Constraints on spatial flatness.

[Chalky]

I vaguely recall a comment made by Philip Helbig a year or 2 ago
(either at SPR or SAR), to the effect that the observational evidence
constrains the universe to be at least nearly flat, but not
necessarily precisely flat.

I would be interested to learn both a ball park figure for what that
nearly flat constraint actually is, numerically, and what the relevant
astrophysical evidence is, which establishes that constraint.

As someone who has never completely understood/come to terms with the
conventional geometrical interpretation of the GR axioms, I would also
appreciate clarification of where that leaves us locally.

Does this mean that when we understand local gravity to be a
consequence of spacetime curvature (for local physics), it also means
that this too should be understood within the constraint that this
geometry is spatially flat (or nearly so), or not?

[Phillip Helbig---remove CLOTHES to reply]

In article
<b6f69229-8df5-4835-a5de-a04e74030296@w1g2000prk.googlegroups.com>,
Chalky <chalkyspam@bleachboys.co.uk> writes:

> I vaguely recall a comment made by Philip Helbig a year or 2 ago
> (either at SPR or SAR), to the effect that the observational evidence
> constrains the universe to be at least nearly flat, but not
> necessarily precisely flat.
>
> I would be interested to learn both a ball park figure for what that
> nearly flat constraint actually is, numerically, and what the relevant
> astrophysical evidence is, which establishes that constraint.

As a matter of principle, we can never prove that the universe is
perfectly flat, even if it is. However, we can prove, in principle,
that the universe is not perfectly flat, assuming it is not. Basically,
if the sum of lambda (cosmological constant, "dark energy", "smooth
tension") and Omega (density parameter) is 1, then the universe is
spatially flat. Both quantities, and their sum, have some error
associated with them, so that we don't know EXACTLY what the sum is. We
can only say whether that error contains the flat case or rules it out.

At present, the perfectly flat case is still allowed. Maybe the
universe is perfectly flat, we don't know. Future observations will
decrease the size of the error. If the universe is perfectly flat, then
that will always be compatible with the observations. If not, at some
point the error will become so small that we can rule out the flat case.

Observationally, the main constraint are observations of the CMB. The
interesting thing, though, is that the flat case (not just A flat case,
but a PARTICULAR flat case, namely with lambda about 0.7 and Omega about
0.3) seems to fit ALL the observations. Various combinations of
observations give different errors, of course, but all seem compatible
with one another. Do a search on "cosmic data fusion", "combined
cosmological constraints" etc. Depending on the observations used, the
errors etc I believe the error on the sum is of the order of a few per
cent. Calculating a meaningful error is almost as difficult as doing
the observations in the first place!

> As someone who has never completely understood/come to terms with the
> conventional geometrical interpretation of the GR axioms, I would also
> appreciate clarification of where that leaves us locally.

I'm not sure what you mean here. The curvature mentioned above is a
local curvature, but if we believe the universe is homogeneous, then it
has the same value everywhere. (Note: the GLOBAL topology is not
constrained by just the sum of lambda and Omega, though other CMB
observations can constrain it.)

> Does this mean that when we understand local gravity to be a
> consequence of spacetime curvature (for local physics), it also means
> that this too should be understood within the constraint that this
> geometry is spatially flat (or nearly so), or not?

Don't confuse the curvature of space (discussed above) with the
curvature of spacetime (the GR approach to gravity).

[Chalky]

In addition to my already submitted response, there are a couple
additional points which I would like to address below.

On Dec 28, 7:20*pm, hel...@astro.multiCLOTHESvax.de (Phillip Helbig---
remove CLOTHES to reply) wrote:

> As a matter of principle, we can never prove that the universe is
> perfectly flat, even if it is. *However, we can prove, in principle,
> that the universe is not perfectly flat, assuming it is not. *Basically,
> if the sum of lambda (cosmological constant, "dark energy", "smooth
> tension") and Omega (density parameter) is 1, then the universe is
> spatially flat. *Both quantities, and their sum, have some error
> associated with them, so that we don't know EXACTLY what the sum is. *We
> can only say whether that error contains the flat case or rules it out.

