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Mar 16, 2000, 3:00:00â€¯AM3/16/00

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At least without the hypothesis of huge amounts of unobservable matter,

General Relativity leads to the conclusion that our universe is infinite

with a hyperbolic non-Euclidean geometry.

General Relativity leads to the conclusion that our universe is infinite

with a hyperbolic non-Euclidean geometry.

The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

Einstein wrote that when he was young he intuitively agreed with Kant's

views of space and time based on "forms of intuition" (i.e. a basis for

visualization and spacial imagination). He may have drawn the conclusion

that Kant's views correspond to an early stage of development, and that

the direction from Kant's space-time-philosophy to the axiomatic views

of Hilbert, Russel and others is progressive.

In fact however, the axiomatic views are essentially the same as

already presented by Euclid more than two thousand years ago. Hilbert's

axiomatization of geometry is essentially no better than Euclid's.

The only essential error of Kant however, was his assumption that space

is limited to three dimensions. Questions of 4-dimensional geometry are

even very elegant examples of what he called "synthetic a priori"

jugdements.

So a three-dimensional space with constant negative curvature should be

impossible whereas a three-dimensional space with constant positive

curvature is simply the surface of a four-dimensional sphere.

A quote from Wolfang Rindler, Essential Relativity, 1977, p.109:

"If we lived in a three sphere S3 of curvature 1/a^2 and drew concentric

geodesic spheres around ourselves, their surface area would at first

increase with increasing geodesic radius r (but not as fast as in the

Euclidean case), reaching a maximum at 4*pi*a^2, with included volume

pi^2*a^3, at r = 0.5*pi*a. After that, successive spheres contract until

finally the sphere at r = pi*a has zero surface area and yet contains

all our space: its surface is, in fact, a single point, our "antipode".

The total volume of the three-sphere is finite, 2*pi^2*a^3, and yet

there is no boundary."

I suppose that the analogous cases in both a two sphere S2 and a three

sphere S3 cannot be consistently answered, if the curvature is constantly

-1/a^2 instead of 1/a^2.

Wolfgang Gottfried G.

http://members.lol.li/twostone/E/physics1.html

http://members.lol.li/twostone/a5.html (Physik und Erkenntnistheorie)

Mar 16, 2000, 3:00:00â€¯AM3/16/00

to

z@z <z...@z.lol.li> wrote in message news:8aqrmp$ngn$1...@pollux.ip-plus.net...

> At least without the hypothesis of huge amounts of unobservable matter,

> General Relativity leads to the conclusion that our universe is infinite

> with a hyperbolic non-Euclidean geometry.

>

> The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

> At least without the hypothesis of huge amounts of unobservable matter,

> General Relativity leads to the conclusion that our universe is infinite

> with a hyperbolic non-Euclidean geometry.

>

> The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

Infinity in one direction only, like "line vector".

--

Morpheus

Mar 16, 2000, 3:00:00â€¯AM3/16/00

to

In article <8aqrmp$ngn$1...@pollux.ip-plus.net>, z@z <z...@z.lol.li> writes

>At least without the hypothesis of huge amounts of unobservable matter,

>General Relativity leads to the conclusion that our universe is infinite

>with a hyperbolic non-Euclidean geometry.

>

>The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

...

>At least without the hypothesis of huge amounts of unobservable matter,

>General Relativity leads to the conclusion that our universe is infinite

>with a hyperbolic non-Euclidean geometry.

>

>The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

It's perfectly possible for the universe to be finite, but unbounded.

Given a finite age, an infinite extent would appear to be impossible.

--

Jeremy Boden

Mar 17, 2000, 3:00:00â€¯AM3/17/00

to

> At least without the hypothesis of huge amounts of unobservable

matter,

> General Relativity leads to the conclusion that our universe is

infinite

> with a hyperbolic non-Euclidean geometry.

>

> The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

matter,

> General Relativity leads to the conclusion that our universe is

infinite

> with a hyperbolic non-Euclidean geometry.

