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Some questions on the Cosmological Principle

  1. Nov 12, 2015 #1
    Firstly, I am just a very interested layman so please forgive my ignorance and non mathematical approach.

    As I understand it, the cosmological principle states that on larger scales the universe is homogeneous and isotropic. So could someone help me to understand the following:

    1) Does this imply that the universe is infinite? As any observer must be able to see the same from any point in space?

    2) Wouldn't this also suggest that the gravitational forces acting upon galaxies (or groups of galaxies) across large distances cancel out, so we should have a static universe on larger scales not an expanding one? (EDIT: If we were to ignore dark energy)

    3) Assuming a big bang, then doesn't that imply there must be an edge to our universe?

    Thanks for any help.
     
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  3. Nov 12, 2015 #2

    DaveC426913

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    'Finite yet boundless' is the phrase often used.

    The surface of a sphere is homogeneous and isotropic, yet it is not infinite in extent, nor does it have a boundary.

    The universe is thought to be finite in a higher dimension. i.e. it wraps around
     
  4. Nov 12, 2015 #3

    PeterDonis

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    No. As DaveC points out, a spatially finite but unbounded universe, such as a 3-sphere, also satisfies the principle.

    No. Look up the Friedmann equations.

    No. Either the universe is spatially infinite, or finite but unbounded. In either case there is no edge.
     
  5. Nov 12, 2015 #4
    Thanks for the reply. I sort of get that but find it very difficult to understand in 3 dimensions. I understood observations showed that the universe was pretty flat. Also, if there is a slight positive curvature, which would make the universe round on larger scales, then the universe would be much bigger and possibly older than we have calculated?
     
  6. Nov 12, 2015 #5

    PeterDonis

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    It's the same as in 2 dimensions, just with one more dimension added. :wink: If you can see how a 2-sphere has a finite area but no boundary, then a 3-sphere having a finite volume but no boundary works the same way.

    "Pretty flat" is not the same as "absolutely flat". The best fit to our observations is a spatially flat and infinite universe, but the "error bars" are large enough that a spatially finite universe with a very, very large radius of curvature is still possible.

    No. Our estimates of the age of the universe do not depend on our estimates of its spatial size/finiteness.
     
  7. Nov 12, 2015 #6

    DaveC426913

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    Especially since we believe the observable universe is about 92 billion light years across, yet only 13.7 billion years old. :wink:
     
  8. Nov 12, 2015 #7
    I was just watching some of Leonard Susskind's lectures on cosmology, but haven't quite understood the Friedmann equations yet. I was more thinking of Newton's (shell) theorem and how that is applied to a homogeneous and isotropic universe and it just seemed that everything would cancel out.
     
  9. Nov 12, 2015 #8
    I suppose the way I am looking at it is, if the universe is slightly curved then taking the 2d sphere analogy, if I place a number of galaxies equidistant around the equator of the sphere and assume there is no other matter, then the gravitational forces acting on the galaxies cancel out and they don't move wrt each other. Of course the sphere can 'expand' and the distance between the galaxies still remains equidistant, but in that analogy I can't see how gravity has anything to do with the sphere expanding as the Friedmann equations would suggest.
     
  10. Nov 12, 2015 #9
    Sorry, my mind just doesn't grasp that! :) I can see how that works on the surface of a sphere, which is 2 dimensional, but we live in 3d space. So I can't see how there is another dimension 'expanding' which in turn causes our 3d 'space' to expand and the distance between galaxies grow.
     
  11. Nov 12, 2015 #10

    marcus

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    I like the question, but I can't think of a succinct response right at the moment. :redface: Hopefully someone else will.

    Around 1850 Georg Riemann showed how you could describe a 2-sphere without referring to any surrounding 3d space. You could describe the curvature and distances entirely from the POV of a 2d amoeboid creature living in the 2d sphere. All existence was concentrated on that infinitesimally thin 2d spherical surface. There was no "inside" or "outside" or "3rd dimension". Respect and gratitude to Riemann. He contributed a lot else besides, to mathematics. Oh, it was not limited to the 2-sphere. All kinds of manifolds with different shapes could exist and be described and studied without assuming any higher dimension space for them to live in. they did not need to be "embedded" in order to exist.