True. Having said that, we can arguably now rule flatness _in_ on
philosophical grounds, with the assistance of Occam's Razor.
After all, most physicists remained happy, for more than half a
century, in relation to the shaving off of "Einstein's greatest
blunder", based on weaker observational constraints than this.

> Observationally, the main constraint are observations of the CMB. *

That is handy to know, since I am already familiar with those
conclusions.

>The interesting thing, though, is that the flat case (not just A flat case,
> but a PARTICULAR flat case, namely with lambda about 0.7 and Omega about
> 0.3) seems to fit ALL the observations.

Not exactly true, at least for (Sn1a) standard candle evidence.

Yes, it is very close to the best _flat_ fit you can get by
optimally adjusting the retrofit parameters of EFE, but this does not
mean it actually does fit, unless you are prepared to reject all of
the statistically strongest observational evidence which demonstrates
a lack of fit. (You can always get a square peg into a round hole if
you are prepared to whittle away at the sharp edges vigorously enough)

Kowalski et all applied a 3 sigma outlier cut to a GR model where
Omega was allowed to float, for optimum (blind) fit to data. The
result was a best fit model which exhibited strong curvature. To get a
reasonable fit to WMAP, they had to apply a 2 sigma outlier cut to the
remaining 315 supernovae which had passed every other test for
positively identifying them as type 1a.

You don't have to be a stats whiz kid to work out from this that at
least 15 supernovae must have thereby been rejected, for no better
reason than that they demonstrated the strongest statistical evidence
that GR theory did not fit the facts.

I am not suggesting here that this means space is strongly curved. I
prefer to conclude from this that GR is not the embodiment of
mathematical perfection that some people still assume.

> The curvature mentioned above is a
> local curvature, but if we believe the universe is homogeneous,

But we don't believe that, except as a large scale approximation. We
merely use it as a simplifying assumption during theoretical
cosmological derivations.

IIRC, computer simulations of matter distribution on the scale of the
universe, indicate that this approximation is only valid on scales
greater than ~10% of its radius.

> then it
> has the same value everywhere. *

This is merely a theoretical assumption, AFAICT.

Given that we are talking about an unexplained "dark physics"
parameter which had been thoroughly debunked by the '70's, and only
resurrected when we found that long distance observational evidence
still did not fit GR theory, that assumption would seem to be
particularly weak, especially on scales less than 0.1 R (unless you
know of specific evidence to the contrary, of course :-).

[Chalky]

On Dec 28, 7:20 pm, hel...@astro.multiCLOTHESvax.de (Phillip Helbig---
remove CLOTHES to reply) wrote:
> Depending on the observations used, the
> errors etc I believe the error on the sum is of the order of a few per
> cent.

That sounds like an extremely tight constraint on deviation from
flatness on the scale of the universe to me, and better news than I
was expecting.

> In article
> <b6f69229-8df5-4835-a5de-a04e74030...@w1g2000prk.googlegroups.com>,
>
> Chalky <chalkys...@bleachboys.co.uk> writes:

> > As someone who has never completely understood/come to terms with the
> > conventional geometrical interpretation of the GR axioms, I would also
> > appreciate clarification of where that leaves us locally.
>
> I'm not sure what you mean here.

I was merely enquiring whether that spatial curvature (or lack of it)
also applies close to, for example, the surface of the Sun, or the
event horizon of a black hole.

I, in my turn, am not sure whether you have addressed that question,
or not, thus far.

> The curvature mentioned above is a local curvature,

Now you are really confusing me. We have known, to an accuracy of
better than a few per cent, that the sum of the internal angles of a
triangle add up to 180 degrees, for millennia.

