Inflation and curvature

This thread is an extension of the one about the shape of the universe. One can show the FRW solutions to GR are not flat, ie that the Riemann curvature tensor does not vanish.

The same is true of de Sitter space, the solution of GR that provides a model of the inflationary phase that the evidence points to having happened in the early universe. The metric of de Sitter space can be written

\(ds^2 = -dt^2 + e^{2 t /a}\delta_{i j}dx^i dx^j\)

This metric describes a time dependant space where flat spatial sections expand exponentially. You can also rewrite the metric so that it is static but then the spatial sections are no longer flat.

Cue farsight: how is it possible that de Sitter is flat?

 

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prometheus,
I mine as well post my reply to your answer here. I'm genuinely interested in this topic. If it can be shown that our universe is open than I would like to know more about it. I looked around the net yet couldn't find any evidence for an open de Sitter universe.
http://encyclopedia.com/doc/1080-EnstiendeSitteruniverse.html

Obviously, by "FRW solution" you mean "the Standard Model of modern cosmology" and de Sitter space can be one of three probable solutions so long as you neglect the presents of ordinary matter. Now I don't know if this applies to encyclopedia.com's vague definition for a de Sitter universe but it's a known fact that, through time, anything that is relatively close to us will appear to recede away from us at a continually decreasing rate. At present (13.7Gy) H equals 71.337 km/sec/Mpc, but when the universe reaches an age of 9.77x10+14yr, H will equal 1 m/sec/Mpc (assuming the universe is flat). As Far As I Know the observations of distant Ia supernova show a universe that is accelerating away from us at a constant rate (relative to distance) and, based on their cosmological red-shift, these observations agree with the assumption that the universe is flat.
http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf
.
Could you please provide a current link to any modern observation showing our universe as an open interpretation of de Sitter space based on the FRW solutions? I hope the links you addressed to Farsight were not the "modern measurement of the CMB" you spoke of (one was published in 1998 and the other in 2001).

 

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It's flat in the sense that a de Sitter universe contains flat physical space. A de Sitter "space" is not the same as physical space. It's a mathematical object that describes the evolution of physical space over time. At any point in time within that evolution, the physical space is flat and homogeneous. Since a gravitational field is inhomogeneous space (Einstein 1920), there is no overall gravity constraining the expansion of the universe.

 

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Edit: Einstein de Sitter universe is not the same as de Sitter (confusingly). Einstein de Sitter is what you get if you assume the universe contains only matter (no radiation or dark energy) and the density is exactly the critical density so the universe is flat. Here things recede at an ever decreasing rate. de Sitter space is a very different beast - if you have a universe with no matter or radiation but only dark energy then you will get a space that is expanding exponentially. When I've been talking about the universe being open, if dark energy is a cosmological constant that will be the same for ever then what will happen is our universe will approach a de Sitter universe asymptotically - we do have matter and radiation but eventually they will be so dilute as to be negligible.

Edit again: see above. When I say FRW solution that's what I mean - the solutions to GR that were found by Friedman, Robertson and Walker to GR when you assume the universe is homogeneous and isotropic. I'm not a cosmologist so I can't go too much further than that.

As I said before, I was mixing up the standard model of cosmology and a newer measurement of dark energy from the CMB. Let me be clear: if you assume dark energy does not exist then the density of the universe means the curvature of the universe is very close to flat. However, there is dark energy which is something making the universe's expansion accelerate. If this goes on forever then the universe will approach a de sitter space, but there are other theories about what will happen and indeed, what dark energy actually is.

Now I'm done repeating myself...

Well, I think I was pretty clear about what I was posting - the original discovery paper of dark energy and a review. I never claimed they were hot off the press papers. In physics quite often there is quite a lot of time between the discovery of something and a proper analysis of it, especially if it requires a big pile of cash to put a probe in orbit. Case in point is the CMB. Discovered in 1964. First real study was in 1983 with the RELIKT anisotropy probe. AFAIK dark energy has been discovered (obv.) but not studied experimenatally at all yet. There are lots of theoretical papers that farsight wouldn't like on the subject but no data yet. Here's an arbitrarily chosen review from 2008 just for the sake of putting one in.

