Mike wrote:
How do the measurements (WMAP, etc.) fit with the notion of an
infinite or finite universe and in relation to GR?

At present, the best measurements on a cosmological scale show that the
universe is spatially flat to within a few percent. And no evidence is
seen for multiple images of a single object or region.

The implications directly affect answers to your question. If a) the FRW
manifolds of General Relativity are a good model of the universe at
cosmological distances, and b) the universe is EXACTLY spatially flat,
then it cannot be spatially compact. However if there is a small
positive spatial curvature, then it could be compact (with a small
negative spatial curvature it still cannot be spatially compact).

So the measurements and GR are consistent with the universe being
spatially infinite. But they cannot rule out the universe being
spatially closed with a scale beyond our cosmological horizon (~14 giga
lightyears).

If GR is not a good model of the universe at cosmological scales, then
we don't know much until a better model can be formulated. While there
are phenomena that bring the validity of GR into question, none have
been established to the point of being generally accepted as a
refutation of GR.

Tom Roberts

In sci.physics Tom Roberts <tjroberts...@sbcglobal.net> wrote:

Mike wrote:
How do the measurements (WMAP, etc.) fit with the notion of an
infinite or finite universe and in relation to GR?
At present, the best measurements on a cosmological scale show that the
universe is spatially flat to within a few percent. And no evidence is
seen for multiple images of a single object or region.
The implications directly affect answers to your question. If a) the FRW
manifolds of General Relativity are a good model of the universe at
cosmological distances, and b) the universe is EXACTLY spatially flat,
then it cannot be spatially compact.

This is not quite right.  A spatially flat universe can still have the
topology of a three-torus, which is spatially compact.

There are a number of searches underway for nontrivial topology.
including this possibility.  So far, there is no evidence for anything
except an infinite, spatially noncompact universe, but nontrivial
topology can't yet be excluded.

Steve Carlip

Jeff?Relf who wrote:

rect-Al wrote:

hanson wrote:

rect-Al wrote:

hanson wrote:

rect-Al wrote:

hanson wrote:

rect-Al wrote:

hanson wrote:

rect-Al wrote:

hanson wrote:

Thanks for the laughs, schmuck... ahahaha... ahahahanson

carlip-nospam@physics.ucdavis.edu wrote:
In sci.physics Tom Roberts <tjroberts137@sbcglobal.net> wrote:
a) the FRW
manifolds of General Relativity are a good model of the universe at
cosmological distances, and b) the universe is EXACTLY spatially flat,
then it cannot be spatially compact.

This is not quite right. A spatially flat universe can still have the
topology of a three-torus, which is spatially compact.

Hmmm. I know that a flat 3-space can have the topology of a 3-torus. But
I thought the spatially-flat FRW manifold was unique [#], with the
spatial topology R^3.

Is it possible to "cut and fold" that manifold to
become spatially compact, and still satisfy the field equation?

IOW: are
there actually multiple flat FRW manifolds with different topologies?

[#] up to isometry.

Topologically, I have no doubt that the spatial submanifold can be cut
into a cube with opposite faces identified, thus creating a flat
3-torus. Such manipulations have no effect on the validity of the field
equation except possibly in neighborhoods of the joins, and for the
initial consistency conditions the field equation includes. The original
manifold has sufficient spacelike Killing vectors that I don't think the
neighborhoods of the joins have any problems,

so this is really a
question about those initial consistency conditions, and whether the
periodic boundary conditions of a 3-torus can satisfy them. Perhaps
those same Killing vectors imply this,

In sci.physics Tom Roberts <tjroberts137@sbcglobal.net> wrote:
a) the FRW
manifolds of General Relativity are a good model of the universe at
cosmological distances, and b) the universe is EXACTLY spatially flat,
then it cannot be spatially compact.

This is not quite right. A spatially flat universe can still have the
topology of a three-torus, which is spatially compact.

There are a number of searches underway for nontrivial topology.
including this possibility. So far, there is no evidence for anything
except an infinite, spatially noncompact universe, but nontrivial
topology can't yet be excluded.

carlip-nos...@physics.ucdavis.edu wrote:

This is not quite right. A spatially flat universe can still have the
topology of a three-torus, which is spatially compact.

Hmmm. I know that a flat 3-space can have the topology of a 3-torus. But
I thought the spatially-flat FRW manifold was unique [#], with the
spatial topology R^3.

Is it possible to "cut and fold" that manifold to
become spatially compact, and still satisfy the field equation? IOW: are
there actually multiple flat FRW manifolds with different topologies?

[#] up to isometry.