> > Does this mean that when we understand local gravity to be a
> > consequence of spacetime curvature (for local physics), it also means
> > that this too should be understood within the constraint that this
> > geometry is spatially flat (or nearly so), or not?

> Don't confuse the curvature of space (discussed above) with the
> curvature of spacetime (the GR approach to gravity).

I don't think I am.

I am merely trying to confirm that, to the limits of observational
accuracy, spacetime curvature as induced by massenergy, can be
interpreted exclusively as a consequence of the inclusion of the
dimension of time within such manifolds.

[Phillip Helbig---remove CLOTHES to reply]

In article <gjat0d$4tt$1@fb07-hees.theo.physik.uni-giessen.de>, Chalky
<chalkyspam@bleachboys.co.uk> writes:

> > As a matter of principle, we can never prove that the universe is
> > perfectly flat, even if it is. However, we can prove, in principle,
> > that the universe is not perfectly flat, assuming it is not. Basically,
> > if the sum of lambda (cosmological constant, "dark energy", "smooth
> > tension") and Omega (density parameter) is 1, then the universe is
> > spatially flat. Both quantities, and their sum, have some error
> > associated with them, so that we don't know EXACTLY what the sum is. We
> > can only say whether that error contains the flat case or rules it out.
>
> True. Having said that, we can arguably now rule flatness _in_ on
> philosophical grounds, with the assistance of Occam's Razor.
> After all, most physicists remained happy, for more than half a
> century, in relation to the shaving off of "Einstein's greatest
> blunder", based on weaker observational constraints than this.

True, but they were wrong. The simplicity argument, Occam's razor etc
were also the reason many people believed so strongly in the Einstein-de
Sitter universe that they ignored observational data to the contrary for
as long as possible (or even longer, in some cases). Thus, the
simplicity argument doesn't have a good history in cosmology. (Back in
the 70s, Carl Sagan collaborated with Schlovskii on a book about
extraterrestrial life, and mentions in the introduction that he has let
stand without comments some statements to the fact that dialectical
materialism requires there to be life on every planet. OK, Occam's
razor isn't that bad, but it is sharp and one has to be careful not to
cut oneself with it.)

On the other hand, there is a history of various principles, paradigms
etc in cosmology. In the past, this was due to the fact that there were
little observational data (in the words of Malcolm Longair, when he was
starting out there were only two-and-one-half facts in cosmology).
However, today cosmology is a data-driven science. Why philosophise
about something when one can measure it?

Of course, I'm a fan of Occam's razor and it is a good tool and is the
primary reason I reject many "alternative cosmologies". However, is
flatness SO much simpler that it is absurd to consider alternatives,
even if a perfectly flat case matches the data?

In the past, especially before computers, a simple model was often
assumed just to make calculations easier. That's really not necessary
today.

> >The interesting thing, though, is that the flat case (not just A flat case,
> > but a PARTICULAR flat case, namely with lambda about 0.7 and Omega about
> > 0.3) seems to fit ALL the observations.
>
> Not exactly true, at least for (Sn1a) standard candle evidence.
>
> Yes, it is very close to the best _flat_ fit you can get by
> optimally adjusting the retrofit parameters of EFE, but this does not
> mean it actually does fit, unless you are prepared to reject all of
> the statistically strongest observational evidence which demonstrates
> a lack of fit. (You can always get a square peg into a round hole if
> you are prepared to whittle away at the sharp edges vigorously enough)
>
> Kowalski et all applied a 3 sigma outlier cut to a GR model where
> Omega was allowed to float, for optimum (blind) fit to data. The
> result was a best fit model which exhibited strong curvature. To get a
> reasonable fit to WMAP, they had to apply a 2 sigma outlier cut to the
> remaining 315 supernovae which had passed every other test for
> positively identifying them as type 1a.
>
> You don't have to be a stats whiz kid to work out from this that at
> least 15 supernovae must have thereby been rejected, for no better
> reason than that they demonstrated the strongest statistical evidence
> that GR theory did not fit the facts.

