"de sitter effect" and cosmology history

[quantumtopology]

Lately I've been doing some research about the first years of modern cosmology, around the second decade of 20th century, and trying whenever I can to go to the sources or the original papers of Einstein and de Sitterand also Hubble and to my surprise I find a rather different story from that told in modern cosmology books, mostly in what relates to the expanding universe issue, looks like someone around the 30's decided that the "expanding universe" was the only valid hypothesis and discarded any other interpretations of redshift. I found that de Sitter in 1917 predicted redshifts in relation with distance in his model B "de sitter universe" and even called it "spurious radial velocity", and Hubble himself in his 1929 often cited paper atributed the redshifts to this "de sitter effect", and this paper is the one I've seen cited dozens times in cosmology books like the one "in wich Hubble discovered the expansion of the universe". I Know Eddington had a lot to do with this a bit twisted retelling of how things happened but also Einstein and de Sitter came in later years to conform with this official cosmology. The fact is that I wondered If someone else finds this odd too.

[modest]

I found that de Sitter in 1917 predicted redshifts in relation with distance in his model B "de sitter universe" and even called it "spurious radial velocity", and Hubble himself in his 1929 often cited paper attributed the redshifts to this "de sitter effect", and this paper is the one I've seen cited dozens times in cosmology books like the one "in wich Hubble discovered the expansion of the universe".

 

You are correct. Einstein and de Sitter had some rather long debates about the nature of the universe—all well documented. De Sitter predicted that mass would "scatter". His metric predicted that and quite rightly so. That is why Hubble picked up on it.

 

The thing those old papers will not tell you is that both Einstein's original model and de Sitter's original model are both solutions of FLRW cosmology and as soon as Einstein and de Sitter realized that, they both endorsed Friedmann and Lemaître's solution.

 

In other words, de Sitter's universe is not a distinctly different solution to cosmology than standard cosmology, it is just a slightly different metric describing one scenario of the same thing.

 

I can get precise quotes and links when I have a bit more time.

 

~modest

[quantumtopology]

Thanks for your input Modesto.

But my point is highlighted by your answer as I perceive you are suggesting that ultimately there is one and only one possible cosmlogy, the FLWR cosmology that I guess you identify with the current official model. And what I was trying to stablish is that historically there is not such a thing as a FLWR cosmology but that this was a posterior construct with a clear purpose of imposing a dogma: the only acceptable cosmology is that based on an expanding universe that logically brings along the BigBang Theory.This is what I think that historycally (and empirically) doesn't hold up if you look at it closely and without preconceptions.

To start with Friedmann: he wrote between 1922 and 1925 (year of his untimely death) several brilliant papers with exact solutions to Einstein field equations, not only dynamical solutions but also stationary ones with curvatures positive negative and null; treating all of them with great equity.

Robertson and Walker were great mathematicians that improved technical points of the metric and the line elements of a determined solution( the dynamical one).

We are left with Lemaitre, a priest from belgium who studied with Eddington for a year and that independently of the other three proposed the model we readily identify with the expanding universe and the seed of what later was called Big Bang theory, the primoeval atom. I think it is hard to argue against the fact that Lemaitre was influenced by his religious beliefs in composing this theory, a fact himself never hid when he presented it as a theory of the creation of the universe.

Don't you think there is alternatives to be explored and tested before we leave this matter settled? Specially with all the troubles the present model is experiencing: dark energy dark matter, cosmological constant, initial singularity, galactic ages.....

[coldcreation]

[snip]

Don't you think there is alternatives to be explored and tested before we leave this matter settled? Specially with all the troubles the present model is experiencing: dark energy dark matter, cosmological constant, initial singularity, galactic ages.....

 

You bring up some good points. I would recommend glancing through the Redshift z thread where the de Sitter effect (both the static and dynamic interpretations) was discussed quite extensively.

 

There, you will find discussions about the relation of the de Sitter effect with the CMB, from post 321 onwards. You will find in that thread the relation of the de Sitter effect to time dilation, DE, CDM, globally curved spacetime, and the list goes on. De Sitter was mentioned in the Redshift z thread over 460 times (I just did a quick search).

 

De Sitter is discussed extensively throughout that thread, though, the discussion really picks up from about post 160, 161 and so on through the following pages (up to and including the last page).