>

> The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

In General Relativity it can because the General Relativity says so. One

solution is that space is infinite and time is semi-infinite (i.e., past

is finite but future is infinite). Another solution is that both space

and time are finite. If a non-zero cosmological constant or exotic

matter

is allowed there are also solutions that have finite time with infinite

space or infinite time with finite space.

> Hilbert's axiomatization of geometry is essentially no better than

> Euclid's.

Euclid's axiomatization is too incomplete by modern standards. For

example,

the parallel postulate which is intended to differentiate the Euclidean

geometry from other possible geometries is also true in Minkowskian

geometry. Euclid obviously believed that the real world is Euclidean,

unlike some philosophers who seem to have preferred Lobachevskian

geometry.

Euclid apparently was not aware of any other possibility and therefore

failed to see any need of postulates that differentiate from them.

Already

in his first problem of equlateral triangle he needs an assumption he

never

made: he draws two circles, each through the centre of the other, and

then

uses the intersection of the two circles. But he never postulates or

proves

that the circles intersect.

> So a three-dimensional space with constant negative curvature should

be

> impossible whereas a three-dimensional space with constant positive

> curvature is simply the surface of a four-dimensional sphere.

Mathematically both are possible. The curvature of the physical space is

not exactly constant, but in large scales it looks like approximately

constant. Whether it is negative or positive shall be determined from

observations. Until we know we cannot regard any possibility as

impossible.

> A quote from Wolfang Rindler, Essential Relativity, 1977, p.109:

>

> "If we lived in a three sphere S3 of curvature 1/a^2 and drew

concentric

> geodesic spheres around ourselves, their surface area would at first

> increase with increasing geodesic radius r (but not as fast as in

the

> Euclidean case), reaching a maximum at 4*pi*a^2, with included

volume

> pi^2*a^3, at r = 0.5*pi*a. After that, successive spheres contract

until

> finally the sphere at r = pi*a has zero surface area and yet

contains

> all our space: its surface is, in fact, a single point, our

"antipode".

> The total volume of the three-sphere is finite, 2*pi^2*a^3, and yet

> there is no boundary."

>

> I suppose that the analogous cases in both a two sphere S2 and a three

> sphere S3 cannot be consistently answered, if the curvature is

constantly

> -1/a^2 instead of 1/a^2.

A space with a constant positive curvature can be modelled as a sphere

in an Euclidean space. Similarly a space with a constant negative

curvature

can be modelled as a sphere in an Minkowskian space. The latter sphere

is

infinite.

Mikko

Mar 17, 2000, 3:00:00â€¯AM3/17/00

to

"z@z" wrote:

> The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

> The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

First consider the half line: t>=0. This is "infinite" in the sense

that t is unbounded, and yet at any point there is "finite age"

because the distance from t=0 to the point is finite.

In GR things are more complicated. Your question becomes sharper

when one points out that at the big bang space was "compressed"

into a point singularity, so if at a finite time space was of

infinite extent then it seems space had to expand infinitely

fast (in some sense). I believe this is easily remedied: in the

big-bang models space is finite at any finite time after the

big bang. As one goes to increasing time, there are two

possibilities: 1) the space can expand without bound and time can

increase wothout bound; 2) space remains finite and contracts

back to a point within a finite time. At present we don't really

know which of those possibilities applies to our universe.

> So a three-dimensional space with constant negative curvature should be

> impossible whereas a three-dimensional space with constant positive

> curvature is simply the surface of a four-dimensional sphere.

A 3-d space with constant negative curvature is a hyperbaloid.

But this is GR and you are trying to discuss Riemannian geometry,

not the semi-Riemannian geometry of GR. In GR the spacetime of

constant positive curvature is called deSitter spacetime, and it

has a compact 3-space but an infinite time axis (RxS^3); it is

essentially the surface of a hyperbaloid in a 5-d manifold. The

spacetime with constant negative curvature is called anti-deSitter

spacetime, and it is also a compact 3-space and an infinite time

axis (RxS^3); it is also the surface of a hyperbaloid in a 5-d

manifold, but this time that 5-d manifold has two timelike

directions. For more details see Hawking and Ellis, _The_Large_

_Scale_Structure_of_Space-Time_.

