Big-Bang Singularity & Finite Time in an Infinite Universe

By Wolfgang G. Gasser


"singularities and infinite universes"  - 1999-09-07

By David Scarth:

In the cosmological model, there are three possible outcomes after the big bang:

1.      A finite, unbounded universe with positive space-time curvature.

2.      An infinite, unbounded, flat universe (zero curvature).

3.      An infinite, unbounded universe with negative space-time curvature.

In all these cases, the universe expands from a singularity at the big bang.

I can visualize the first case (sort of!), but I have a difficulty with the second and third. At the singularity, the universe is compressed into an infinitely small volume. In the next instant, the universe has infinite volume. I have difficulty in seeing how it can change from infinitely small to infinitely large in one instant.

A singularity always involves a discontinuous transition similar to the transition from zero to one.

Expansion of the universe is equivalent to shrinking of all matter-systems such as particles and galaxies (as long as we assume that there is nothing outside the universe). So nothing prevents us from assuming that the volume of the universe has had always the same size as today.

Under this assumption however, we can recognize that the big-bang outcome of a finite universe is not as different from the outcome of an infinite universe as it seems at first sight.

By the way, can black holes explode? If not, this would strongly suggest that BIG BANG is impossible.


"Finite time in an INFINITE space-time continuum?" 2000-03-16

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?

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 spatial 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" judgements.

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/a2 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 a2, with included volume pi2 a3, 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 pi2 a3, 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/a2 instead of 1/a2.


"Finite time in an INFINITE space-time continuum?" 2000-03-22

By Wolfgang:

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

By Tom Roberts:

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 Special Relativity there is symmetry between future and past. The time transformation t' = gamma (t - x v/c2) entails that in frame F' moving at v with respect to 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 with respect to us by gamma d v/c2.

Because General Relativity is based on SR in this respect, at least the most obvious reasonings lead to the conclusion, that an infinite spatial 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)?

According to my epistemological beliefs, point singularities in reality are

o    impossible like physical quantities resulting from division by zero,

o    impossible like movements from the future to the past,

o    impossible like an imaginary-number time,

o    dubious like a virtual particle zoo,

o    and so on.

Big Bang seems to me merely the modern variant of an ordinary creation myth (certainly the most sophisticated ever constructed).

I prefer admitting to myself that I don't know, to believing in questionable conclusions drawn from even more questionable premises.

By Wolfgang:

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.

By Tom Roberts:

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 a two-dimensional manifold with constant negative curvature. The impossibility of the latter should entail the impossibility of former.

In the case of a 2-dim surface of curvature 1/a2 with a = 1 m, the circumference of the circle with radius r can easily be calculated (because we know that this 2-dim manifold is the surface of 3-dim 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-dim surface of negative curvature -1/a2. 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-dim surface with constant negative curvature would require an infinite number of spatial dimensions, wouldn't it?


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