 

 

 

If you have anything (new?) to add to any of those remarks, or observations regarding the contents there, or if you are familiar with any recent work (not covered there) that you would be willing to share with us, in the form of links or references perhaps, I for one would certainly be interested.

 

I await your remarks either in this thread or in the other if you have any direct questions regarding the contents of that thread.

 

I do have one question: Are you suggesting that the observed redshift might be a de Sitter effect (or de Sitter-like effect) in a stationary (non-expanding) universe?

 

That is pretty much what the other thread was about. I'm not entirely convinced that the case is closed, though some may feel that way.

 

It's fascinating stuff, that's for sure.

 

 

CC

[quantumtopology]

Hello Coldcreation, thank you for the reference to the redshift thread, I've only been able to read a few posts so far but I really find it to be great stuff, anyway you seem to display a knowledge that is way much superior to mine as I am basically a layman with a big interest in physics and specially General relativity and Cosmology. But it is weird because some of what you say about hyperbolic space, redshift , de sitter... strikes a chord with insights I've had when thinking about these themes.

I basically am critic with the actual cosmology model, I guess since I don't have a formal training in physics I haven't been indoctrinated enough to buy it without questioning it.

So I started searching for alternatives and coming from a different field (Biochemistry) I found something that at first seemed outrageous to me, no physicist even considered nothing even slightly critical with the official model, later on I learned that it is almost impossible to publish anything questioning the dogma, what's more nobody outside the model is even allowed to have telescope time to test alternatives or receive funds for research. I finally got used to this strange state of things for a science but it still puzzles me, even more so that aparently very few people seems to care about it. And don't get me started with the fact that everytime someone says something along these lines in forums is inmediately insulted, called crackpot , crank, etc. So I am positively surprised to find this forum and threads like yours.

To answer you briefly (I have some reading to do it is a huge thread) I think there is good motives to consider a de sitter-like effect based on a empirically testable in principle universe with a constant negative curvature(hyperbolic). The reasoning goes like this: if cosmological redshift can be explained by a geometrical stretching of wavelength as I believe it can be mathematically done and we eventually prove our universe has this configuration, then the expanding universe hypothesis to explain redshift is unnecesary.

So it's up to observational findings. For example I'm thinking of angular size-redshift relation, or the supposed accelaration of expansion which could as well atributed to hyperbolical space geometry.

Recently I've read about an astronomer named Vahe Gurzadyan who seems to be making a good case for hyperbolical geometry based on WMAP and boomerang experiments. Did you know about it? Sorry I don't have references handy but if you google it probably will find something.

Also I'm working on a connexion between hyperbolical geometry and "dark matter" but at the moment it is not really developed.

Well that's enouh for now, looking forward to a fruitful correspondence:)

[modest]

But my point is highlighted by your answer as I perceive you are suggesting that ultimately there is one and only one possible cosmlogy, the FLWR cosmology that I guess you identify with the current official model.

 

You can think of FLRW cosmology like a rainbow and a de Sitter universe like the color red. The latter is contained in the former.

 

You can describe a de Sitter universe in the FLRW metric by setting the matter content to zero and the cosmological constant positive. A de Sitter universe is a specific example of a FLRW universe. Therefore, cosmologists don't have a problem with de Sitter's universe, they just use a different, and much more intuitive metric to describe it. In fact, our universe is probably turning into a de Sitter universe meaning that one day his model will describe the universe quite well.

 

The important thing to recognize is that things do physically recede from one another in a de Sitter universe. This was not immediately recognized when he developed the model meaning there are some old papers from the 20's which might mislead a person as to the nature of the de Sitter effect. Eddington explains:

 