    So the analog, a 3-sphere, does not have to have an inside or an outside or a "4th dimension". Just think about the experience of the 3d creature roaming around in it, measuring distances and angles of triangles etc , comparing radius and volume of 3d balls. circumnavigating. It doesn't have to be "embedded" in order for him to report his experiences exploring the geometry of his 3-sphere world. At least I think it doesn't.

    It can expand just by all the internal distances getting larger---it doesn't have to be embedded, or have a center point.
     
  12. Nov 12, 2015 #11

    PeterDonis

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    The shell theorem says that, if the configuration of matter outside some region is spherically symmetric, you can ignore its gravitational effect on the matter inside the region. But that doesn't mean everything cancels out. See below.

    No, that won't work, because the configuration is not symmetric in a circle (since we're in 2d instead of 3d, the shell theorem applies to a circle instead of a sphere) around a given point. You would have to have a homogeneous distribution of galaxies everywhere on the 2d sphere. Then, if you pick any point on the 2d sphere and consider a small circular region around it ("circular region" meaning a region of the surface of the 2d sphere bounded by a circle drawn on the 2d sphere at some radius along the 2d sphere around a given point), the distribution of matter outside that small circular region is symmetric ("circularly symmetric", by analogy with "spherically symmetric" in the 3d space case), so that distribution has no net gravitational effect on the matter inside the small circular region. However, different pieces of matter inside the small circular region can still have effects on each other. See further comments below.

    Exactly; all of the above was talking about the sphere at some instant of time. But it doesn't say anything about how the size of the sphere itself changes with time.

    The Friedmann equations don't say that gravity is what's causing the expansion. The expansion itself is due to initial conditions and inertia; the model assumes that the universe started out expanding. The shell theorem doesn't rule that out; in the 2d sphere analogy, the shell theorem doesn't prevent the 2d sphere from having some given expansion rate at a given moment of time. In Newtonian terms, gravity affects acceleration, not velocity; we can give the 2d sphere any velocity we like at some instant of time as far as gravity is concerned.

    What the Friedmann equations tell you is how the rate of expansion changes with time, depending on the matter and energy present. In the 2d sphere analogy, it tells you how the rate of expansion of the sphere changes with time. If ordinary matter and radiation are present, that rate of expansion will decrease with time, and that makes intuitive sense because the matter and radiation causes attractive gravity. Your question basically is, how is that consistent with the shell theorem?

    Let's go back to the 2d sphere analogy, and let's assume that the 2d sphere is expanding. From the viewpoint of an observer at some point on the 2-sphere (i.e., living in some particular galaxy), this expansion appears as all the other galaxies moving away from him. If we choose a small circular region around his galaxy, so small that no other galaxies are in it, then we see that the gravitational effects of all those other galaxies on him cancel out; he experiences no net force, and stays where he is, which makes sense. But this analysis can't tell us anything about whether, or how, the motion of other galaxies relative to him will change with time.

    In order to investigate that, we have to pick a circular region that is large enough to contain at least one other galaxy besides his. Let's suppose the galaxies around him are symmetric, so that if we enlarge the circle just enough, it will contain some number of galaxies evenly distributed around the edge of the circular region. The gravitational force of all these galaxies on our observer will still cancel, so he will still stay where he is. But the force his galaxy exerts on all the other galaxies does not cancel out; each of those galaxies will be pulled towards him by the gravity of his galaxy. Since all those other galaxies are moving away from him, the effect of the pull of his galaxy will be to decelerate them. And since the same analysis applies no matter which galaxy we pick to be at the center of a circular region, we conclude that the expansion as a whole will decelerate. And that is what the Friedmann equation describes (with one spatial dimension added), for the case of a universe filled with ordinary matter and energy.