Each cosmological test has its own best fit and its own error zone (for
definiteness, think of a 2-sigma contour surrounding the best fit in the
lambda-Omega plane). For various reasons, the actual best fits might be
different, influenced by statistical noise or the fact that a certain
test is more sensitive in a certain direction than another (on this
note, the CMB and the SNIa results provide almost orthogonal constraints
in the lambda-Omega plane). As long as one best fit is within the
errors of the other, I don't see a problem.

> > The curvature mentioned above is a
> > local curvature, but if we believe the universe is homogeneous,
>
> But we don't believe that, except as a large scale approximation. We
> merely use it as a simplifying assumption during theoretical
> cosmological derivations.

The local curvature is constant on the same scale as the universe is
homogeneous.

> Given that we are talking about an unexplained "dark physics"
> parameter which had been thoroughly debunked by the '70's,

What debunking are you referring to? It was dropped to give
mathematically simpler models when there was no strong evidence for it.
I don't see that as debunking.

> and only
> resurrected when we found that long distance observational evidence
> still did not fit GR theory,

The cosmological constant IS part of GR theory. The universe doesn't
care what Einstein later thought about it. Had someone else come up
with GR, they might have had the cosmological constant in from the
start. The contingency of the historical development of science is
interesting, but the universe doesn't care about it.

[Phillip Helbig---remove CLOTHES to reply]

In article
<fccb96d1-a939-4f1f-a01c-43b132718c54@i24g2000prf.googlegroups.com>,
Chalky <chalkyspam@bleachboys.co.uk> writes:

> On Dec 28, 7:20 pm, hel...@astro.multiCLOTHESvax.de (Phillip Helbig---
> remove CLOTHES to reply) wrote:
> > Depending on the observations used, the
> > errors etc I believe the error on the sum is of the order of a few per
> > cent.
>
> That sounds like an extremely tight constraint on deviation from
> flatness on the scale of the universe to me, and better news than I
> was expecting.

Assuming a simple topology, only a positive curvature implies a finite
universe. Thus, it would be nice to know the curvature enough to know
if the universe is finite or not. If it is exactly 0, of course, we
will never know this.

> > > As someone who has never completely understood/come to terms with the
> > > conventional geometrical interpretation of the GR axioms, I would also
> > > appreciate clarification of where that leaves us locally.
> >
> > I'm not sure what you mean here.
>
> I was merely enquiring whether that spatial curvature (or lack of it)
> also applies close to, for example, the surface of the Sun, or the
> event horizon of a black hole.

Local concentrations of matter will have local effects.

> I, in my turn, am not sure whether you have addressed that question,
> or not, thus far.

No, not yet (except for the previous sentence).

> > The curvature mentioned above is a local curvature,
>
> Now you are really confusing me. We have known, to an accuracy of
> better than a few per cent, that the sum of the internal angles of a
> triangle add up to 180 degrees, for millennia.

Yes, but only locally. :-) Sorry for the confusion. What I meant is
that the spatial curvature normally mentioned in cosmology is a local
curvature (even if measured globally). There can be a GLOBAL curvature,
e.g. a torus which is flat locally but not Euclidean globally.

Whether the angles of a triangle the size of the universe add up to 180
degrees is the interesting question. Even if they don't, then locally
they will (as nearly as we can tell), just as a flat map is OK for your
bicycle tour but not for looking at the route of an around-the-world
trip.

> > > Does this mean that when we understand local gravity to be a
> > > consequence of spacetime curvature (for local physics), it also means
> > > that this too should be understood within the constraint that this
> > > geometry is spatially flat (or nearly so), or not?

Flat space is a good approximation.

> > Don't confuse the curvature of space (discussed above) with the
> > curvature of spacetime (the GR approach to gravity).
>
> I don't think I am.
>
> I am merely trying to confirm that, to the limits of observational
> accuracy, spacetime curvature as induced by massenergy, can be
> interpreted exclusively as a consequence of the inclusion of the
> dimension of time within such manifolds.