Allusion has been made to the fact that the recession of the galaxies in the present theory of the expanding universe is not precisely the effect forseen by de Sitter. It may be well to explain the manner of the transition. The phenomenon that is generally called the “de Sitter effect” was a rather mysterious slowing down of time at great distances from the observer; atomic vibrations would be executed more slowly, so that their light would be shifted to the red and imitate the effect of a receding velocity. But besides discovering this, de Sitter examined the equations of motion and noticed that the real velocities of distant objects would probably be large; he did not, however, expect these real velocities to favour recession rather than approach. I am not sure when it was first recognized that the complication in the equations of motion was neither more nor less than a repulsive force proportional to the distance; but it must have been before 1922. Summarizing the theory at that date, I wrote—“De Sitter’s theory gives a double explanation of this motion of recession: first, there is the general tendency to scatter according to the equation [math]d^2r/ds^2 = 1/2 \lambda r[/math]; second, there is the general displacement of spectral lines to the red in distant objects due to the slowing down of atomic vibrations which would be erroneously interpreted as motion of recession.” I also pointed out that it was a question of definition whether the later effect should be regarded as a spurious or a genuine velocity. During the time that its light is traveling to us, the nebula is being accelerated by the cosmcal repulsion and acquires an additional outward velocity exceeding the amount in dispute; so that the velocity, which was spurious at the time of emission of the light, has become genuine by the time of its arrival. Inferentially this meant that slowing down of time had become a very subsidiary effect compared with cosmical repulsion; but this was not so clearly realized as it might have been. The subsequent developments of Freedmann and Lemaitre were geometrical and did not allude to anything so crude as “force”; but, examining them to see what has happened, we find that slowing down of time has been swallowed up in the cosmical repulsion; it was a small portion of the whole effect (a second order term) which had been artificially detached by the earlier methods of analysis.

 

Theories of the Universe: From ... - Google Book Search

 

So, if you're looking for a static universe where redshift arrives by some other effect than physical recession this model is not the solution. De Sitter and Einstein and everyone else has since realized this.

 

Also, the idea that there was a dogma slanted towards expansion is backwards. Most everyone assumed that the universe would be static and it took a good deal of evidence to change people's minds about that.

 

~modest

[quantumtopology]

Good points Modesto, and I agree with you that the desitter universe is not the exactly the right model for a stationary universe, I only used it as a historical departure point and as an example of how interpretations change depending on the context. In 1917 de sitter presented his model as a static one,15 years later Eddington was saying that de Sitter was the first to suggest an expanding universe based in his perhaps biased view of the 1917 "A. Einstein’s Theory of Gravitation and its Astronomical Consequences" paper.

 

I also agree with you that inthe firt years thre was a strong bias towards a static, finite universe, being this practically the only reason for his having to add a term (cosmological constant) to his equation.Moreover was the reason he didn't accept at first the de Sitter's model on the grounds that it was not finite.Aside from this in the 20's and 30's of the past century there was indeed resistence to the expansion idea (wich I find normal giving its implications). It's funny that even Eddington (you'll probably know this it's wiki material) couldn't accept the logical step of a universe in expansion, that at some initial moment there was a point like singularity, he thought our universe was expanding from an initial Einstein universe.

So the historical twisting I'm talking about came afterwords, championed by people like George Gamow.

In the interesting Eddington quote you brought to our attention he is already taking for granted that there is "cosmic repulsion" and it is by comparison to this big value that he finds the de Sitter effect "swallowed up" by the whole effect and being a very small proportion of it. Or at least that is how I undertand it.

Regards

[modest]

Quantumtopology, I recommend you read this paper in its entirety:

The Origins of the Velocity-Distance Relation JOURNAL FOR THE HISTORY OF ASTRONOMY V. 10, P. 133, 1979

 

If you miss the latter pages like (eg. 153 and 154) then you will miss the important part regarding the de Sitter effect.

 

I should correct you in that Eddington was not taking things for granted. He showed deductively that things do scatter in a de Sitter universe and the scattering overwhelms the de Sitter effect. This is shown rather well by Tolman in 1929:

 

On the Astronomical Implications of the de Sitter Line Element for the Universe

 

This physical scattering is how expansion reveals itself in de Sitter's metric. It was, however, not immediately recognized as a consequence of his model. Because of this failure of the early analysis you will find some old papers (early 20's) that consider de Sitter's universe static. But, things in his universe most certainly are not static. They are forced away from one another which is what we now call expansion, but was at the time referred to (in relation to de Sitter's model) as a tendency for things to scatter.

 

~modest

[quantumtopology]

Hi Modesto

I think I mostly agree with the account of facts of the first paper, but I think we differ in interpretation, I think that de Sitter changed his mind because he thought (wrongly) that models A and B were the only possible solutions to static universes and by elimination he thought the dynamical solutions had to be chosed. Sadly de Sitter died only 2 or 3 years after this so it is useless speculating if he might have changed his mind again.

I have to find time to read the second lnk.