    There isn't. The 3d space we live in does not have to be embedded in any higher-dimensional space in order to expand. Neither does a 2d sphere need to be embedded in a 3-space in order to expand; the fact that we visualize a 2-sphere embedded in a 3-space, or the fact that the 2-sphere we happen to live on is embedded in a 3-space, does not mean any 2-sphere must be embedded in a 3-space. Logically, mathematically, and physically, that simply is not a requirement.
     
  13. Nov 13, 2015 #12
    Ok, thanks for that. I can't say I fully understand that but will have a read on related material.

    I guess where my mind struggles is that I can see the natural properties of a 2d surface on a sphere lend itself to having a finite area and no boundaries, but I don't know of any 3d shape / geometry that is the same. All 3d shapes have boundaries, so that implies that our 3d universe must have a boundary or be infinite.
     
  14. Nov 13, 2015 #13
    I suppose what I was thinking was if we have 3 galaxies all of the same mass and all equidistant along some x axis, then the two outside galaxies would move towards the middle galaxy. I could place another 2 galaxies at either side of the end galaxies which would cancel out their effect, but the result would still be that the outside 2 galaxies would move towards the centre. I could keep adding galaxies in this way but the net effect would always be movement towards the centre unless there were an infinite amount of galaxies.

    However taking the 2d analogy, I can place a finite amount of galaxies on the surface of there sphere so they are isotropic and hence all the effects of gravity cancel out. (I've put more detail below)

    So in that case, the only way we could have a big crunch, would be if the galaxies were not all isotropic or some galaxies had a lot more mass than others.

    Thanks for the great explanation. But here is where I am still struggling to understand:

    Taking the 2d sphere analogy, if I place a finite amount of galaxies on the surface of the sphere in such a way that all galaxies are isotropic, i.e. all equidistant and all with the same mass, and I assume no expansion or other forces at work, then the net effect must be that all gravitational forces cancel out. No galaxies move towards each other. It can't be any other way.

    So if the sphere does expand and the distance between galaxies grow, as the surface still remains isotropic, then there is still no net effect of gravity that could exert a force on the expansion. Does that make sense?

    Now I could imagine that if we could assume another dimension (inside the sphere) and that gravity could permeate through that other dimension, then all the mass from all the galaxies would also be centred in the middle of the sphere. And that force would of course have an effect on the expansion. So if there was some initial force that 'inflated' the sphere in such a way that the force of expansion was just slightly greater than the net effect of gravity from all the galaxies accelerating towards the centre of the sphere (or in other words the outward force was just slightly greater than the escape velocity), then I could see how the rate of expansion would be effected and slow over time but never reaching zero. Which in some way I say analogous to our universe.

    Of course that is just my simple way of looking at it and in any case I'd have no idea how that would transform to a 3d universe.
     
  15. Nov 13, 2015 #14
    Reading this thread was useful for me too.
     
  16. Nov 13, 2015 #15

    PeterDonis

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    No, it isn't. If you take a universe like that, which is not expanding or contracting at some instant, then in the next instant, it will start contracting. That's what the Friedmann equations say.

    Deriving that result from the simplified model I gave, that you quoted, is easy: just consider the same circular region around a chosen galaxy, just large enough that there are some number of other galaxies at the edge of the region. In what you quoted, I assumed those galaxies were all moving away from the chosen galaxy, because that's the case in our actual universe. In that case, the other galaxies will decelerate, relative to the chosen galaxy, because of the chosen galaxy's gravity.

    But if we assume, instead, that all the other galaxies are at rest relative to the chosen galaxy, then the chosen galaxy's gravity will cause them to accelerate inward, towards the chosen galaxy. And since that is true regardless of which galaxy we choose, the universe as a whole must begin contracting.

    Yes, it can. You have not responded to the argument I actually made; you're just ignoring it. Don't. Go back and re-read it.
     
  17. Nov 13, 2015 #16
    Sorry, didn't mean to ignore it, just didn't understand it fully.

    I'm not sure it was but I am probably not making myself clear, sorry. The issue I am struggling to understand is how can an isotropic and homogeneous universe with matter it 'contract' unless there is a finite amount of matter in it. (see my rationale below) And a finite amount of matter suggests that there are boundaries, at least as far as how matter can be distributed in that universe.