I'm not sure exactly what you mean here. Maybe someone else can chime
in.

[carlip-nospam@physics.ucdavis.edu]

Chalky <chalkyspam@bleachboys.co.uk> wrote:

[...]
> I was merely enquiring whether that spatial curvature (or lack of it)
> also applies close to, for example, the surface of the Sun, or the
> event horizon of a black hole.

The spatial curvature people refer to when they say "the universe is
nearly spatially flat" is the average curvature, averaged on the same
scale that one can say "the universe is nearly sptially homogeneous."

(There's a lot of current research on the question of what, exactly, it
means to average small scale curvature to get this large scale result.
It's not trivial, both because the large scale curvature is measured
at a constant cosmological time, with an uncertain relationship to
locally measured times, and because there are always ambiguities
in averaging tensors. Most people think the averaging process
shouldn't have much physical effect, but a few think it might have
important implications for how we interpret observations.)

Steve Carlip

[Chalky]

In the hope that this may be of assistance to anyone else tempted to
chime in:

On Dec 30, 5:37*pm, hel...@astro.multiCLOTHESvax.de (Phillip Helbig---
remove CLOTHES to reply) wrote:
> In article
> <fccb96d1-a939-4f1f-a01c-43b132718...@i24g2000prf.googlegroups.com>,
>
> Chalky <chalkys...@bleachboys.co.uk> writes:

> > *I am merely trying to confirm that, to the limits of observational
> > accuracy, spacetime curvature as induced by massenergy, can be
> > interpreted exclusively as a consequence of the inclusion of the
> > dimension of time within such manifolds.
>
> I'm not sure exactly what you mean here. *Maybe someone else can chime
> in.

I mean, are we justified in concluding that, despite the curvature of
spacetime (sic), 3D _space_, per se , remains "flat" (ie
Euclidean) , to within the limits of observational evidence for both
local and global dynamics.

[Chalky]

On Dec 30, 5:37 pm, hel...@astro.multiCLOTHESvax.de (Phillip Helbig---
remove CLOTHES to reply) wrote:
> In article
> <fccb96d1-a939-4f1f-a01c-43b132718...@i24g2000prf.googlegroups.com>,
>
> Chalky <chalkys...@bleachboys.co.uk> writes:

> > I was merely enquiring whether that spatial curvature (or lack of it)
> > also applies close to, for example, the surface of the Sun, or the
> > event horizon of a black hole.
>
> Local concentrations of matter will have local effects.

Is that a yes or a no?

> > I, in my turn, am not sure whether you have addressed that question,
> > or not, thus far.
>
> No, not yet (except for the previous sentence).

So, it is still no, not yet, AFAICT :-)

> > > The curvature mentioned above is a local curvature,
>
> > Now you are really confusing me. We have known, to an accuracy of
> > better than a few per cent, that the sum of the internal angles of a
> > triangle add up to 180 degrees, for millennia.
>
> Yes, but only locally. :-) Sorry for the confusion. What I meant is
> that the spatial curvature normally mentioned in cosmology is a local
> curvature (even if measured globally).

Let us try to keep this simple. I understood a 3% curvature margin of
error to mean + - 3% deviation from Euclidean spatial geometry on the
scale of the universe. Is that correct?

There can be a GLOBAL curvature,
e.g. a torus which is flat locally but not Euclidean globally.

I know.

However, I am not particularly interested in potential science
fiction scenarios that can be conjured up by creative use of Riemann
geometry.

I am only really interested in real physics that can be supported by
empirical observational evidence.

> Whether the angles of a triangle the size of the universe add up to 180
> degrees is the interesting question.

> Flat space is a good approximation.

You seem to imply there is something more accurate.

If so, what is it?