Regards

[modest]

Quantumtopology, this is a long post so let me first say that I find cosmology a fascinating subject and I thank you for the pleasure of discussing such an interesting facet in the cosmic puzzle.

 

I think that de Sitter changed his mind because he thought (wrongly) that models A and B were the only possible solutions to static universes and by elimination he thought the dynamical solutions had to be chosed.

 

You're probably referring to this quote of de Sitter's:

 

Both the solutions [A (Einstein's) and B (de Sitter's)] must be rejected, and as these are the only statical solutions of the equations… the true solution represented in nature must be a dynamical solution.

 

from 1931 and page 154 of the previous link. Where he says "of the equations" he is referring to general relativity. Tolman showed that there were only 3 static solutions to the field equations. Coldcreation might have a link to that proof—I recall him mentioning it in a previous post—but, I do not have it off hand.

 

The three static solutions are models A and B (Einstein's original model and de Sitter's original model) as well as an empty Minkowski spacetime which as a cosmological model would mostly go by the name "Milne model" now. Saying that these models are "static" solutions to the field equations has a very specific meaning which may not be what one would intuitively think of as static. I'll explain.

 

In general relativity space has an energy density which causes gravity. The energy density is nothing more than looking at some large volume of space and seeing how much energy there is in that volume. It is literally a density of energy. The energy comes from the mass inside the volume and/or the cosmological constant inside the volume.

 

A relativistic model of cosmology is said to be "static" if the energy density does not change over time. In other words, if you look at a very large volume of space and it has X-amount of energy then it will also have X energy tomorrow and X energy a billion years from now. If the model is said to be static then the energy of a given volume of space doesn't change over time.

 

Three cosmological models that are solutions of general relativity satisfy this requirement:

 

1. Model A: Einstein's original model of 1917. It has mass and a positive cosmological constant. The mass is attracted toward itself by virtue of gravity's attractive nature and is repelled away from itself by virtue of the cosmological constant. The result is a balance between the two, like standing a pencil on its point.
    
   
2. Model B: de Sitter's original model of 1917. It has no mass and a positive cosmological constant. Because this model has no mass, there is no contribution to the energy density from mass, only from the cosmological constant. The cosmological constant is a property of space which means that a given volume will have a given amount of energy from it, and that value will never change—it is a constant. The model is therefore "static" in the sense that the energy density of space does not change over time. If, however, one tried to put mass in a de Sitter universe it would no longer be static because the mass would expand changing the energy density of the volume.
    
   
3. Minkowski spacetime (eg Milne model): This cosmological model has no mass and no cosmological constant. The energy density is zero and does not change over time because it remains zero.
   

 

Here is the best quote I could find talking about the energy density's time invariance in relation to the empty models: http://books.google.com/books?id=1TXO7GmwZFgC&pg=PA29&lpg=PA29&dq=%22generally,+one+does+not+have+a+choice+of+foliation+if+it+is+to+respect%22&source=bl&ots=dAOgegCV8e&sig=0h2WUJiCdSl5CT9hS-u8LzcpVG8&hl=en&ei=c-voS4OAB4eQNoO15PsJ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBIQ6AEwAA#v=onepage&q=%22generally%2C%20one%20does%20not%20have%20a%20choice%20of%20foliation%20if%20it%20is%20to%20respect%22&f=false.

 

We can reject the second two models as being realistic descriptions of our universe simply because they have no mass. This is, in fact, the reason de Sitter rejected his own model, saying "I had come to the conclusion that my solution “B” could not be accepted as an adequate explanation, as it supposes the universe to be empty, or at least emptiness to a good approximation.

 

The idea of emptiness being static is remarked on in the paper I linked in my last post on page 139:

In the third paper de Sitter set forth his ideas on the implications of the General Theory for the large scale properties of the universe. Here he explained his own solution to the field equations: the empty universe, which, by virtue of having nothing in it to move, was also, like Einstein’s model, static.

It is trivially static because there is nothing to move.

 

This leaves us with one non-trivial static solution to general relativity which is Einstein's original model A. But, such a universe is not stable. The Friedmann equations demonstrate why:

 

Loitering universes reach a point with H = 0 and also zero acceleration, known as an Einstein solution (which lies at infinity on this graph because the critical density is zero). Along the green line the Einstein point is in the future, while it is in the past along the red line. At the Einstein point the repulsive negative pressure from the cosmological constant exactly balances the gravitational attraction of density:

 

[math]\rho_m = 2 \rho_{\Lambda}[/math]

 

This is a delicate and unstable balance.