> [quote about drawing concentric circles of increasing radius]

> I suppose that the analogous cases in both a two sphere S2 and a three

> sphere S3 cannot be consistently answered, if the curvature is constantly

> -1/a^2 instead of 1/a^2.

Yes, they can if one described them consistently. Note that there

are no "two sphere S2 and three sphere S3 with negative curvature" --

such topologies are inconsistent with a metric of constant negative

curvature. I believe the Riemannian 2-surface of constant negative

curvature has a topology of RxS^1, and the Riemannian 3-surface has a

topology of RxS^2. In both cases, if one drew concentric circles of

increasing radius, there is no upper bound on either their radius or

circumference -- these manifolds are not compact. As you noted in

your quote from Rindler, the 2-d and 3-d spaces of constant positive

curvature are compact, and there is a finite limit on both the radius

and circumference of such a series of concentric circles.

These surfaces of constant negative curvature can be difficult to

visualize, because the two-surface (RxS^1) cannot be embedded

isometrically in a 3-d flat space (it requires 4-d space).

Tom Roberts tjro...@lucent.com

Mar 22, 2000, 3:00:00â€¯AM3/22/00

to

: = Tom Roberts in http://www.deja.com/=dnc/getdoc.xp?AN=599037871

:: = Wolfgang G. in http://www.deja.com/=dnc/getdoc.xp?AN=598305929

:: = Wolfgang G. in http://www.deja.com/=dnc/getdoc.xp?AN=598305929

:: The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

:

: First consider the half line: t>=0. This is "infinite" in the sense

: that t is unbounded, and yet at any point there is "finite age"

: because the distance from t=0 to the point is finite.

In SR there is symmetry between future and past. The time transformation

t' = gamma * (t - x*v/c^2) entails that in frame F' moving at v wrt us,

all is past in one direction and future in the other. An object at

distance d, moving radially away from us with velocity v is in the past

wrt us by gamma*(d*v/c^2).

Because GR is based on SR in this respect, at least the most obvious

reasonings lead to the conclusion, that an infinite spacial extension

entails infinite past.

: In GR things are more complicated. Your question becomes sharper

: when one points out that at the big bang space was "compressed"

: into a point singularity, so if at a finite time space was of

: infinite extent then it seems space had to expand infinitely

: fast (in some sense).

Does the concept 'point' make sense even without space and time? What

is a "point singularity" if there is no embedding space (or time)?

With respect to what does the extent of the universe change from zero

to infinity (or to a finite value)? [1]

According to my epistemological beliefs, point singularities in reality

are impossible like physical quantities resulting from division by zero,

impossible like movements from the future to the past, impossible like

an imaginary-number time, dubious like a virtual particle zoo, and so

on. Big Bang seems to me merely the modern variant (certainly the most

sophisticated ever constructed) of an ordinary creation myth.

I prefer admitting to myself that I don't know, to believing in

questionable conclusions drawn from even more questionable premises.

:: So a three-dimensional space with constant negative curvature should

:: be impossible whereas a three-dimensional space with constant positive

:: curvature is simply the surface of a four-dimensional sphere.

: But this is GR and you are trying to discuss Riemannian geometry,

: not the semi-Riemannian geometry of GR.

The "semi-Riemannian geometry of GR" with negative curvature is

certainly even more complicated, less intuitive and more questionable

than than a two-dimensional manifold with constant negative curvature.

The impossility of the latter should entail the impossibility of

former.