(Please, please don't fob me off with 2 dimensional analogies drawn on
a 3D sphere. [Just because I said I never really came to terms with
the geometrical interpretation of the GR axioms, that does not
necessarily mean I am an idiot])

[Chalky]

On Dec 30, 11:20*pm, carlip-nos...@physics.ucdavis.edu wrote:
> Chalky <chalkys...@bleachboys.co.uk> wrote:
>
> [...]
>
> > I was merely enquiring whether that spatial curvature (or lack of it)
> > also applies close to, for example, the surface of the Sun, or the
> > event horizon of a black hole.
>
> The spatial curvature people refer to when they say "the universe is
> nearly spatially flat" is the average curvature, averaged on the same
> scale that one can say "the universe is nearly spatially homogeneous." *

Thanks. This does fit with what I always thought it meant. However,
perhaps I should put my question slightly differently. On this large
scale we know that space is flat, or nearly so. On the large scale,
however, we also know that spacetime is strongly curved.

Now, on the small scale we know that spacetime is still strongly
curved in the vicinity of extremely massive objects. But what about
space here? Is it strongly curved, or nearly flat, or is it the case
that we can't really say one way or the other, because we are embedded
in, and interact with spacetime, not space?

> (There's a lot of current research on the question of what, exactly, it
> means to average small scale curvature to get this large scale result.
> It's not trivial, both because the large scale curvature is measured
> at a constant cosmological time.

Yes, that is the rub. So when we talk about the universe _now_ we are
not talking about anything we can actually see, but something we might
be able to see if we could communicate information at infinite
speed..........Or have I got that wrong?

>*Most people think the averaging process *
> shouldn't have much physical effect, but a few think it might have
> important implications for how we interpret observations.)
>
> Steve Carlip

[Phillip Helbig---remove CLOTHES to reply]

In article <gjfelo$3qe$1@fb07-hees.theo.physik.uni-giessen.de>, Chalky
<chalkyspam@bleachboys.co.uk> writes:

> I mean, are we justified in concluding that, despite the curvature of
> spacetime (sic), 3D _space_, per se , remains "flat" (ie
> Euclidean) , to within the limits of observational evidence for both
> local and global dynamics.

No. For example, the classical gravitational-lensing effect, bending of
light near a massive body, is due both to the curvature of space and to
time running slower in a deeper gravitational potential ("curvature of
space-time"). In his first calculations, Einstein considered only the
latter effect and got half the correct value. However, it is a very
small effect. Even massive galaxies produce image separations (a rough
measure for the strength of the gravitational lens effect) of a few
arcminutes. That's only, roughly speaking, an effect at the level of
one-tenth per mil or so. (This is much smaller than the observationally
allowed global deviation from flatness.)

[Jonathan Thornburg [remove -animal to reply]]

Chalky <chalkyspam@bleachboys.co.uk> wrote:
> Now, on the small scale we know that spacetime is still strongly
> curved in the vicinity of extremely massive objects. But what about
> space here? Is it strongly curved, or nearly flat, or is it the case
> that we can't really say one way or the other, because we are embedded
> in, and interact with spacetime, not space?

Mathematically, if we start with a given (4-dimensional) spacetime
which is "well-behaved" (I think "globally hyperbolic" is the right
property), we can consider foliations of it by 3-dimensional spacelike
slices.
[Such a foliation is just a partition of the spacetime
into a 1-parameter family of (disjoint) 3-dimensional
spacelike slices. In practice we usually want the slices
to depend smoothly on the parameter (a.k.a. the time coordinate).]
Then we can investigate the the 3-curvature of those slices. Such
foliations are highly non-unique, i.e. there are generally many
different such foliations ("slicings") of a given spacetime.
The choice of a foliation is equivalent (up to trivial 1-dimensional
reparameterizations) to the choice of a time coordinate (i.e. choose
the slices to be time_coordinate=constant).