Topologically, I have no doubt that the spatial submanifold can be cut
into a cube with opposite faces identified, thus creating a flat
3-torus. Such manipulations have no effect on the validity of the field
equation except possibly in neighborhoods of the joins, and for the
initial consistency conditions the field equation includes. The original
manifold has sufficient spacelike Killing vectors that I don't think the
neighborhoods of the joins have any problems, so this is really a
question about those initial consistency conditions, and whether the
periodic boundary conditions of a 3-torus can satisfy them. Perhaps
those same Killing vectors imply this, but that is well beyond my
knowledge of GR.

There are a number of searches underway for nontrivial topology.
including this possibility. So far, there is no evidence for anything
except an infinite, spatially noncompact universe, but nontrivial
topology can't yet be excluded.

Yes. And given the existence of a cosmological horizon, I doubt that
nontrivial topology can ever be excluded (it could be closed at a scale
well beyond our horizon, and thus be invisible to us).

On Apr 28, 4:13 pm, Tom Roberts wrote:

carlip-nos...@physics.ucdavis.edu wrote:
This is not quite right.  A spatially flat universe can still have the
topology of a three-torus, which is spatially compact.

Boy, you guys are amazing.  So, we have spatially curved, spatially
flat, and now spatially compact?  What other spatially possibilities
are there?

Hmmm. I know that a flat 3-space can have the topology of a 3-torus. But
I thought the spatially-flat FRW manifold was unique [#], with the
spatial topology R^3.

The Schwarzschild metric allows both gravitational time dilation and
curvature in space, but the FLRW metric only allows a curvature in
space.  If you believe in the Schwarzschild metric, the FLRW metric
must be false.  On the other hand, if you believe in the FLRW metric,
the Schwarzschild metric must be false.  Since there is no Newtonian
limit for the FLRW metric, this metric must be wrong.

On Apr 28, 12:05 am, Tom Roberts <tjroberts...@sbcglobal.net> wrote:





Mike wrote:
How do the measurements (WMAP, etc.) fit with the notion of an
infinite or finite universe and in relation to GR?

At present, the best measurements on a cosmological scale show that the
universe is spatially flat to within a few percent. And no evidence is
seen for multiple images of a single object or region.

The implications directly affect answers to your question. If a) the FRW
manifolds of General Relativity are a good model of the universe at
cosmological distances, and b) the universe is EXACTLY spatially flat,
then it cannot be spatially compact. However if there is a small
positive spatial curvature, then it could be compact (with a small
negative spatial curvature it still cannot be spatially compact).

So the measurements and GR are consistent with the universe being
spatially infinite. But they cannot rule out the universe being
spatially closed with a scale beyond our cosmological horizon (~14 giga
lightyears).

If GR is not a good model of the universe at cosmological scales, then
we don't know much until a better model can be formulated. While there
are phenomena that bring the validity of GR into question, none have
been established to the point of being generally accepted as a
refutation of GR.

Tom Roberts

Thanks for the answer but my question now is why should we need a
model to determine if it is infinite or finite? Actually, that would
be a determination of what the model implies and not what actually is.

Mike- Hide quoted text -

- Show quoted text -

In sci.physics.relativity Tom Roberts <tjroberts137@sbcglobal.net> wrote:
so this is really a
question about those initial consistency conditions, and whether the
periodic boundary conditions of a 3-torus can satisfy them. Perhaps
those same Killing vectors imply this,

They do.

Mike wrote:
How do the measurements (WMAP, etc.) fit with the notion of an
infinite or finite universe and in relation to GR?

At present, the best measurements on a cosmological scale show that the
universe is spatially flat to within a few percent. And no evidence is
seen for multiple images of a single object or region.

The implications directly affect answers to your question. If a) the FRW
manifolds of General Relativity are a good model of the universe at
cosmological distances, and b) the universe is EXACTLY spatially flat,
then it cannot be spatially compact. However if there is a small
positive spatial curvature, then it could be compact (with a small
negative spatial curvature it still cannot be spatially compact).

So the measurements and GR are consistent with the universe being
spatially infinite. But they cannot rule out the universe being
spatially closed with a scale beyond our cosmological horizon (~14 giga
lightyears).

If GR is not a good model of the universe at cosmological scales, then
we don't know much until a better model can be formulated. While there
are phenomena that bring the validity of GR into question, none have
been established to the point of being generally accepted as a
refutation of GR.

Tom Roberts

On Apr 28, 10:59 pm, Koobee Wublee wrote:

Boy, you guys are amazing. So, we have spatially curved, spatially
flat, and now spatially compact? What other spatially possibilities
are there?

