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PeterDonis
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They can't be. There are no spatial points in the universe with location infinity.If plus and minus infinity can be a physical thing
They can't be. There are no spatial points in the universe with location infinity.If plus and minus infinity can be a physical thing
The number line is infinite, but the difference of any two numbers is finite.
Thanks guys. I thought that I had fully internalized the idea that infinity is not a big number. But you've made my understand that my concept is still flawed.It is a bit of pure mathematics. It sounds contradictory but it isn't. It may require adjusting your intuition of what is meant by an infinite set.
I'm not sure "observable" makes sense during a time whenSo...,
What was the size of the observable universe at Baryogenesis epoch?
I'm not sure "observable" makes sense during a time when
- the universe was opaque to light, and
- the universe was expanding much faster than the speed of light.
1) NoDoes the infinite volume help with the flatness problem?
Measurements look flat because the volumes are so huge?
Also should the flatness look flatter as time goes on? The larger a sphere (for example) becomes the flatter the region you happen to be in looks? Geometrically?
It's particularly straightforward in terms of conformal time ##\eta## (defined by ##c dt = a d\eta##) and co-moving distance ##\chi## (defined by ##r = a\sin{\chi}## for positive curvature or ##r= a\sinh{\chi}## for negative curvature), in which case the particle horizon is defined by a line where ##\chi## varies in direct proportion to ##\eta##.
Not restricted to light. Neutrinos and gravitational waves may also be suitable observational media in principle. For very early times, one can linearly extrapolate backwards using the LCDM model to estimate the sizes of the Hubble radius (R) and the particle horizon radius (Dpar), in e.g.The size of the observable universe is set by the particle horizon, which is as far away as light could have travelled, in principle, between a suitable spacetime origin and the current cosmic time ##t##. So I don't believe the initial "opacity" of the universe is relevant to this definition.
[ https://www.livescience.com/how-real-is-the-multiverse ; includes recent updates on the theory.]Physicists suspect that eternal inflation is generic, meaning a consequence of most, if not all, models of inflation. So, following this suspicion, if inflation is correct, then eternal inflation is also likely correct, and the multiverse might be real.
What kind of proof do have for that statement?The universe does not exist inside anything, it's all there is.
It is a definition.What kind of proof do have for that statement?
Perhaps a more precise statement would be that every phenomenon that we know of can be modeled without embedding spacetime in something else. Thus Occam's Razor currently disfavours anything "beyond" or "outside" spacetime.What kind of proof do have for that statement?
From that statement alone, the mediocrity principle (basically "we are not special") would suggest that there are things beyond or outside of our space-time.Perhaps a more precise statement would be that every phenomenon that we know of can be modeled without embedding spacetime in something else. Thus Occam's Razor currently disfavours anything "beyond" or "outside" spacetime.
Actually, the answer is definitely yes. There are analogs of the Nash embedding theorems for pseudo-Riemannian manifolds.But thinking about it in that simplistic way is nevertheless misleading. Take the expansion itself. Often people curvature of space-time must be relative to some "uncurved" outside description. For an example of how this might occur, consider the curvature of the Earth's surface. This curvature exists because the Earth's surface is a two-dimensional surface embedded in three-dimensional universe. The question arises: can the curvature of General Relativity be described by imagining it embedded in some larger number of dimensions? The answer is most definitely no.
It does require a large number of dimensions for the general case ( for a smooth embedding; for a C1 embedding, very few extra dimensions are needed) but these dimensions would be unobservable, just as a being within a 2-sphere is unaware of an embedding in 3-space. Embedding doesn’t change the geometry of a manifold, it just provides a different way of describing it. It also allows definition of things which are presumably unobservable and embedding dependent like extrinsic curvature.You can "flatten" certain geometries in GR by describing them in a higher number of (flat) dimensions. But I don't think this is something you can do generally. And even if you could, it would require a large number of large extra dimensions, something which I'm pretty sure is ruled out by observations.
Quite a few authors for GR first introduce embedding and extrinsic curvature first, then derive intrinsic curvature. Personally, I always found this a waste of time, but some teachers prefer it pedogogically. But, hey, this is the way Gauss went about it. Who am I to question Gauss?All this is to say, the only way that curvature even makes sense in General Relativity is to describe the curvature internally only, without reference to some kind of uncurved background. This extends to our concepts of the universe as well: even if there are things "outside" of our space-time, the relationship between those structures and our own is completely different from what you'd expect from the naïve view.
Where and when did Gauss learn GR?!Quite a few authors for GR first introduce embedding and extrinsic curvature first, then derive intrinsic curvature. Personally, I always found this a waste of time, but some teachers prefer it pedogogically. But, hey, this is the way Gauss went about it. Who am I to question Gauss?
Gauss developed theory of extrinsic and intrinsic curvature of surfaces and showed equivalence of embedding approach and intrinsic approach.Where and when did Gauss learn GR?!
A man ahead of his time!Gauss developed theory of extrinsic and intrinsic curvature of surfaces and showed equivalence of embedding approach and intrinsic approach.
I don't get how to reconcile these two ideas:
- the universe may be infinite in size
- the observable universe grew from a very small volume to a very large volume
Does not rhyme. Also, meter is off.I don't know, but I've been told, that if the curvature is non-zero that it increases over time.
What you can say is that given the volume enclosing some amount of matter (galaxies) in the present, then the volume enclosing that same matter decreases as you go back in time to the initial state we call the big bang. In a pure FLRW solution (i.e. no inflation, etc.) there is no lower bound to this volume. Despite this, the universe as a whole can still be infinite at all times. It's just a feature of 'infinite' - no matter how many times you cut it in half, it is still infinite.The idea is that the volume has always been infinite. So it isn't possible to compare two sizes directly. We CAN however measure and compare the densities at two different times.
This is not generally true. In a wide range of pure FLRW solutions, the curvature scalar R grows without bound as time is followed backwards. I am not aware of any general tendency for curvature to increase over time.I don't know, but I've been told, that if the curvature is non-zero that it increases over time. So for it to be so close to zero now it must have been much closer to zero back then. Or so they say.
This is referring to a particular definition of spatial curvature (and in a particular type of model, see below), not spacetime curvature. As @PAllen posted, the spacetime curvature of any expanding FRW solution will decrease with time.I've been told, that if the curvature is non-zero that it increases over time.
I see. So it isn't so much that it's impossible, but rather that it's ridiculously complicated, and disfavored for that reason. At least when plain GR is concerned.Actually, the answer is definitely yes. There are analogs of the Nash embedding theorems for pseudo-Riemannian manifolds.
It does require a large number of dimensions for the general case ( for a smooth embedding; for a C1 embedding, very few extra dimensions are needed) but these dimensions would be unobservable, just as a being within a 2-sphere is unaware of an embedding in 3-space. Embedding doesn’t change the geometry of a manifold, it just provides a different way of describing it. It also allows definition of things which are presumably unobservable and embedding dependent like extrinsic curvature.
Quite a few authors for GR first introduce embedding and extrinsic curvature first, then derive intrinsic curvature. Personally, I always found this a waste of time, but some teachers prefer it pedogogically. But, hey, this is the way Gauss went about it. Who am I to question Gauss?
I am wondering if someone can help me with the size of the universe at
- Quark epoch (10^{-5} seconds)
- Photon epoch (380,000 years)
Thank you very much @JimJCWUsing Jorrie’s calculator and PLANCK Data (2015), the size of the observable universe is calculated to be
R = 0.87 Mly at t = 0.38 Myr
From the extrapolation of a linear log(R) vs. log(t) plot, the size of the observable universe is estimated to be
R = 3 km at t = 10 µs
@Jorrie