B The observable and non-observable parts of the Universe

[PAllen]

Mathematically that doesn't imply what you might intuitively want it to imply.

For example, let's assume your $t=0$ scenario is valid. You claim to have a set of infinite extent. But, you are unable to find any two points whose distance is non-zero. This is a contradiction.

The mathematical answer is probably that in the limit you no longer have a valid manifold. You can't talk about distances unless you have a valid metric. And, in the limit, you no longer have a valid metric.

I’m actually not suggesting those satements to imply anything other than what they say. Just that those statements are important to understand and somewhat counterintuitive. Formulating them as Ned did, I agree is not mathematically valid, but I believe he really meant these underlying statements and improperly simplified them for a nontechnical article.

 

[PeterDonis]

You claim to have a set of infinite extent.

No, that's not what he claimed. The limiting procedure @PAllen described does not construct a set and say that it is "a spacelike slice at time $t = 0$". It only constructs a number and says that it has the same value (infinity) at $t = 0$ as it has for all $t > 0$. Of course this is a sloppy heuristic description, but it can be made precise without having to make any claims about an infinite set of points existing at $t = 0$.

That said, there is also a different way of proceeding, which is to use the method used to construct Penrose diagrams to add a "boundary" corresponding to $t = 0$ to the FRW spacetime manifold. This procedure does in fact lead to a boundary with an infinite set of distinct points in it, each one corresponding to a particular 3-tuple of comoving coordinates, i.e., to a distinct "spatial point" in each spacelike slice in the region $t > 0$. Physically, this corresponds to the fact that, in the limit $t \rightarrow 0$, every single spatial point in the FRW universe becomes causally disconnected from every other. We can then use the induced metric (if it can be called that--see further comments below) on the boundary to compute zero spatial volume for this infinite set of points (because, as I noted earlier, the sum of an infinite set of zeroes is still zero).

You claim to have a set of infinite extent. But, you are unable to find any two points whose distance is non-zero. This is a contradiction.

No, it isn't. It simply says that this...

You can't talk about distances unless you have a valid metric. And, in the limit, you no longer have a valid metric.

...is correct in the sense that there is no valid Riemannian metric on the infinite set of points at $t = 0$ constructed by the method I described above. But you do have a valid manifold, because having a valid manifold does not require having a valid Riemannian metric. And you do have a perfectly well-defined mathematical formula that assigns the value zero to the integral $\int \int \int a(t = 0) dx dy dz$ over this manifold. And this formula is derived from a thingie that looks like a metric, just not a Riemannian one.

So at this point we have a situation which, although describable consistently using math, has no description in ordinary language that really does justice to it.

 

[PeroK]

But you do have a valid manifold, because having a valid manifold does not require having a valid Riemannian metric. And you do have a perfectly well-defined mathematical formula that assigns the value zero to the integral $\int \int \int a(t = 0) dx dy dz$ over this manifold. And this formula is derived from a thingie that looks like a metric, just not a Riemannian one.

What is the definition of this metric "thingie"?

 

[PeterDonis]

What is the definition of this metric "thingie"?

$a(t) \left( dx^2 + dy^2 + dz^2 \right)$ for $t = 0$

 

[PeroK]

$a(t) \left( dx^2 + dy^2 + dz^2 \right)$ for $t = 0$

But $a(0) = 0$?

 

[PeterDonis]

But $a(0) = 0$?

Yes, that's why the thingie I wrote is not a valid Riemannian metric for $t = 0$. But it's still a perfectly well-defined mathematical formula.

 

[PAllen]

Most FLRW cosmologies have an irremovable singularity in that not only is there geodesic incompleteness, but curvature invariants become infinite, which prevents an extension of the manifold to remove geodesic incompleteness. However, the special case of a(t)=t, the Milne cosmology is interesting in relation to what @PeterDonis is discussing. Here, geodesic incompleteness is removable because curvature is identically zero everywhere, and a coordinate transform takes you to Minkowski coordinates. Here, the limiting t=0 surface (not considered part of the Milne model) is simply the light cone from t=0, spatial origin. This is topologically R3, because it is S2XR plus the central point. However, metrically, any way of computing its volume is zero because one component of the volume element must be light like. Here we are using the Minkowski metric which is perfectly well defined, and is a true (pseudoriemannian) metric.

 

[PainterGuy]

Thank you all of you!

I see that post #13, #14, #15 were more direct replies to my post. I appreciate and value your advice, and could see your point. There were many points about the big bang which had bothered me for years but I'm glad that I have the basic answers and can picture it better.

There is no doubt that a book should be the primary source of learning, and it gives you a detailed and coherent view of a subject. At the same you need to devote more time to go through a book which some of us find it hard for several reasons.

I have another question cosmic microwave background radiation, CMBR, which is closely related to the observable universe. Could I ask it here or start a new thread?

Thank you!

 

[PAllen]

Thank you all of you!

I see that post #13, #14, #15 were more direct replies to my post. I appreciate and value your advice, and could see your point. There were many points about the big bang which had bothered me for years but I'm glad that I have the basic answers and can picture it better.

There is no doubt that a book should be the primary source of learning, and it gives you a detailed and coherent view of a subject. At the same you need to devote more time to go through a book which some of us find it hard for several reasons.

I have another question cosmic microwave background radiation, CMBR, which is closely related to the observable universe. Could I ask it here or start a new thread?

Thank you!

Start a new thread.