The Schwarzschild metric allows both gravitational time dilation and
curvature in space, but the FLRW metric only allows a curvature in
space. If you believe in the Schwarzschild metric, the FLRW metric
must be false. On the other hand, if you believe in the FLRW metric,
the Schwarzschild metric must be false. Since there is no Newtonian
limit for the FLRW metric, this metric must be wrong.

Nice - now you appear to believe there is only one solution to general
relativity.

Now, spacetime can be cut just like a diamond. Brilliant!

You (plural) allow topology to find a zoo-ful of invisible
pico-scopic particles. The same tool should also yield the
macroscopic structure of the cosmos. This might be the end of
traditional mathematics in physics if you consider topology as a
discipline of mathematics. <shrug

On Apr 29, 12:17 am, Eric Gisse <jowr...@gmail.com> wrote:

On Apr 28, 10:59 pm, Koobee Wublee wrote:
Boy, you guys are amazing.  So, we have spatially curved, spatially
flat, and now spatially compact?  What other spatially possibilities
are there?

Spatially curved, spatially flat, spatially compact, why not spatially
spongy or spatially elastic?  Only in the minds of topologists.

The Schwarzschild metric allows both gravitational time dilation and
curvature in space, but the FLRW metric only allows a curvature in
space.  If you believe in the Schwarzschild metric, the FLRW metric
must be false.  On the other hand, if you believe in the FLRW metric,
the Schwarzschild metric must be false.  Since there is no Newtonian
limit for the FLRW metric, this metric must be wrong.

Nice - now you appear to believe there is only one solution to general
relativity.

We have been through this before.  My position has always been that
the field equations yield an infinite number of solutions where each
solution is independent of the others.

 Each solution describes a
completely different geometry.  Thus, each solution describes a
universe.  The Schwarzschild solution is not unique.

 It manifests
black holes.  There are other solutions that do not manifest black
holes but also degenerate into Newtonian limit.

 One such example is
Schwarzschild’s original solution which is not the same as the
Schwarzschild metric.  <shrug

If you BELIEVE IN the Schwarzschild metric and Schwarzschild’s
original solution are the same, you have also to explain how the FLRW
metric is the same as the Schwarzschild metric.

 I don’t think you
can.  Thus, you have to accept the Schwarzschild metric is not the
same as Schwarzschild’s original solution.  <CHECKMATE ONCE AGAIN

On Apr 29, 9:39 am, Koobee Wublee < wrote:

Spatially curved, spatially flat, spatially compact, why not spatially
spongy or spatially elastic? Only in the minds of topologists.

They all have well defined meanings. Don't be mad because you don't
know what they are.

We have been through this before. My position has always been that
the field equations yield an infinite number of solutions where each
solution is independent of the others.

There are an infinite number until boundary and/or initial conditions
are imposed,

in which case there is a unique solution. PDE and ODE
uniqueness theorems are quite explicit on the subject.

Each solution describes a
completely different geometry. Thus, each solution describes a
universe. The Schwarzschild solution is not unique.

Often stated but never proven.

Every time you crafted a "non-unique"
solution for conditions Birkhoff's theorem assumes, I was able to
obtain a coordinate transformation back to Schwarzschild.

Since you
still don't understand what it means for the metric to be a tensor, it
makes sense that you still get confused over this basic point.

It manifests
black holes. There are other solutions that do not manifest black
holes but also degenerate into Newtonian limit.

So...?

One such example is
Schwarzschild’s original solution which is not the same as the
Schwarzschild metric. <shrug

-1, Wrong. This has been explained to you before - Schwarzschild's
original solution is simply a translation of the modern Schwarzschild
solution. The explicit coordinate transformation between the two was
given to you, and the calculation showing they are equivalent was
performed. You have no excuse for being that ignorant.

If you BELIEVE IN the Schwarzschild metric and Schwarzschild’s
original solution are the same, you have also to explain how the FLRW
metric is the same as the Schwarzschild metric.

Why?

The Schwarzschild solution is a vacuum solution, the FRW solution is
not. Why would they be the same?

I don’t think you
can. Thus, you have to accept the Schwarzschild metric is not the
same as Schwarzschild’s original solution. <CHECKMATE ONCE AGAIN

No, I don't - the explicit transformation has been given to you.

Not
my fault or problem that you don't know how to transform the
components of a tensor from one coordinate system to another.

The universe is created in the collective mind of topology. It allows
the topologists to achieve godhood. Thus, after separating their ways
during the renaissance, science and religion finally come back
together.