 

Never mind the fact that the flat slicing of de Sitter is only a half covering of the full space and never mind the fact that I can calculate the curvature of de Sitter space and it's not zero (for the metric I wrote above the Ricci scalar is \(\frac{12}{\alpha^2}\))?

What is certainly true is that the spatial part of the space is flat, in other words at an instant in time the spatial part of the metric is flat but that doesn't matter because by the time I do anything in de Sitter the space has expanded. The point I was looking at is no longer where it was and everything is going crazy. de Sitter is certainly not flat in any sense - you are wrong.

 

There we go then. I'm right.

Are you for real? It matters. Space is flat. It's homogeneous. There's no gravitational field in it, because inhomogeneous space is what a gravitational field is. The universe has always comprised homogeneous space. It has never been constrained by gravity. If it had, then when it was very small and very dense, it wouldn't have expanded, now would it? And yet here we are.

The relationship between the universe and a gravitational field is one of expanding space, or space dilation, to gravitational time dilation. CMBR photons are redshifted, photons climbing out of a gravitational field are redshifted. If you go back to my original post #62 on the other thread, you can consider the early universe to be like the flat space in a void at the centre of a planet. This is where gravitational time inflation is at a maximum. So in the very early universe, comprised of space with an innate stress-energy or pressure, everything within would have been moving very slowly compared to the initial expansion, which would look very rapid in comparison. There is no yardstick to measure this by within the universe, so to us it would look like inflation.

 

So you completely ignore the more specific part of my post. Why is that? Do you understand that coordinates in general do not cover a whole manifold? Do you understand what the Ricci scalar represents or how you get it? If you understood GR like you purport to then the answer to these questions would be yes...

..and then this suggests that you clearly don't understand GR.

umm... yes. I'm not going to cut myself to prove it though.

Let me restate my little example. Suppose you are floating in de sitter space and now somewhere off in space you see another astronaut in some aribitrary direction. Now suppose you take your laser pointer and point it at the astronaut. If you send a burst of photons at the astronaut assuming the space is flat, by the time the light has gone from you to him he will have moved because the space has expanded.

Saying the spatial sections of the metric are flat is all well and good, but it can have no effect on physics because by the time you do an experiment the spatial sections will have changed. If you like we can rewrite the de Sitter metric so that it is static but the spatial sections are curved - remember physics doesn't depend on what coordinates you use.

Are you asking me how the universe started? I don't know, and I don't think anyone knows. There are plenty of theories (legitimate theories that is, not like your theories) of course.

Of course the universe is constrained by gravity - it's just that gravity is a very weak force compared to the others that we know about.

This is a very nice piece of plausible waffle for people that don't know any physics that will then hopefully think of you as being clever, a visionary even. The unfortunate fact for you is that what you write is tripe. In de Sitter space time is not simply dilated like you can get in special relativity, points in space are physically moving apart. The common analogy is the surface of a balloon being inflated. All points on the balloon are moving away from each other.

Another sad point for your waffle is that inflation predates the CMB by nearly 400,000 years, so the CMB provides no information about inflation. The universe has been expanding in a less extreme way since the CMB was formed which has caused the redshift. Simply put, the predictions of mainstream cosmological theory match far better with what we observe, whereas your predictions are just words that prove nothing.

 

I understand all that. And it's all besides the point, because as you said yourself, the spatial part of the space is flat. So space is flat. That's it, prometheus. End of story. And all your outrage and abuse won't change it one bit.

So what? This space is at all times homogeneous. It's flat. There's no gravity in it.

Physics depends on reality, not on how you write a metric.

No I'm not. I don't know either.

You still don't get it do you? What I've been telling about homogeneous flat space is what Einstein said. I'm not making it up.

No it isn't. Because WMAP has determined that space is flat, which means its homogeneous, and there is no gravity in homogeneous space. And let's face it, if the universe was constrained by gravity, how come it ever expanded?

Ah, a well crafted and reasoned argument to counter what I said. Not. You should try. But of course you won't.

And that analogy is the wrong one. A better analogy is a compressed elastic sphere. Remove whatever it was that kept it compressed, and it expands. All points in the interior of the sphere are moving apart from one another.

