Re: Constraints on spatial flatness.

[Chalky]

[[Mod. note -- When approving this article a few minutes ago, I
mistakenly deleted the last line. My apologies to all for the
mixup! Here is the full version. -- jt]]

On Jan 1, 12:05 pm, Igor Khavkine <igor...@gmail.com> wrote:

> Before your question can be answered, you have to first fix what you
> mean by "3D space", as different ways to slice 4D space-time into 3D
> space-like surfaces give different notions of "3D space" even for the
> same 4D space-time.
>
> In cosmology, there is a specific way to slice space-time that is
> commonly used. That is, 3D space is defined as slices of of constant
> cosmological time, which measures the proper time counted from the Big
> Bang along world-lines of observers stationary with respect to the
> homogeneous matter-energy background. It is the curvature of these
> spatial slices that is observed to be approximately zero (on scales
> where the homogeneity assumption is warranted).
>
> Now, what you said about "inclusion of time" and "such manifolds",
> does not make a whole lot of sense to me. So, I'll go ahead and guess
> that you are wondering about models that are alternative to standard
> general relativity that still fit cosmological observations. Sound
> about right?

Yes.

> Well, it's never possible to exclude all alternatives,
> but to date general relativity and homogeneous cosmological models fit
> observations better than other proposals, which do not include either
> relativity or homogeneity as assumptions.

This approach is relativistic, but pregeometric, and not published (to
date), being developed from ch.44 of MTW.

As for homogeneity, yes, this seems to be a vital assumption on the
large scale.
On the smaller scale, this could be debatable.

> Unfortunately, it's not
> clear what specific kinds of alternatives you are interested in.
> Perhaps you can be more specific.

Bit tricky, as unpublished, but hopefully the above will suffice for
now.

> Then you might get a more specific
> answer then.

This discussion is already progressing fruitfully, afaiac, helping me
to clarify a few things at the interface between geometry and (my
understanding of) pregeometry. I thank all contributors for that.

[Igor Khavkine]

On Jan 1, 9:07 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
> On Jan 1, 12:05 pm, Igor Khavkine <igor...@gmail.com> wrote:

> > Before your question can be answered, you have to first fix what you
> > mean by "3D space", as different ways to slice 4D space-time into 3D
> > space-like surfaces give different notions of "3D space" even for the
> > same 4D space-time.

[...]
> > Now, what you said about "inclusion of time" and "such manifolds",
> > does not make a whole lot of sense to me. So, I'll go ahead and guess
> > that you are wondering about models that are alternative to standard
> > general relativity that still fit cosmological observations. Sound
> > about right?
>
> Yes.

OK. When and whether to discuss such alternatives is up to you. In the
mean time, I'd like see you acknowledge that you have a good grasp on
the distinctions between "3D space" and "4D space-time", as well as
between "curvature" and "topology". Understanding these distinctions
is very important for asking informed questions about relativistic
cosmology and benefiting from the answers.

Igor

[Igor Khavkine]

On Jan 1, 9:07 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
> On Jan 1, 12:05 pm, Igor Khavkine <igor...@gmail.com> wrote:

> > Before your question can be answered, you have to first fix what you
> > mean by "3D space", as different ways to slice 4D space-time into 3D
> > space-like surfaces give different notions of "3D space" even for the
> > same 4D space-time.

[...]
> > Now, what you said about "inclusion of time" and "such manifolds",
> > does not make a whole lot of sense to me. So, I'll go ahead and guess
> > that you are wondering about models that are alternative to standard
> > general relativity that still fit cosmological observations. Sound
> > about right?
>
> Yes.

OK. When and whether to discuss such alternatives is up to you. In the
mean time, I'd like see you acknowledge that you have a good grasp on
the distinctions between "3D space" and "4D space-time", as well as
between "curvature" and "topology". Understanding these distinctions
is very important for asking informed questions about relativistic
cosmology and benefiting from the answers.