On Apr 29, 2:21 pm, Eric Gisse wrote:

On Apr 29, 9:39 am, Koobee Wublee < wrote:
Spatially curved, spatially flat, spatially compact, why not spatially
spongy or spatially elastic?  Only in the minds of topologists.

They all have well defined meanings. Don't be mad because you don't
know what they are.

Sure, they all do just like every verse in the bible has at least one
interpreted meaning.  <shrug

We have been through this before.  My position has always been that
the field equations yield an infinite number of solutions where each
solution is independent of the others.

There are an infinite number until boundary and/or initial conditions
are imposed,

What are these boundary and/or initial conditions?

in which case there is a unique solution. PDE and ODE
uniqueness theorems are quite explicit on the subject.

This is utter bullsh*t.  You were told of these and don’t even bother
to understand it.  <shrug

 Each solution describes a
completely different geometry.  Thus, each solution describes a
universe.  The Schwarzschild solution is not unique.

Often stated but never proven.

The solutions yielded by a set of differential equations have no more
extra meaning than the solutions of a quadratic equation such as (x^2
- 3 x + 2 = 0).  In this case, x = 1 or 2.  Unless you can prove (1 > 2), the solutions are independent of each other.  However, 1 is a
transform from 2, and 2 is a transform from 1.  Even if they are a
transform of each other, 1 is still not 2, and 2 is still not 1.  This
should be a very basic mathematical concept, but I see that it bothers
and most physicists.  Why?

 You have an excuse because you are a multi-
year super senior.   How about the professors and PhD’s?  Correct me
if I am wrong.  The 1st year algebra student should have already
understood this very simple but basic mathematical concept.  <shrug

Every time you crafted a "non-unique"
solution for conditions Birkhoff's theorem assumes, I was able to
obtain a coordinate transformation back to Schwarzschild.

This has never been done.  You are dreaming again.  <shrug

Since you
still don't understand what it means for the metric to be a tensor, it
makes sense that you still get confused over this basic point.

As I said many times over, the metric cannot be a tensor.

 It is merely a matrix.

 That is because the geometry must be invariant.  To
describe the geometry, you need the metric and the coordinate system.

Since coordinate system depends on each observer, the metric must vary
with the coordinate system to yield an invariant geometry.  This very
basic concept in geometry and methodology of experimentation still
eludes you and all physicists.  The nonsense about the metric being
invariant has been around since Ricci’s time.  Mysticism is always
wisdom in disguise, is it not?

 It manifests
black holes.  There are other solutions that do not manifest black
holes but also degenerate into Newtonian limit.

So...?

So what?  Black holes or no black holes are serious matters.  <shrug

One such example is
Schwarzschild’s original solution which is not the same as the
Schwarzschild metric.  <shrug

-1, Wrong. This has been explained to you before - Schwarzschild's
original solution is simply a translation of the modern Schwarzschild
solution. The explicit coordinate transformation between the two was
given to you, and the calculation showing they are equivalent was
performed. You have no excuse for being that ignorant.

What gives you the right to call the Schwarzschild metric more unique
than Schwarzschild’s original metric?  Why is Schwarzschild metric not
a transformation of Schwarzschild’s original metric instead?

If you BELIEVE IN the Schwarzschild metric and Schwarzschild’s
original solution are the same, you have also to explain how the FLRW
metric is the same as the Schwarzschild metric.

Why?

Why not?

The Schwarzschild solution is a vacuum solution, the FRW solution is
not. Why would they be the same?

So?  Thus, the interior of the sun behaves like an FLRW metric, right?

I don’t think you
can.  Thus, you have to accept the Schwarzschild metric is not the
same as Schwarzschild’s original solution.  <CHECKMATE ONCE AGAIN

No, I don't - the explicit transformation has been given to you.

You don’t even know how to transform from Schwarzschild’s original
solution to the Schwarzschild metric.  <shrug

Not
my fault or problem that you don't know how to transform the
components of a tensor from one coordinate system to another.

Yes, it is not your fault that you are a multi-year super senior.  I
seem to have heard of that excuse many times over though.  Since the
metric cannot be a tensor, there is no point of transforming the
metric into something that is still the same metric.  <shrug

The universe is created in the collective mind of topology.  It allows
the topologists to achieve godhood.  Thus, after separating their ways
during the renaissance, science and religion finally come back
together.

You are so sore loser.  You have been checkmated many times over and
still refuse to accept defeat.  That is also why you are still a super
senior.  Is the University of Alaska really that difficult to get a
degree from?  I thought they pay you to go to college over there.