If the density is slightly too high, the universe will collapse from the Einstein point, following the green line down to a big crunch at the Einstein-de Sitter point. If the density is too low, the expansion takes off and the universe follows the red line to the de Sitter point of total domination by the Cosmological constant.

 

Solving the Friedman Equation

 

Physical evidence also rules out Einstein's model A (a loitering universe) because such a universe would not have Hubble's law regarding distance and redshift which our universe does. There is a lot of other evidence such as the lack of old stars and the CMBR.

 

 

The conclusion therefore can only be that there is no cosmic solution to general relativity that is static and consistent with our universe. This is demonstrably true and it would be no less true if de Sitter had lived longer. On this point he was not mistaken and it is a credit to scientists like de Sitter and Einstein that they accept the work of others and abandon their own beliefs and hard work when evidence falsifies that work and supports the work of another (like Friedmann and Lemaitre).

 

It can be shown that any cosmic model which is static (in the sense that there is no distance-velocity relationship) and contains mass would have to contradict general relativity. To go down that road we would need a whole new theory of gravity, or at least to somehow change the hugely successful and extremely-well tested theory of gravity that we have now.

 

~modest

[quantumtopology]

Glad we share a fascination about cosmology, Modesto.

Some questions, I understand the 3 static models (A,B and Milne) are derived with the Einstein field equations in the version with the added cosmological constant term lambda.

So are you considering static solutions without the lambda contant, just with the original equation? I say this because I found this interesting link on comology history

http://www.spsnational.org/radiations/2008/ecp_bigbang2.pdf where in page 28 there is a solution with the original equation that presents us a hyperbolic static solution Einstein rejected because it was an infinite universe.

So what I am picturing( so please take it like a speculative toy universe so that you are encouraged to pinpoint all of its inconsistencies) would be a static solution of GR without cosmological constant with a hyperbolic geometry that would allow the redshift-distance relation, with mass, and constant energy density.

could the instability problem vanish without the cosmological constant?

[coldcreation]

[...]

So what I am picturing( so please take it like a speculative toy universe so that you are encouraged to pinpoint all of its inconsistencies) would be a static solution of GR without cosmological constant with a hyperbolic geometry that would allow the redshift-distance relation, with mass, and constant energy density.

could the instability problem vanish without the cosmological constant?

 

quantumtopology your the perfect candidate to delve into this book:

 

The Static Universe: Exploding the Myth of Cosmic Expansion http://www.amazon.com/Static-Universe-Exploding-Cosmic-Expansion/dp/0986492620/ref=ntt_at_ep_dpi_2

 

By

Hilton Ratcliffe

 

Hilton is actually a member of Hypography, though I haven't seen him post lately. Probably very busy writing another book, or promoting the two already published.

 

There are a couple of interesting threads here where Hilton actively participated, as I'm certain modesto will recall. Some of our debates were heated, albeit always friendly and courteous.

 

Hilton, always a gentleman and a scholar (well, unless provoked), is non-mainstream, as you can imagine, with the title of his recent book, and too of his prior release:

 

The Virtue of Heresy: Confessions of a Dissident Astronomer http://www.amazon.com/Virtue-Heresy-Confessions-Dissident-Astronomer/dp/1419695568/ref=ntt_at_ep_dpi_1

 

But he raises some very interesting points and is exceedingly active in the arena of observational cosmology, as well as other areas, i.e., astronomy, astrophysics. He's a good read. Check'm out.

 

 

Here is one of his threads:

 

Is Newtonian Mechanics an advantage or a limitation in astrophysics?

 

In short, Hilton's idea is that GR is not needed to explain observations, that Newtonian mechanics will do just fine. At least that's what I gather. I have not read his latest book, The Static Universe, so I'm not sure if there are any new developments on that front.

 

 

 

CC

[quantumtopology]

Thanks for the links. I took a brief look. However, I for the moment, adhere to GR or at least to the pre-cosmological contant version of GR, I mean, the theory might be in some sense incomplete but it is my belief that it is the best we have to explain gravitation.

Perhaps I have given an impression of myself as more radical than I actually am.

