Finite time in an INFINITE space-time continuum?
z@z
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)
Morpheus
> 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
Jeremy Boden
>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
Mikko Levanto
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
Tom Roberts
> 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
z@z
:: = 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
Tom Roberts
> 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
oN
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?
>
>
Mikko Levanto
news:8blj9f$e2l$1...@SOLAIR2.EUnet.yu...
> WRONG!
> Time does not exist.
When did you come to that conclusion?
Mikko