In the case of a 2-d surface of curvature 1/a^2 with a = 1 m, the

circumference of the circle with radius r can easily be calculated

(because we know that this 2-d manifold is the surface of 3-d sphere

with radius a = 1 m). The circumference increases at first with

increasing geodesic radius r, reaching the maximum of 2pi*a = 6.28 m

at r = pi/2*a = 1.57 m. Then successive circumferences contract

until zero at r = pi*a = 3.14 m. After that the same pattern recurs

forever.

If non-Euclidean geometries are as consistent as (multi-dimensional)

Euclidean geometries, then the function expressing the circumference

of a circle depending on radius r must exist also in the case of a

2-d surface of negative curvature -1/a^2. What is the circumference

in the case of e.g. a = 1 m and r = 1 km?

: These surfaces of constant negative curvature can be difficult to

: visualize, because the two-surface (RxS^1) cannot be embedded

: isometrically in a 3-d flat space (it requires 4-d space).

I think that a 2-d surface with constant negative curvature would

require an infinite number of spacial dimensions, wouldn't it?

Wolfgang Gottfried G.

[1] http://www.deja.com/=dnc/getdoc.xp?AN=522160425

[2] http://www.deja.com/=dnc/getdoc.xp?AN=570732454

http://www.deja.com/=dnc/getdoc.xp?AN=571349086

http://www.deja.com/=dnc/getdoc.xp?AN=571795934

Mar 23, 2000, 3:00:00â€¯AM3/23/00

to

"z@z" wrote:

> In SR there is symmetry between future and past. [...]

> In SR there is symmetry between future and past. [...]

While your discussion is unintelligible to me, in some sense there is

indeed such a symmetry in SR.

> Because GR is based on SR in this respect, at least the most obvious

> reasonings lead to the conclusion, that an infinite spacial extension

> entails infinite past.

To conclude that you need to assume that the local properties of

spacetime determine its global properties. That is a false assumption.

>

> : In GR things are more complicated. Your question becomes sharper

> : when one points out that at the big bang space was "compressed"

> : into a point singularity, so if at a finite time space was of

> : infinite extent then it seems space had to expand infinitely

> : fast (in some sense).

> Does the concept 'point' make sense even without space and time?

No.

> What

> is a "point singularity" if there is no embedding space (or time)?

I really meant it in the limit from later times as one approaches

the singularity.

> With respect to what does the extent of the universe change from zero

> to infinity (or to a finite value)? [1]

With respect to an observer inhabiting the spacetime.

> According to my epistemological beliefs, point singularities in reality

> are impossible like physical quantities resulting from division by zero,

Yes, most physicists agree with this, AFAIK. The expectation is that

GR breaks down near such a singularity, and an as-yet-unknown theory

of quantum gravity is needed to describe what happens.

> Big Bang seems to me merely the modern variant (certainly the most

> sophisticated ever constructed) of an ordinary creation myth.

But it has the advantage of describing a host of experimental

and observational data, unlike most other such myths.

> The "semi-Riemannian geometry of GR" with negative curvature is

> certainly even more complicated, less intuitive and more questionable

> than than a two-dimensional manifold with constant negative curvature.

> The impossility of the latter should entail the impossibility of

> former.

You know not whereof you speak.

> If non-Euclidean geometries are as consistent as (multi-dimensional)

> Euclidean geometries, then the function expressing the circumference

> of a circle depending on radius r must exist also in the case of a

> 2-d surface of negative curvature -1/a^2. What is the circumference

> in the case of e.g. a = 1 m and r = 1 km?

Offhand, I don't know. It is computable....

> I think that a 2-d surface with constant negative curvature would

> require an infinite number of spacial dimensions, wouldn't it?

No. Four, IIRC.

Tom Roberts tjro...@lucent.com

Mar 26, 2000, 3:00:00â€¯AM3/26/00

to

WRONG!

Time does not exist.

Time does not exist.

z@z <z...@z.lol.li> wrote in message 8bb94i$2tr$1...@pollux.ip-plus.net...