My only aim is to gain knowledge about ways to falsify inthe popperian sense the expanding solutions to GR, as opposed to accept it like a dogma. And right now , I think there is ground to do it given the many problems of the standard model. It is also my belief that in the near future , with observational advances we are going to be able to make a more informed choice.

[modest]

There are a couple of interesting threads here where Hilton actively participated, as I'm certain modesto:) will recall. Some of our debates were heated, albeit always friendly and courteous.

 

Yup. A very nice and thought-provoking poster ... even if I did disagree wholeheartedly with everything he said :hihi:

 

Some questions, I understand the 3 static models (A,B and Milne) are derived with the Einstein field equations in the version with the added cosmological constant term lambda.

So are you considering static solutions without the lambda contant, just with the original equation?

Both.

 

There should be three static solutions. For ease of readability let's use de Sitter's original shorthand:

Model A: Einstein's spherical universe with matter content and cosmological constant

Model B: De sitter's hyperbolic universe with no matter content and cosmological constant

Model C: Special relativistic Minkowski spacetime (eg the Milne model) with no matter content and no cosmological constant.

Models A and B arise from the modified field equations (with Lambda) and model C comes from the original field equations without Lambda. Both with and without the cosmological constant, these are the only three static solutions.

I say this because I found this interesting link on comology history

http://www.spsnational.org/radiations/2008/ecp_bigbang2.pdf where in page 28 there is a solution with the original equation that presents us a hyperbolic static solution Einstein rejected because it was an infinite universe.

 

Indeed, that's a very good find. It's directly on point. Let me quote the relevant part:

 

[math]P^{\prime} = -1/2(\rho + P)A^{\prime} \quad \quad (17)[/math]

 

where primes denote derivatives with respect to r.

 

An isotropic and homogeneous universe means that the energy density and pressure are uniform throughout space, so [math]\rho = \rho_0 = \mbox{constant}[/math] and [math]P = P_0 = \mbox{constant}[/math]. Therefore [math]P^{\prime} = 0[/math], so that Eq. (17) requires [math](\rho + P)A^{\prime} = 0[/math] which leads to an interesting decision.

http://www.spsnational.org/radiations/2008/ecp_bigbang2.pdf

(page 28)

 

and it really does lead to an interesting decision. The paper goes on to describe two solutions that make [math]P^{\prime} = 0[/math]. There are, however, three. The two that are given in the paper are,

[math]A^{\prime} = 0[/math]

and

[math]\rho + P = 0[/math]

These options go on to derive two 'static' metrics, one hyperbolic and the other spherical. They are none other than models A and B. Einstein's universe is spherical while de Sitter's is hyperbolic.

 

The option not discussed in the paper is [math]\rho = P = 0[/math] which is none other than the third and final static metric—Minkowski spacetime (model C).

 

I think the paper would have done well to explain that the two options Einstein rejected ended up being models B and C. And, also let me reiterate that models B and C are only "static" in a trivial sense.

 

So what I am picturing( so please take it like a speculative toy universe so that you are encouraged to pinpoint all of its inconsistencies) would be a static solution of GR without cosmological constant with a hyperbolic geometry that would allow the redshift-distance relation, with mass, and constant energy density. could the instability problem vanish without the cosmological constant?

 

The problem is that the cosmological constant is necessary to balance the attractive force of gravity if the solution is going to be static. If there is any mass, any energy density, in the universe then it would not remain static. It would collapse back in on itself. The purpose of Lambda is not so much to make the universe spherical (although Einstein very much liked that consequence as it avoided any boundary issues) but rather to satisfy its static nature in the presence of a positive [math]P[/math] (the energy density of matter).

 

The problem is that this is very much like trying to stand a pencil on its tip. If, for example, some of the mass in Einstein's static universe were to be converted to photons in a nuclear process—in a star perhaps—then that energy content would change from [math]P[/math] to [math]\rho[/math] tipping the universe toward a collapse and an eventual big crunch. :(

 

~modest

[quantumtopology]

Hi Modest

I find your explanations clarifying. Let's see if can come up with something we both agree on;) ( and if we can't at least we'll have a good time and might even learn something wich i'm already doing)

First point, you identify maybe too easily Milnes's model with the one derived with Einstein equations without the lambda. A quick look to what wikipedia has to say about the Milne's model and you find that it says that they resemble only superficially in that they are both hyperbolic and static but Milne's doesn't even follow general relativity.