Igor

[Chalky]

On Jan 3, 9:52*am, Igor Khavkine <igor...@gmail.com> wrote:
> On Jan 1, 9:07 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
>
> > On Jan 1, 12:05 pm, Igor Khavkine <igor...@gmail.com> wrote:
> > > Before your question can be answered, you have to first fix what you
> > > mean by "3D space", as different ways to slice 4D space-time into 3D
> > > space-like surfaces give different notions of "3D space" even for the
> > > same 4D space-time.
>
> [...]
>
> > > Now, what you said about "inclusion of time" and "such manifolds",
> > > does not make a whole lot of sense to me. So, I'll go ahead and guess
> > > that you are wondering about models that are alternative to standard
> > > general relativity that still fit cosmological observations. Sound
> > > about right?
>
> > Yes.
>
> OK. When and whether to discuss such alternatives is up to you. In the
> mean time, I'd like see you acknowledge that you have a good grasp on
> the distinctions between "3D space" and "4D space-time",

Absolutely. 4D space-time is strongly curved on the scale of the
observable universe, sufficiently so for topologies such as PH
describes to be worth contemplating in this context.

3D space is not, and such considerations are irrelevant unless one
wishes to contemplate a gigantic spatial multiverse that extends far
beyond our observational event horizon. Arguably, that is the domain
of metaphysics, not physics, as such speculation is impossible to
check observationally.

> as well as
> between "curvature" and "topology".

Hmm, I think I understand the distinction, but further clarification
would not go amiss, particularly if my above response suggests I
don't.

> Understanding these distinctions
> is very important for asking informed questions about relativistic
> cosmology and benefiting from the answers.
>
> Igor

[Igor Khavkine]

On Jan 1, 9:07 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
> On Jan 1, 12:05 pm, Igor Khavkine <igor...@gmail.com> wrote:

> > Before your question can be answered, you have to first fix what you
> > mean by "3D space", as different ways to slice 4D space-time into 3D
> > space-like surfaces give different notions of "3D space" even for the
> > same 4D space-time.

[...]
> > Now, what you said about "inclusion of time" and "such manifolds",
> > does not make a whole lot of sense to me. So, I'll go ahead and guess
> > that you are wondering about models that are alternative to standard
> > general relativity that still fit cosmological observations. Sound
> > about right?
>
> Yes.

OK. When and whether to discuss such alternatives is up to you. In the
mean time, I'd like see you acknowledge that you have a good grasp on
the distinctions between "3D space" and "4D space-time", as well as
between "curvature" and "topology". Understanding these distinctions
is very important for asking informed questions about relativistic
cosmology and benefiting from the answers.

Igor

[Igor Khavkine]

On Jan 4, 3:56 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
> On Jan 3, 9:52 am, Igor Khavkine <igor...@gmail.com> wrote:
> > On Jan 1, 9:07 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
> > > On Jan 1, 12:05 pm, Igor Khavkine <igor...@gmail.com> wrote:

> > OK. When and whether to discuss such alternatives is up to you. In the
> > mean time, I'd like see you acknowledge that you have a good grasp on
> > the distinctions between "3D space" and "4D space-time",
>
> Absolutely. 4D space-time is strongly curved on the scale of the
> observable universe, sufficiently so for topologies such as PH
> describes to be worth contemplating in this context.
>
> 3D space is not, and such considerations are irrelevant unless one
> wishes to contemplate a gigantic spatial multiverse that extends far
> beyond our observational event horizon. Arguably, that is the domain
> of metaphysics, not physics, as such speculation is impossible to
> check observationally.
>
> > as well as
> > between "curvature" and "topology".
>
> Hmm, I think I understand the distinction, but further clarification
> would not go amiss, particularly if my above response suggests I
> don't.

Well, further (or repeated) clarifications might be necessary. You've
once again, brought "3D space" into the discussion without specifying
what you mean by that. This point was brought up several times in the
thread, most recently by Stephen Carlip in his last post. There is no
unique way to define "3D space" given a 4D space-time. And when I say
not unique, I actually mean that there are infinitely many
possibilities. So, unless you specify which possibility you have in
mind, you're only confusing everyone.