    Would you mind helping me understand this better as I think it is where a lot of my confusion comes from. Accepting what the Friedmann equations say, when I think about our universe 'contracting', I think of it as if we place all the galaxies on some imaginary coordinate system. The ratio of the distances between the galaxies always stays the same but the actual distances between them reduce. But for this to happen then doesn't it mean that there must be a finite amount of matter / energy in the universe? Otherwise it would be impossible to trace the universe back to a singularity, as the universe would never stop contracting if there were an infinite amount of matter.

    Also, taking the 2d analogy again, when we talk about contraction I assumed it meant that the galaxies are not moving around on the surface of the sphere, like above it meant that the distances between them that are getting smaller. But for that to happen then the contraction is moving the galaxies towards the centre of the sphere, so the sphere is 'shrinking', which in the analogy would mean the galaxies are moving into a space that was previously in a different dimension. So are we saying that as our 3d universe contracts that it is moving into a space that was previously in a different dimension too?

    Maybe if you could help clear these two points up then I think the rest would fall into place. Again sorry for the my confusion.
     
  18. Nov 13, 2015 #17

    PeterDonis

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    No, it doesn't. This has already been addressed. The fact that you find it difficult to imagine a universe with spatial sections that are 3-spheres, containing a finite amount of matter but without any boundary, does not mean that model is not perfectly valid and self-consistent. It is.

    This is a reasonable description, yes.

    No. There are self-consistent contracting models with finite spatial slices (3-spheres without boundary, as I noted above), and also with infinite spatial slices. Both are perfectly valid models.

    This is not correct. Contracting models with infinite spatial slices still have a singularity. The singularity theorems of Hawking & Penrose, proved in the late 1960's and early 1970's, cover this case.

    What may be confusing you is the claim often made in pop science treatments that the singularity has "zero size". That is not correct. The singularity is not actually part of spacetime; when we talk about a singularity in such models, what we are really talking about is a mathematical limit that can be taken, and the fact that certain quantities increase without bound in that limit, while others remain finite. In the case of a collapsing universe (with either finite or infinite spatial slices), invariants related to the spacetime curvature increase without bound when the limit is taken, but the proper time along the worldline of any "comoving" observer does not; it converges to a finite value in the limit. The latter property (which is called "geodesic incompleteness"--there are geodesics, in this case the freely falling worldlines of comoving observers, which end at a finite value of their affine parameter) is what makes the singularity a "singularity". But since the singularity is not actually part of spacetime, there is nowhere, in the model with infinite spatial slices, where an infinite slice suddenly "turns into" a zero-size slice. Every spatial slice that is actually part of the spacetime is infinite in this model.

    Correct.

    Yes.

    No. In the 2d analogy, the "space" inside and outside of the 2-sphere is not physically real. It's just a crutch that we need in order to visualize what is going on. The 2-sphere is a perfectly valid mathematical space in its own right and does not need to be embedded in a higher-dimensional space in order to exist. In the 2-d version of the model, the 2-sphere is the only "space" that exists, and it is perfectly possible for it to "contract" or "expand" without needing to be embedded in any other space.

    No. See above.
     
  19. Nov 13, 2015 #18
    ok, got that now. thanks.

    Ok, thanks. Not really understanding this (see below).

    Correct, I have no idea what you mean by spatial sections that are 3-sphere. So will need to do some research. However my point was not based on my lack of understanding per se, but on observations and logic. If the following is acceptable:

    then I imagine we place our current universe against some imaginary coordinate system as above. But as the universe contracts, and the distances between galaxies reduce, then for each time interval the I compare new positions of the galaxies to original starting points what would I see? I would see all the galaxies moving towards me I assume. And just and it is valid for any observer to see all galaxies moving away from them, then it must follow that is just as valid for any observer to see all galaxies moving towards them. But how can all the galaxies reduce to an infinite set of different coordinate points? As I understand it, there is nothing in 3d geometry that would allow this. If something is shrinking from all directions that it must end up at a single point... no?