 

Correct me if I'm wrong, de sitter option was d+p=0 but also d=P=0 but adding lambda so that the universe had curvature (as I see it explained in the next paper of the series of wich the link was part 2.

 

So it makes sense that the milne model with d=p=0 without lambda is not allowed in GR since it has no general curvature and it is minkowskian.

 

So far so good, It seems like we are running out of options as you say. Let me think it over. Looks like we would have to go into solutions that for Einstein(and conventional cosmology) looked unphysical if I wanna continue toying with the idea of a stationary solution.

to be continued...

[modest]

Hi Modest

I find your explanations clarifying. Let's see if can come up with something we both agree on;) ( and if we can't at least we'll have a good time and might even learn something wich i'm already doing)

 

Sure

 

First point, you identify maybe too easily Milnes's model with the one derived with Einstein equations without the lambda. A quick look to what wikipedia has to say about the Milne's model and you find that it says that they resemble only superficially in that they are both hyperbolic and static but Milne's doesn't even follow general relativity.

 

You are correct. I confused things. You should consider model C to be: Minkowski spacetime. Milne put matter in his special relativistic Minkowski spacetime which doesn't work. Not only is it inconsistent with general relativity, it isn't even internally consistent.

 

Minkowski spacetime has no mass or cosmological constant. It would be the third 'static' solution... actually... let me try to find Tolman's proof of that...

 

...

 

Here we go: "On the Possible Line Elements for the Universe--Tolman--1929"

 

EDIT -->

 

Sorry, here's a proper link: http://www.pnas.org/content/15/4/297.long The static requirement is introduced on the second page.

 

<-- /EDIT

 

Correct me if I'm wrong, de sitter option was d+p=0 but also d=P=0

 

A' = 0 would go on to derive Einstein's model. p=P=0 would go on to derive Minkowski spacetime and p+P=0 would give de Sitter's model. You'll see it said explicitly at the end of page 107 and start of 108 here:

 

Introduction to gravitation - Google Books

 

EDIT-->

 

Here's a much better link explaining: http://books.google.com/books?id=1ZOgD9qlWtsC&pg=PA335&lpg=PA335&dq=%22lead+respectively+to+the+einstein,+to+the+de+sitter,+and+to+the+special+relativity+line+elements%22&source=bl&ots=qSFqI96FUA&sig=lQUC4JIOpCf48SvKYew-ar2aoeg&hl=en&ei=TBjrS568C8H6lwezq-ycBA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBIQ6AEwAA

 

<-- /EDIT

 

but adding lambda so that the universe had curvature (as I see it explained in the next paper of the series of wich the link was part 2.

 

Adding lambda doesn't necessarily give curvature. Mostly curvature is the result of mass—the less mass a universe has the more hyperbolic it will be. The more mass is added, the more spherical it becomes.

 

De Sitter's model is hyperbolic because it has no mass. Adding the cosmological constant allowed Einstein to add mass yet keep the solution static. Adding the mass caused the positive curvature.

 

So it makes sense that the milne model with d=p=0 without lambda is not allowed in GR since it has no general curvature and it is minkowskian.

 

Usually Minkowski space is considered hyperbolic, you can see from the wiki link.

 

EDIT-->

 

I should explain. Curvature means that a slice of constant proper time is curved. If you look at this image: http://www.astro.ucla.edu/~wright/omega0sr.gif (from Net Wright's tutorial) of a Minkowski universe you can see the horizontal lines (of constant proper time) are curved up toward the outside. That is the negative, hyperbolic, curvature of an empty (or very low mass) universe.

 

<-- /EDIT

 

Looks like we would have to go into solutions that for Einstein(and conventional cosmology) looked unphysical if I wanna continue toying with the idea of a stationary solution.

 

Yeah, certainly a static solution that is consistent with general relativity wouldn't be able to agree with observation.

 

If it's the age of the universe (that is, having a beginning at the big bang) that you find troublesome then perhaps an expanding model that doesn't have a big bang beginning would be of interest. Fred Hoyle's Steady State model is consistent with General Relativity and is quite a bit closer to observation than any static solution.

 

~modest

[modest]

I apologize. I made several edits to the post. I should have read the links that I was linking and proofed the post better.

 

I'm curious if you have a link to this:

as I see it explained in the next paper of the series of wich the link was part 2

I couldn't find part 3.

 

~modest