Now, the simplest way to get a notion of 3D space from a 4D space-time
is to pick a function (called a time function), which must satisfy
some
regularity conditions, whose level sets are space-like 3D surfaces.
Each
one of these surfaces is called a 3D space (at fixed time). These
surfaces are slices and the whole procedure slicing. Quite clearly,
there are infinitely many choices of the time function that will yeild
different inequivalent slicings. If there is something that's not
clear
about slices or slicings, now is the time to ask, as I think it might
be
very close to the core of the confusion here.

Now, you are up on slicing, you can tell us what kind of particular
slicig you are interested in. Perhaps you are interested in the
standard
cosmological slicing that I described earlier:

I wrote in a pervious post:
> In cosmology, there is a specific way to slice space-time that is
> commonly used. That is, 3D space is defined as slices of of constant
> cosmological time, which measures the proper time counted from the Big
> Bang along world-lines of observers stationary with respect to the
> homogeneous matter-energy background. It is the curvature of these
> spatial slices that is observed to be approximately zero (on scales
> where the homogeneity assumption is warranted).

But you still have to be specific. Either agree to use this one, or
specify your own.

Hope this helps.

Igor

[Igor Khavkine]

On Jan 4, 3:56 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
> On Jan 3, 9:52 am, Igor Khavkine <igor...@gmail.com> wrote:
> > On Jan 1, 9:07 pm, Chalky <chalkys...@bleachboys.co.uk> wrote:
> > > On Jan 1, 12:05 pm, Igor Khavkine <igor...@gmail.com> wrote:

> > OK. When and whether to discuss such alternatives is up to you. In the
> > mean time, I'd like see you acknowledge that you have a good grasp on
> > the distinctions between "3D space" and "4D space-time",
>
> Absolutely. 4D space-time is strongly curved on the scale of the
> observable universe, sufficiently so for topologies such as PH
> describes to be worth contemplating in this context.
>
> 3D space is not, and such considerations are irrelevant unless one
> wishes to contemplate a gigantic spatial multiverse that extends far
> beyond our observational event horizon. Arguably, that is the domain
> of metaphysics, not physics, as such speculation is impossible to
> check observationally.
>
> > as well as
> > between "curvature" and "topology".
>
> Hmm, I think I understand the distinction, but further clarification
> would not go amiss, particularly if my above response suggests I
> don't.

Well, further (or repeated) clarifications might be necessary. You've
once again, brought "3D space" into the discussion without specifying
what you mean by that. This point was brought up several times in the
thread, most recently by Stephen Carlip in his last post. There is no
unique way to define "3D space" given a 4D space-time. And when I say
not unique, I actually mean that there are infinitely many
possibilities. So, unless you specify which possibility you have in
mind, you're only confusing everyone.

Now, the simplest way to get a notion of 3D space from a 4D space-time
is to pick a function (called a time function), which must satisfy
some
regularity conditions, whose level sets are space-like 3D surfaces.
Each
one of these surfaces is called a 3D space (at fixed time). These
surfaces are slices and the whole procedure slicing. Quite clearly,
there are infinitely many choices of the time function that will yeild
different inequivalent slicings. If there is something that's not
clear
about slices or slicings, now is the time to ask, as I think it might
be
very close to the core of the confusion here.

Now, you are up on slicing, you can tell us what kind of particular
slicig you are interested in. Perhaps you are interested in the
standard
cosmological slicing that I described earlier:

I wrote in a pervious post:
> In cosmology, there is a specific way to slice space-time that is
> commonly used. That is, 3D space is defined as slices of of constant
> cosmological time, which measures the proper time counted from the Big
> Bang along world-lines of observers stationary with respect to the
> homogeneous matter-energy background. It is the curvature of these
> spatial slices that is observed to be approximately zero (on scales
> where the homogeneity assumption is warranted).

But you still have to be specific. Either agree to use this one, or
specify your own.

Hope this helps.

Igor