Big Bang in a Flat Universe

I never had a problem with a Big Bang in a universe with positive
curvature, because traced back light trajectories in all directions
meet together at a point, a finite distance in the past.

However, now that astronomers seem to have decided we live in a flat
universe, I do.

Flat means obeys Euclidean geometry which means that convergence point
is an infinite distance away in the past (or non existent).

Do we have another paradox here?


Chalky

Thus spake Chalky
I never had a problem with a Big Bang in a universe with positive
curvature, because traced back light trajectories in all directions
meet together at a point, a finite distance in the past.

However, now that astronomers seem to have decided we live in a flat
universe, I do.

Flat means obeys Euclidean geometry which means that convergence point
is an infinite distance away in the past (or non existent).

Do we have another paradox here?


No. Flat here means obeying Euclidean space geometry. It is still
possible for space to expand from one time to another.



Regards

--
Charles Francis
substitute charles for NotI to email

Oh No wrote:
Thus spake Chalky
I never had a problem with a Big Bang in a universe with positive
curvature, because traced back light trajectories in all directions
meet together at a point, a finite distance in the past.

However, now that astronomers seem to have decided we live in a flat
universe, I do.

Flat means obeys Euclidean geometry which means that convergence point
is an infinite distance away in the past (or non existent).

Do we have another paradox here?


No. Flat here means obeying Euclidean space geometry. It is still
possible for space to expand from one time to another.

I don't understand this. If space is flat but time is curved, the only
way you can see flat space is to restrict your attention to one time.
This gives you the inner surface of a sphere to look at. However, the
surface of a sphere is _the classic_ textbook example of where
Euclidian geometry rules do _not_ work. So how can you turn round and
now tell me that they do?

Chalky

Thus spake Chalky
Oh No wrote:
Thus spake Chalky
I never had a problem with a Big Bang in a universe with positive
curvature, because traced back light trajectories in all directions
meet together at a point, a finite distance in the past.

However, now that astronomers seem to have decided we live in a flat
universe, I do.

Flat means obeys Euclidean geometry which means that convergence point
is an infinite distance away in the past (or non existent).

Do we have another paradox here?


No. Flat here means obeying Euclidean space geometry. It is still
possible for space to expand from one time to another.

I don't understand this. If space is flat but time is curved, the only
way you can see flat space is to restrict your attention to one time.
This gives you the inner surface of a sphere to look at. However, the
surface of a sphere is _the classic_ textbook example of where
Euclidian geometry rules do _not_ work. So how can you turn round and
now tell me that they do?


If I may correct one thing, we don't say time is curved. Time is one
dimensional, and off the top of my head I don't think the definition of
curvature applies in one dimension. I probably ought to look it up in a
text book and make sure, but I haven't done that. What we actually say
is that space is flat but space-time is curved, meaning that it does not
obey the rules of Minkowski space-time.

You are right that in order to discuss the curvature of space we
restrict ourself to one time. But this does not necessarily give us the
surface of a sphere to look at. There are in fact three possibilities
for a homogeneous isotropic cosmology. When we look at any 2 dimensional
plane we may find:

1. The geometrical rules of a sphere. This is positive curvature.
2. Euclidean geometry. This is zero curvature, or a flat universe.
3. The geometrical rules of a saddle. This is negative curvature.

The inner or outer surface of a sphere makes no difference to the
geometrical rules which apply, btw. The geometrical rules on a sphere
are those of positive curvature.



Regards

--
Charles Francis
substitute charles for NotI to email

Oh No wrote:
Thus spake Chalky
Oh No wrote:
Thus spake Chalky
I never had a problem with a Big Bang in a universe with positive
curvature, because traced back light trajectories in all directions
meet together at a point, a finite distance in the past.

However, now that astronomers seem to have decided we live in a flat
universe, I do.

Flat means obeys Euclidean geometry which means that convergence point
is an infinite distance away in the past (or non existent).

Do we have another paradox here?


No. Flat here means obeying Euclidean space geometry. It is still
possible for space to expand from one time to another.

I don't understand this. If space is flat but time is curved, the only
way you can see flat space is to restrict your attention to one time.
This gives you the inner surface of a sphere to look at. However, the
surface of a sphere is _the classic_ textbook example of where
Euclidian geometry rules do _not_ work. So how can you turn round and
now tell me that they do?


If I may correct one thing, we don't say time is curved. Time is one
dimensional, and off the top of my head I don't think the definition of
curvature applies in one dimension.

This is a question of semantics which probably doesn't matter provided
we all know what we are talking about.

I probably ought to look it up in a
text book and make sure, but I haven't done that. What we actually say
is that space is flat but space-time is curved, meaning that it does not
obey the rules of Minkowski space-time.

You are right that in order to discuss the curvature of space we
restrict ourself to one time. But this does not necessarily give us the
surface of a sphere to look at. There are in fact three possibilities
for a homogeneous isotropic cosmology. When we look at any 2 dimensional
plane we may find:

1. The geometrical rules of a sphere. This is positive curvature.
2. Euclidean geometry. This is zero curvature, or a flat universe.
3. The geometrical rules of a saddle. This is negative curvature.

