter w could change. We don't know for sure that the effect is due to a constant.

But in some conversations, depending on whom you are with, you don't have to choose. You can consider both possibilities, that it is a constant Lambda in the law of gravity and doesn't have to be explained, any more than any other constant. And on the other hand the possibility that it is not a constant but the effect arises from some exotic energy field, which could be subject to gradual change over time and all the (wild?) speculation that ensues.
Both possibilities can be acknowledged as long as you are talking with people who don't need a quick answer.

I guess I am partial to treating it simply as a constant in the law, as long as I don't see any compelling evidence against that. As my "default" case. But I certainly want to acknowledge immediately that there could be a quintessence field and so forth.