For Schwarzschild spacetime, the usual slicing is the one associated
with the Schwarzscdhild time coordinate. With this slicing, space
(more precisely, the spatial slices of constant Schwarzschild time)
is indeed 3-curved near a massive body (for example a planet, star,
or black hole); this 3-curvature accounts for 1/2 of the calculated
light deflection for light passing near the body.

However, if we use Painleve-Gullstrand coordinates, then space
(more precisely, the spatial slices of constant Painleve-Gullstrand
time) is indeed 3-flat.

Thus we see that the 3-curvature of space varies depending on how we
choose our slicing, or equivalently how we choose our time coordinate.

For Kerr spacetime with a nonzero spin parameter, I believe there
are no slicings which are 3-flat everywhere, or even everywhere
outside the event horizon. But I am not quite sure of this.

[I don't know the proof, but I *am* sure that there are
no maximal (trace of 3-extrinsic curvature tensor = 0)
slicings of Kerr, but that's a somewhat different question.]

--
-- From: "Jonathan Thornburg [remove -animal to reply]" <jthorn@astro.indiana-zebra.edu>
Dept of Astronomy, Indiana University, Bloomington, Indiana, USA
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

[Calvin D. Ritchie]

There's a new book by Weinberg, "Cosmology", Oxford University Press,
2008, which discusses many of the points that you raise. There are
four sections to the book: (1) The Expansion of the Universe; (2) The
Cosmic Microwave Radiation Background; (3) The Early Universe, and;
(4) Inflation. In my non-expert opinion, it is up to the usual
Weinberg standard of excellence.

Don

[Igor Khavkine]

On Dec 31, 6:23 am, Chalky <chalkys...@bleachboys.co.uk> wrote:
> On Dec 30, 5:37 pm, hel...@astro.multiCLOTHESvax.de (Phillip Helbig) wrote:

> There can be a GLOBAL curvature,
> e.g. a torus which is flat locally but not Euclidean globally.

Point of terminology: there are two distinct ideas here, curvature and
topology. Both a torus and Euclidean space are flat, locally,
globally, or what will you. However, Euclidean space is topologically
trivial (in any reasonable sense), while the torus is not. The non-
triviality of the torus's topology can be evidenced by the presence of
non-contractible loops, of non-exact but closed differential forms,
etc.

> > Whether the angles of a triangle the size of the universe add up to 180
> > degrees is the interesting question.
> > Flat space is a good approximation.
>
> You seem to imply there is something more accurate.

Hmm, perhaps you're reading more into Phillip's comments than there is
to them. To be more precise, here's what homogeneous cosmological
models with flat slices of constant cosmological time imply for
triangles on the scale of the universe: the internal angles of a
triangle with space-like edges contained in a slice of constant
cosmological time do add up to 180 degrees, with high accuracy; the
internal angles of a triangle with time-like edges spanning
appreciable stretches of cosmological time generally do not add up to
180 degrees, at the same level of accuracy.

Hope this helps.

Igor

[Igor Khavkine]

On Dec 31, 4:40 am, Chalky <chalkys...@bleachboys.co.uk> wrote:
> On Dec 30, 5:37 pm, hel...@astro.multiCLOTHESvax.de (Phillip Helbig) wrote:
> > Chalky <chalkys...@bleachboys.co.uk> writes:
> > > I am merely trying to confirm that, to the limits of observational
> > > accuracy, spacetime curvature as induced by massenergy, can be
> > > interpreted exclusively as a consequence of the inclusion of the
> > > dimension of time within such manifolds.
>
> > I'm not sure exactly what you mean here. Maybe someone else can chime
> > in.
>
> I mean, are we justified in concluding that, despite the curvature of
> spacetime (sic), 3D _space_, per se , remains "flat" (ie
> Euclidean) , to within the limits of observational evidence for both
> local and global dynamics.

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? 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. Unfortunately, it's not
clear what specific kinds of alternatives you are interested in.
Perhaps you can be more specific. Then you might get a more specific
answer then.