AFAIK these are standard GR textbook examples of 4 dimensional
spacetime geometry, not of the 3 dimensional geometry of space. To get
a representation of the latter, I imagine you can consider an observer
embedded in the surface of that geometry. If you consider a lookback
time of 1 year, then the volume of space enclosed could then be
represented as a circle with radius 1.

I think Chalky's point is this:

If you take the simplest example of a 4 sphere, then such a circle
would represent a total surface area of 4 pi at such a small lookback
time. As the lookback time increases, the total surface area of the
visible sample slice initially increases as t squared, then as somewhat
less, and then starts to decrease, until we approach zero again at the
big bang.

However, with a flat or saddle shaped 4 surface, no such eventual
convergence occurs, thus no big bang.

Now, you have responded by saying that space is flat but spacetime is
curved. Chalky has thus replied that this doesn't make sense since the
spatial surface of the sphere of observation around us clearly has
positive curvature. I am sure you would agree with this spatial
conclusion locally.

HTH
John Bell

Chalky wrote:

I don't understand this. If space is flat but time
is curved, the only way you can see flat space is
to restrict your attention to one time. This
gives you the inner surface of a sphere to look
at.

Why? Einstein showed us that simultaneaty is
"observer dependent"; I don't think the concept of
"3D space at a single time" makes much sense at all
in a model where General Relativity is considered to
hold sway, only in an idealized math of a 3D object
with no time dimension at all.

However, the surface of a sphere is _the classic_
textbook example of where Euclidian geometry rules
do _not_ work.

Not exactly. Geometry _restricted to_ the infinitely
thin surface of that sphere is non-Euclidean.

Geometry done looking at right angle to that sphere,
using light rays in the interior of a ball with that
sphere as its surface, which is where the light
waves are travelling with which you are concerned
about convergence, is quite ordinary; if your space
is "flat", the geometry is Euclidean.

So how can you turn round and now tell me that
they do?

Because you haven't understood the applicability of the term
"non-Euclidean geometry" correctly. It is geometry
done embedded in a space which is itself curved, not
geometry done while looking at an object which is
itself curved but embedded in a space which is not
curved.

HTH

xanthian.

Kent Paul Dolan wrote:

Chalky wrote:

I don't understand this. If space is flat but time
is curved, the only way you can see flat space is
to restrict your attention to one time. This
gives you the inner surface of a sphere to look
at.

Why?

Ah! Follow me down more deeply.

Einstein showed us that simultaneaty is
"observer dependent";

Precisely. This is covered in SR.

I don't think the concept of
"3D space at a single time" makes much sense at all
in a model where General Relativity is considered to
hold sway,

Absolutely. All we have is the concept of 2D space relative to that
observer, which is the surface of a sphere. The third dimension of
space = the dimension of time, when c=1, is not directly observable,
and only obtained by inference.

only in an idealized math of a 3D object
with no time dimension at all.

Precisely.

However, the surface of a sphere is _the classic_
textbook example of where Euclidian geometry rules
do _not_ work.

Not exactly. Geometry _restricted to_ the infinitely
thin surface of that sphere is non-Euclidean.

That is what I said, or, at least, meant to say.

Geometry done looking at right angle to that sphere,
using light rays in the interior of a ball with that
sphere as its surface,

Which is what we are talking about

which is where the light
waves are travelling with which you are concerned
about convergence, is quite ordinary; if your space
is "flat", the geometry is Euclidean.

I don't follow this line. We have just agreed that the surface of the
visible sphere is non-Euclidean. We can only break out if that
positively curved plane by invoking the dimension of time within our
observations, which then gives us non-Euclidean again, if we are going
to end up with a Big Bang.

So how can you turn round and now tell me that
they do?

Because you haven't understood the applicability of the term
"non-Euclidean geometry" correctly.

This is quite possibly true.

It is geometry
done embedded in a space which is itself curved, not
geometry done while looking at an object which is
itself curved but embedded in a space which is not
curved.

This sounds sufficiently complicated and almost 'Zen', that it could be
true. However, I still have reservations. It implies that there is a
three dimensional Euclidean space out there which has physical meaning,
even though we can't observe it. If we were talking about a table in a
room, we could walk round it and confirm this. When we are talking
about the cosmos, we can't. We can only infer a third dimension of
spatial observation by invoking the dimension of time, which then gives
us non-Euclidean geometry again.

I therefore think this discussion still has a ways to go.


HTH
Chalky

Thus spake Kent Paul Dolan
Chalky wrote:

I don't understand this. If space is flat but time
is curved, the only way you can see flat space is
to restrict your attention to one time. This
gives you the inner surface of a sphere to look
at.

Why? Einstein showed us that simultaneaty is
"observer dependent"; I don't think the concept of
"3D space at a single time" makes much sense at all
in a model where General Relativity is considered to
hold sway, only in an idealized math of a 3D object
with no time dimension at all.

Well you are wrong. Cosmic time is defined in any GR or cosmology text
book following from Weyl's postulate. When we talk of an expanding
universe we mean that space expands between one hypersurface of constant
cosmic time and the next.