I didn't suggest that the CMB did provide information about inflation.

You're reduced to abuse because you've been hoist by your own petard: the spatial part of the space is flat. It does you no credit, prometheus.

 

The problem is that the distinction between what bit of space-time is space and what bit is time is coordinate dependent. As Prom says, only in a particular type of coordinates does the spatial part of AdS space become flat, in another set of coordinates its curved. This is precisely what I explained to you before, that a non-zero curvature tensor can be altered by a change of coordinates but all you're doing is shuffling the curvature components around.

Physics should be independent of how we describe it so if your conclusion is coordinate dependent then its not something which has direct physical reality. The issue of whether or not space in the AdS space-time has curvature is coordinate dependent, so your conclusion that such space (in terms of the spatial components) is flat is taking a mathematical result too naively. Funny how you're always telling other not to get lost following mathematics as if its real and now you've done the same (though you don't understand the maths, you're having to rely on someone else to tell you a particular result).

If you had a working understanding of tensor calculus you'd not have made yet another mistake. You'd also not have the problem of getting confused when we say "space" when its short hand for "a space-time manifold". Its lazy physicist terminology but often we refer to space-time by saying "space" because typically those we're talking to will know from the context whether we mean a 3 dimensional region of space-time or all of space-time itself. Since you're not one of those people its only served to confuse you and illustrate your lack of grasp for the topic.

And yet your "The spatial part is flat, end of" is dependent on how you write the metric, something Prometheus has explicitly stated several times and you have failed to grasp.

Please refer to the aforementioned comment on space as in spatial vs space as in space-time.

WMAP hasn't determined its exactly flat but that its very close to flat. And that doesn't mean it isn't expanding or has no dark energy or the like. Dark energy tells us how the size of the universe changes and how the rate of that change changes. Flat now doesn't mean it will stay flat, just as a child on a swing has no vertical motion at the bottom of the swing but only for an instant, then they change from going down to going up, just as the universe may be expanding from being curved inwards to curving outwards.

Oh please, how many times have you refused to engage me in discussion. In fact you're now just ignoring my posts, even after I offered to put myself up for suspension if I didn't play nice but since the proviso was you doing the same you've run away scared.

Its always the same with you, you want people to talk about your work only if they are complementing you, when someone asks you to step up with more than just empty rhetoric you've got nothing.

How many times are you going to demonstrate you have no grasp of GR, even when Prom explicitly states certain things, like the coordinate dependence the curvature of the spatial component of AdS space has? If you could do coordinate transformations you could see it for yourself but you can't. Hell, you can't even grasp it when you're spoon fed it.

There's been no hoisting of anyone other than you Farsight.

 

LOL. Prometheus was talking about de Sitter space, not anti de Sitter space. And he said the spatial part of the space is flat. So that's it, egg on face, game over, Farsight wins again. Now face up to it. Einstein spoke of inhomogeneous space when describing a gravitational field, not an inhomogeneous space-time-manifold. Here it is again:

"This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)..."

The fact that you didn't recognise the distinction is your problem, not some deficiency of mine. Hence you can spare me the humpty-dumpty it means what I say it means and the feather-spitting erudite outraged abuse. And don't back up stuff like this:

...where rewriting the metric makes the expansion of the universe go away. That's garbage and you know it. So don't lecture me about intellectual honesty.

 

LOL. Prometheus was talking about de Sitter space, not anti de Sitter space. And he said the spatial part of the space is flat. So that's it, egg on face, game over, Farsight wins again. Now face up to it. Einstein spoke of inhomogeneous space when describing a gravitational field, not an inhomogeneous space-time-manifold. Here it is again:

"This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)..."

The fact that you didn't recognise the distinction is your problem, not some deficiency of mine. Hence you can spare me the humpty-dumpty it means what I say it means squirming and the feather-spitting erudite outraged abuse. And don't back up stuff like this:

...where rewriting the metric makes the expansion of the universe go away. That's garbage and you know it. So don't lecture me about intellectual honesty.

 

Doesn't matter, AdS and dS have the same relevant properties, just opposite curvatures.

Obviously your desperation to take a shot at me and your lack of knowledge has, yet again, led you to put your foot in it.