>: = Tom Roberts in http://www.deja.com/=dnc/getdoc.xp?AN=599037871

>:: = Wolfgang G. in http://www.deja.com/=dnc/getdoc.xp?AN=598305929

>

>

>:: The question arises: HOW CAN AN INFINTE SPACE-TIME HAVE A FINITE AGE?

>:

>: First consider the half line: t>=0. This is "infinite" in the sense

>: that t is unbounded, and yet at any point there is "finite age"

>: because the distance from t=0 to the point is finite.

>

>In SR there is symmetry between future and past. The time transformation

>t' = gamma * (t - x*v/c^2) entails that in frame F' moving at v wrt us,

>all is past in one direction and future in the other. An object at

>distance d, moving radially away from us with velocity v is in the past

>wrt us by gamma*(d*v/c^2).

>

>Because GR is based on SR in this respect, at least the most obvious

>reasonings lead to the conclusion, that an infinite spacial extension

>entails infinite past.

>

>

>: In GR things are more complicated. Your question becomes sharper

>: when one points out that at the big bang space was "compressed"

>: into a point singularity, so if at a finite time space was of

>: infinite extent then it seems space had to expand infinitely

>: fast (in some sense).

>

>Does the concept 'point' make sense even without space and time? What

>is a "point singularity" if there is no embedding space (or time)?

>With respect to what does the extent of the universe change from zero

>to infinity (or to a finite value)? [1]

>

>According to my epistemological beliefs, point singularities in reality

>are impossible like physical quantities resulting from division by zero,

>impossible like movements from the future to the past, impossible like

>an imaginary-number time, dubious like a virtual particle zoo, and so

>on. Big Bang seems to me merely the modern variant (certainly the most

>sophisticated ever constructed) of an ordinary creation myth.

>

>I prefer admitting to myself that I don't know, to believing in

>questionable conclusions drawn from even more questionable premises.

>

>

>:: So a three-dimensional space with constant negative curvature should

>:: be impossible whereas a three-dimensional space with constant positive

>:: curvature is simply the surface of a four-dimensional sphere.

>

>: But this is GR and you are trying to discuss Riemannian geometry,

>: not the semi-Riemannian geometry of GR.

>

>The "semi-Riemannian geometry of GR" with negative curvature is

>certainly even more complicated, less intuitive and more questionable

>than than a two-dimensional manifold with constant negative curvature.

>The impossility of the latter should entail the impossibility of

>former.

>

>In the case of a 2-d surface of curvature 1/a^2 with a = 1 m, the

>circumference of the circle with radius r can easily be calculated

>(because we know that this 2-d manifold is the surface of 3-d sphere

>with radius a = 1 m). The circumference increases at first with

>increasing geodesic radius r, reaching the maximum of 2pi*a = 6.28 m

>at r = pi/2*a = 1.57 m. Then successive circumferences contract

>until zero at r = pi*a = 3.14 m. After that the same pattern recurs

>forever.

>

>If non-Euclidean geometries are as consistent as (multi-dimensional)

>Euclidean geometries, then the function expressing the circumference

>of a circle depending on radius r must exist also in the case of a

>2-d surface of negative curvature -1/a^2. What is the circumference

>in the case of e.g. a = 1 m and r = 1 km?

>

>

>: These surfaces of constant negative curvature can be difficult to

>: visualize, because the two-surface (RxS^1) cannot be embedded

>: isometrically in a 3-d flat space (it requires 4-d space).

>

>I think that a 2-d surface with constant negative curvature would

>require an infinite number of spacial dimensions, wouldn't it?

>

>

Mar 27, 2000, 3:00:00â€¯AM3/27/00

to

oN <NOS...@EUNET.YU> wrote in message

news:8blj9f$e2l$1...@SOLAIR2.EUnet.yu...

> WRONG!

> Time does not exist.

news:8blj9f$e2l$1...@SOLAIR2.EUnet.yu...

> WRONG!

> Time does not exist.

When did you come to that conclusion?

Mikko

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