    Just to be clear I am not stating any of the current models are wrong, I'm just trying to understand them.
     
  20. Nov 13, 2015 #19

    PeterDonis

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    Go back to the 2d sphere analogy again. If we include time, we don't have just one 2d sphere; we have a whole succession of them, one at each instant of time. Saying that the 2d sphere universe is "expanding" or "contracting" just means that the 2d spheres at different instants have different sizes. Each 2d sphere is a "spatial section" of the universe; the whole universe, from the GR point of view, is the "spacetime" comprised of all the 2d spheres, "stacked up" in the order in which they occur in time.

    The case of the 3-sphere universe is the same, just with one spatial dimension added. The whole spacetime is a succession of 3-spheres, one at each instant of time, whose sizes can vary. Each individual 3-sphere is a spatial section of this spacetime.

    How do you propose to do this comparison? You can't compare new positions to old positions, because there is nothing marking out the old positions; the only "markers" are the galaxies themselves, and by assumption the distances between them are decreasing. Those distances are the only "markers" you have, and saying that the universe is "contracting" is just another way of saying that those distances are decreasing with time.

    I'm not sure what you mean by this. Where in the model do you think this is happening?

    No. The 3-sphere never contracts to a point; the "singularity" that pop science presentations refer to as a point is not actually part of spacetime. It's just an abstract limit that can be taken, as I said before.

    Also, you have to stop thinking of the 3-sphere as embedded in some higher dimensional space. It isn't. It's a space in its own right. It isn't "shrinking from all directions" when it's contracting, like a balloon with the air escaping. It's just that 3-spheres at later instants of time are smaller (have less total volume) than 3-spheres at earlier times. That's all there is to it.
     
  21. Nov 14, 2015 #20
    I think I've got this now, thanks. Well at least I understand it is mathematical model equivalent to the surface of an n+1 dimensional ball. (Looked that up) The only thing that still confuses me is how we can be on the surface of a 3-sphere. Something can be on the surface of a 2-sphere, as it is a natural property of any 3 dimensional object to have a surface. So struggling to see how we can just assume an additional dimension exists?

    Also, I assume you don't mean that each 'slice' exist in time simultaneously?

    By measuring blue shift (Assuming a contracting universe) in the same way cosmologists have measured red shift and deduced the universe is expanding and was smaller in size in the past. So if I measure the blue shifts in a receding universe I can map out the positions of galaxies over time. The data would (as I understand it) tell me that all galaxies are receding 'towards' me. In much the same way we say now that all galaxies (on larger scales) are moving away from us.

    Currently anyone in any galaxy can say that all the galaxies are moving away equally from their point of view. So if Alice is in her galaxy and Bob in a different distant galaxy, then in a contracting universe Alice would observe all the galaxies moving towards her and Bob would observe all the galaxies moving towards him.

    However it is not possible for the all the galaxies to eventually converge at both Alice's and Bob's coordinate positions simultaneously. So at some point in time Alice's coordinate position and Bob's coordinate position must merge as the distance between them reduced to zero. (My rationale for this is below)

    I wasn't actually thinking of pop science to be honest. I was thinking of the 2-sphere analogy. In the 2-sphere analogy as the distance reduced between the galaxies, the galaxies are moving closer and closer to the centre of the 3 dimensional sphere. We agreed that in an earlier post. So at some point in time all the galaxies must converge into a single point, or at least try to. So if the analogy between 2-sphere and 3-sphere holds then it must be the same for our universe. Sort of like if I ask everyone on earth to simultaneously throw a ball into the air, then when the balls fall back to earth, the are all heading for the centre of mass.

    So I assumed all that mass must collapse into a singularity.


    I didn't realise I was? Can you give me an example?

    Ah, ok. That's confused me. We use the balloon analogy to explain the expanding universe with us being on the surface of the balloon. So why doesn't the analogy work in reverse?
     
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