Farsight, you seem to have serious issues with choice quotes. You've done it for Einstein and you've done it for Prom several times. You say "Why do you call Einstein an idiot", ignoring where Prom clarifies himself. You now say "He said the space was flat" ignoring he also said that its coordinate dependent and a different coordinate choice, where its static, is no longer flat in the spatial part.

By quote mining and deliberately avoiding things people say to correct your mistakes you demonstrate you have no interest in honest discussion, you only want to have the most superficial appearance of understanding. Do you think people will only read your post and not the thread? That somehow everyone will miss what Prom explicitly said that its coordinate dependent? All your actions do is make you appear to have no honesty and the 'debating' skills of a 5 year old. Maybe you know this, as it would explain why you avoided taking me up on my offer of an alpha rules discussion.

When have you ever won?

See my comment on how physicists often say 'space' when referring to a 'space-time manifold', as its short hand terminology and typically its when talking to those people who can tell the difference. You can't and all your issues stem from a lack of familiarity.

You obviously need it so until you show some don't expect people to assume you have any.

I'm still waiting for you to retort the lengthy list of crank activities you've undertaken, despite you claiming you're not a crank. Having a little trouble?

 

Two other people that know what they're talking about have pointed out that just because the spatial sections of a metric are flat, the time direction is important and can cause a non zero curvature, as is the case in de Sitter (and AdS) space.

I'm beginning to think you are a bot. I write "rewrite the de Sitter metric," and you automatically chime in with "Physics depends on reality, not on how you write a metric," despite the fact that, in the very same quoted text I say "remember physics doesn't depend on what coordinates you use."

What are you asking then?

I've not read the source for your quote, but if Einstein did say that then he was wrong. Curvature and therefore gravity can certainly exist in homogeneous spaces as was shown by de Sitter and Friedmann, Robertson and Walker. The other alternative is that you simply misunderstood what Einstein was trying to say, as I believe I've suggested before.

This is not true, as AN pointed out. WMAP has shown that the universe is close to flat at this moment in time.

If you want a slightly less sarcastic version of what I said before then how about this: General relativity and what we know about the early universe from the CMB and other sources contradict your claim. The expansion of the universe is not like the time dilation of SR, but it's a physical "points in space getting further apart," effect. You say there was no yardstick to measure things in the early universe - it's not true either. There are at least 4 scales that I can think of off the top of my head that are known to effect the evolution of the early universe, and I'm not a cosmologist by any means. They are the Planck scale, the grand unification scale, the electroweak scale and the QCD scale.

What exactly is the difference? The balloon has a force difference (the air pressure in the balloon vs. the tension) such that the balloon increases in size. Your example has some unspecified force difference where the elastic sphere (note; these are commonly called "balloons" in the public domain) also expands.

In any case, these are just analogies and whether they are good or not so good analogies is beside the point. We have GR to tell us exactly what is happening without the need for analogies.

Good.

Saying you are wrong is hardly abuse. I can be a whole lot more abusive if you like.

I haven't got too much to add to what AN and JamesR have had to say about this except to say the following: \(g_{\mu \nu}\) is a symmetric, rank 2 tensor. You can show that in d dimensions, while a generic rank 2 tensor has \(2^d\) components, the fact that \(g_{\mu \nu}\) is symmetric means it has \(\frac{1}{2}d (d+1)\) components. That is where your "...compelling us to describe its state by ten functions (the gravitation potentials gμν)" comes from - in 4 dimensions (ie, including time) the metric tensor has 10 independent components. If Einstein was only talking about the 3 dimensions of space then he would have talked about the metric containing only 6 independent components.

Oops.

 

With respect James, that isn't true. A reference frame isn't something that actually exists. If you change your motion through the universe you might adopt a new reference frame, but it, and any coordinate system you adopt, are artefacts of your measurement. You do not alter the space of the universe by travelling through it. Your measurements of time and space alter, that's all. For example, if you accelerate towards a star, it doesn't actually flatten to a discoid and move closer to you. You started moving, that's all.