Because you haven't understood the applicability of the term
"non-Euclidean geometry" correctly. It is geometry
done embedded in a space which is itself curved, not
geometry done while looking at an object which is
itself curved but embedded in a space which is not
curved.

In its inception by Riemann, and in early accounts of general
relativity, non-Euclidean geometry was a direct generalisation of the
geometry found in a 2-dimensional surface (e.g. the surface of the
Earth) in a 3-dimensional space, and was typically treated as the
geometry of a curved space embedded in a higher dimensional flat space.
In modern treatments we no longer look at embedding spaces, and we
characterise the geometry according to the geometrical properties the
space itself, but it is still possible in principle to talk of an
embedding space of higher dimensions.


Regards

--
Charles Francis
substitute charles for NotI to email

Thus spake Chalky
Kent Paul Dolan wrote:

Absolutely. All we have is the concept of 2D space relative to that
observer, which is the surface of a sphere. The third dimension of
space = the dimension of time, when c=1, is not directly observable,
and only obtained by inference.

I at least see your problem now. But you are not dealing with space-time
as it is defined in physics.

This sounds sufficiently complicated and almost 'Zen', that it could be
true. However, I still have reservations. It implies that there is a
three dimensional Euclidean space out there which has physical meaning,
even though we can't observe it. If we were talking about a table in a
room, we could walk round it and confirm this. When we are talking
about the cosmos, we can't. We can only infer a third dimension of
spatial observation by invoking the dimension of time, which then gives
us non-Euclidean geometry again.

In physics we do assume that the local three dimensions can be extended
outwards. I dare say you will accept that we can do this on the basis of
observation at least in so far as we can use radar to measure the time
and position of the reflection of signal at a spacecraft or a planet.
This gives us four numbers to describe an event. Its time, its distance
from us, and two angular directions on the sphere. The requirement of
four numbers is what we mean by 4 dimensions. In order to make any
reasonable interpretation of astronomical data at all, we also assume
the general principle of relativity, that local laws of physics are
always and everywhere the same. Thus if we can measure 4 dimensions up
to a certain radius from ourselves, we immediately assume that an
observer at that radius could likewise measure 4 dimensions up to the
same distance from himself. So we immediately assume that the
4-dimensional structure we observe locally extends over the entire
cosmos.

You may want to dispute that in some way, but I must put it to you that
if you do, you are doing philosophy and not physics, and that case you
would be posting on the wrong newsgroup.

Now imagine that we measure a circle at some radius, r, from ourselves,
and imagine six observers, equally spaced about that circle, and ask the
question what distance, d, do they measure between each other? We do not
assume Euclidean geometry. General relativity allows three possibilities
(really more, but three simple possibilities at least)

d r This is characteristic of positive curvature
d = r Euclidean geometry
d r negative curvature.

If all those observers are on lines of "free fall" from the big bang or
a point in the infinite past (in the absence of local gravity effects)
expansion means that d and r are increasing in time.

You may observe from this argument that it is not strictly necessary to
assume that either space or space-time exists in some way. All we
actually have is the use of four numbers to identify locations, or
events as they are called. Strictly speaking that is all that can be
assumed by science, not that there exists a thing called space-time
which somehow has its own physical meaning.

It is true however, that most scientists, and most scientific theories,
do assume such a structure. Discussion of it tends to be controversial,
can become heated and is often avoided. However in what is called the
orthodox interpretation of quantum mechanics, no such structure is
assumed. Therein we only have experimental results. Typically quantum
mechanics applies on the very small scale, but actually it should be
applied whenever it becomes impracticable in principle to measure points
in space time. My own research has lead me to consider another
situation, on the large scale. It is impossible in principle to measure
a space time coordinate in empty space, for the simple reason that if
space is empty there is nothing in it which can be measured.


Regards

--
Charles Francis
substitute charles for NotI to email

"Chalky" wrote:
Kent wrote:

Geometry done looking at right angle to that
sphere, using light rays in the interior of a
ball with that sphere as its surface,

Which is what we are talking about

Then read that again: the light rays you are using
to see are NOT light rays traveling ALONG that
spherical surface at right angles to your line of
sight, and therefore within a non-Euclidean space,
they are light rays penetrating that spherical
surface at right angles (normal to), ninty degrees
away from the rays traveling within that surface,
and therefore within a quite ordinary space, the one
in which you live, with whatever geometry that space
in which you live has.

which is where the light waves are travelling
with which you are concerned about convergence,
is quite ordinary; if your space is "flat", the
geometry is Euclidean.

I don't follow this line. We have just agreed that
the surface of the visible sphere is
non-Euclidean. We can only break out if that
positively curved plane by invoking the dimension
of time within our observations, which then gives
us non-Euclidean again, if we are going to end up
with a Big Bang.

Because you are talking about the convergence of
light rays impinging _on your eye_, and light rays
traveling _within_ that sphere surface which is at
right angles to your line of sight are of necessity
_not_ light rays that will impinge on your eye.

HTH

xanthian.


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