Hope this helps.

Igor

[Calvin D. Ritchie]

Edit/correction to earlier post of 12/31/08 by Calvin D. Ritchie:

There are ten (not four) sections to Weinberg's "Cosmology".
(5) General Theory of Small Fluctuations; (6) Evolution of
Cosmological Fluctuations; (7) Anisotropies in the Microwave Sky; (8)
The Growth of Structure; (9) Gravitational Lenses; (10) Inflation as
the Origin of Cosmological Fluctuations.

[Chalky]

On Dec 31 2008, 6:03 pm, hel...@astro.multiCLOTHESvax.de (Phillip
Helbig---remove CLOTHES to reply) wrote:
> In article <gjfelo$3q...@fb07-hees.theo.physik.uni-giessen.de>, Chalky
>
> <chalkys...@bleachboys.co.uk> writes:
> > I mean, are we justified in concluding that, despite the curvature of
> > spacetime (sic), 3D _space_, per se , remains "flat" (ie
> > Euclidean) , to within the limits of observational evidence for both
> > local and global dynamics.
>
> No. For example, the classical gravitational-lensing effect, bending of
> light near a massive body, is due both to the curvature of space and to
> time running slower in a deeper gravitational potential ("curvature of
> space-time"). In his first calculations, Einstein considered only the
> latter effect and got half the correct value.

Hmm, I recall that Newtonian physics also predicted half the light
bending that Einstein did for starlight grazing the Sun.

Any connection here?

[Phillip Helbig---remove CLOTHES to reply]

In article
<5a8afdd3-6300-433f-ad4a-e6c352fc5b0f@o40g2000prn.googlegroups.com>,
Chalky <chalkyspam@bleachboys.co.uk> writes:

> > No. For example, the classical gravitational-lensing effect, bending of
> > light near a massive body, is due both to the curvature of space and to
> > time running slower in a deeper gravitational potential ("curvature of
> > space-time"). In his first calculations, Einstein considered only the
> > latter effect and got half the correct value.
>
> Hmm, I recall that Newtonian physics also predicted half the light
> bending that Einstein did for starlight grazing the Sun.
>
> Any connection here?

Sort of. One gets the Newtonian result (half the correct value) by
thinking of light as particles which have velocity c at infinity. Near
a body, they are accelerated (to above c---this is OK since we are using
Newtonian physics) and then decelerated when moving away (again reaching
velocity c at infinity). This is just a straightforward scattering
calculation, no curved space, no curved spacetime.

Why is the result the same as that due to spatial curvature or spacetime
curvature (time running slower in a gravitational potential well)?
Basically, for the same reason that Bohr's model of the atom gives a
very good approximation for the hydrogen spectrum, even though we now
know that it is not a good approximation for a hydrogen atom: There are
certain quantities involved, and only certain combinations result in
quantities of the appropriate dimension.

I strongly recommend

@BOOK {MBerry86a,
AUTHOR = "Michael V. Berry",
TITLE = "Cosmology and Gravitation",
PUBLISHER = "Adam Hilger",
YEAR = "1986",
ADDRESS = "Bristol"
}

To address a question in another post: After calculating that spatial
curvature at the surface of the Earth (NOT the curvature of the 2-D
surface of the Earth, whose radius of curvature is of course the radius
of the Earth) to be 6.97 x 10^{-6} times that of the curvature of the
surface of the Earth, the author concludes "it is not entirely
surprising that Gauss's observations on a triangle formed by three
mountains failed to detect the non-Euclidicity of space".

I recommend the whole book. The stuff about cosmology is dated
observationally, but still a good introduction. The stuff on GR is of
course still valid. In particular, chapter 4, especially the last
subchapter, is the next thing you should read. I think this book also
shows well the link between GR and cosmology. (Relativists tend to
think of cosmology as an application of GR, and cosmologists see GR as
just one element in cosmology, so it is rare that one gets a view which
does justice to both sides.)