The curvature that goes away is the abstract mathematical curvature that gets in the way of people considering the shape of the universe. We don't want to determine a "shape" that is an integral over 13.7 billion years. We want to work out what shape it is at this point in its evolution. Thus we take note of the The FLRW metric which "starts with the assumption of homogeneity and isotropy of space". Then we pay attention to what Einstein said about a gravitational field being inhomogeneous space. Hence a homogeneous universe has no overall gravitational field. Since it is comprised of space with an innate stress-energy, and since stress is merely directional pressure, and since there's nothing outside the universe to constrain it, it expands.

Modern relativity perhaps, but not Einstein's relativity. Read The Golden Age of General relativity and note where it says

"Other paradigm shifts included a growing appreciation of the: Role of curvature in general relativity..."

Backing this up is Einstein's gravitational field. Note the line in the abstract that says "The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with". If this isn't enough read The Foundation of the General Theory of Relativity. Einstein gives the equations of motion. He talks of curvilinear motion, but he doesn't actually mention curved spacetime. That was 1916, by 1920 he was describing gravity in terms of inhomogeneous space, but nowadays "Einstein's relativity" is something different to what really was Einstein's relativity.

 

And, as you stated many times on PhysOrg, neither does a metric, they are abstract concepts and thus the physical predictions shouldn't depend on how we write them. This is why Lorentz invariance is important, it removes dependency on inertial reference frames. In addition it also means that when you talk about the spatial part of space-time you're talking about a specific frame, because if you change frames you change what you're calling 'space'. In Newtonian physics there's a very precise unique definition of 'now', but not in relativity, general or special. If you boost to a different frame or do a more general coordinate transformation you change what bit of space-time you're calling space.

Hence, as you've had explained to you many times now, your statement "The spatial bit is flat" is not a valid statement as what you mean by 'spatial bit' is not a coordinate independent concept.

You're preaching to the choir, except you don't understand what you're preaching.

But the notion of 'now' is coordinate dependent, it depends on how you describe things. This is a fundamental concept in relativity which isn't in Newtonian mechanics and its one you've failed to grasp despite claiming you understand relativity.

Do you not see the hypocrisy and irony of you basically canonising Einstein, when you're against the physics he helped develop? You haven't studied GR and you disagree with how the mainstream views the 4 known forces yet you cling to what Einstein said.

Yes, because physics evolves as our understanding and experimental knowledge develops. There's more to quantum theory than Bohr or Schrodinger did or to quantum field theory than Dirac and Feynman. The fact we've developed, improved and corrected GR from Einstein's day is a positive thing, not a negative thing.

You have now gone from not grasping the subtleties of pseudo-Riemannian geometry to not even grasping the basics of the scientific method.

 

Another interesting note: GR is a quite obviously geometric theory to those that know about GR and geometry, however, quantum field theories and in particular gauge theories can be cast geometrically as well - same theory written differently. Einstein's GR is the same GR as Einstein proposed but now we have a better understanding of what it actually means, as you might expect after nearly 100 years of study.

 

No I didn't. Read what I said. It's nothing like what you said.

I'm not agreeing with you at all. A simple change of reference frame doesn't change the curvature of space. You don't change the curvature of space by moving through it, not at all.

I can't. There isn't any. It's flat. If you're enquiring as to the "real shape", I'd say it's spherical because it's isotropic.

Your point was Einstein's relativity does not, because it considers spacetime curvature. How can you possible claim that my response "seems irrelevant"? Again, see Einstein's gravitational field. Note the line in the abstract that says "The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with". If this isn't enough read The Foundation of the General Theory of Relativity. Einstein gives the equations of motion. He talks of curvilinear motion, but he doesn't actually mention curved spacetime. That was 1916, by 1920 he was describing gravity in terms of inhomogeneous space. Irrelevant? Not a bit.

You don't have a better understanding of what it means. As I was saying, in 1920 Einstein described a gravitational field as inhomogeneous space. A gravitational field is synonymous with curved spacetime. Thus if space is homogeneous on a large scale there's no spacetime curvature to the universe as a whole. Spatial curvature is something different, we can come on to that. Meanwhile read Possible local geometries. Pay attention to The geometry of the universe is usually represented in the system of comoving coordinates, according to which the expansion of the universe can be ignored. Now pick a geometry. It really is very simple.