SPACE IS NOT CURVED

A model is a description of the relationships of some things. We mentally build our world model from our observations. At some time in our infancy we add to our model of the world the observation that if we let go of an object it falls down. Whether we realize it or not, we never stop enhancing our world model.

No person and no group of people can be omniscient. There will always be things not yet discovered; there will always be unknowns. Therefore, our world model will never be all-inclusive; it can only be a partial representation of reality. But, in order for the model to be realistic all of its features must correspond to reality.

Sometimes a new observation or a new insight to an old observation does not agree or connect with our world model. In those cases we create a new addition to our model or revise a part of the model. Those additions or revisions must also reflect reality or the model will no longer be realistic.

If we want our model to have general application we frame the model in terms of generalities and exclude specific instances of things or events. To do that we require theories, and especially theories expressible in some form of logic such as mathematics. Because the unknown can be expressed only in terms of the known, theories do not spring into existence from nothingness, so to speak. They arise from their postulates. Postulates are used as a basis for reasoning; they are ideas assumed to be true or believed to be self-evident.

As for things believed or assumed to be true or self-evident, we should remember that “It ain’t necessarily so.” How wrong it was to believe that the Sun and stars spin around the Earth; yet, it seemed so self-evident. The self-evident can easily mislead the self.

A theory cannot validate its own postulates Those postulates are the foundation of the theory. If the foundation fails the theory must come tumbling down. Let us now investigate the theory of space curvature.

According to Einstein, matter causes space to curve. Einstein’s theory for that curvature proceeds in the following way. Assume a spherical region of space with a continuous distribution of matter in it. The density may have local variations within that space. The radius excess of that sphere is a measure of space curvature and is proportional to the mass inside the sphere. Einstein’s resulting formula for excess radius of the sphere contains Newton’s gravitational constant (G), the velocity of light (c), the numbers 2 and 3, and the quantity of mass (M) inside the sphere. The beautifully simple formula is: radius_excess=GM/(3c^2).

The important postulate supporting that theory is the assumed continuous distribution of matter. If matter is concentrated at points, then a formula cannot be derived. So what is meant by a continuous distribution of matter? A continuous distribution implies homogeneity in some sense. We see no empty spaces in a liter of water, so the water is continuously distributed and quite homogeneous. But my U-shaped kitchen does not have a continuous distribution of matter. It has cabinets, a sink, etc. with a large empty space between the sides of the U. Spatially it is not homogeneous.

The Earth, like the liter of water, can be considered to have a continuous distribution of matter. However, what about the atom with its relatively huge gaps between the nucleus and the electrons? There are huge gaps between the Sun and the planets, virtually devoid of matter. The gaps between the stars in our galaxy are enormous and practically devoid of matter. Our local cluster has immense gaps between galaxies. The local supercluster has astronomical gaps between the clusters. The voids between superclusters almost boggle the mind.

Now let us consider how densities have varied during the above journey through the cosmos. There are fewer atoms per cubic meter in the solar system that there are in the Earth. The number of stars per unit volume is much greater in the solar system than it is in the local galaxy because of the great distances between stars. But even the local galaxy contains more stars per unit volume than the local cluster. Every jump to a new level of the cosmological hierarchy has brought with it an enormous increase in the size of the gaps with a correspondingly enormous decrease in density.

An inconsistency in the theory arises from the variation of density with hierarchical level in the cosmos. If the spherical volume we choose is the Earth, with its density, we get one curvature for the space within it. Choosing the solar system, with its much lower density, we get a second curvature for space and with the local galaxy a third, and so on. Now it is inconsistent to claim that the space within the galaxy has the third curvature when portions of that space have the second and first curvatures.

Can any real spherical space have matter more or less uniformly distributed within it?
Perhaps the Earth can, but not the sphere that would enclose the solar system. That sphere would have the mass concentrated to a narrow equatorial plane in the sphere. The same would be true for many galaxies. As for galaxy clusters, the distribution is extremely spotty and irregular.

If we investigate what happens to homogeneity with increasing scale we get a surprise. In this investigation we will not indulge in speculations about what might be. We will consider only observations that have actually been made. We begin with the solar system which consists of things made of atoms, such as the Sun, the planets and their satellites, some asteroids, comets, etc. Each object in that list is composed of atoms.
Hence, there are fewer solar system objects than there are solar system atoms.

Because a galaxy is composed of stars, there must be fewer galaxies than stars. Galaxy clusters are groups of galaxies; there must be fewer clusters than galaxies. Superclusters are groups of galaxy clusters; there must be fewer superclusters than clusters. Increasing scale drastically reduces membership count. Reduced membership count with increasing scale does not increase homogeneity. To the contrary, reduced membership count increases heterogeneity. The cluster counts of the eighteen or so identified superclusters vary greatly, as do the sizes of the void spaces observed between them. Increasing cosmological scale increases heterogeneity.

On a cosmological scale, let us say the scale of merely the local galaxy, the mass of the Earth is effectively concentrated at a point. On the scale of superclusters our galaxy has its mass similarly concentrated. The inconsistency noted above is indicative of the point-like nature of matter in space. No size spherical region would avoid these enormous drops in density, increases of heterogeneity, highly localized variations of the curvature, increases in the appearance of point-like masses, and that inconsistency.

Shortly after Hubble established that those fuzzy nebulae were not in the local galaxy, before clusters and superclusters were discovered, galaxies were the be all and end all for cosmologists. They were thought to be the ultimate building blocks of the universe and to be more or less uniformly distributed in the universe, as stars seem to be in the sky. That may have influenced Einstein to believe that when viewed on a cosmological scale the density of the universe would be everywhere the same.

We now know that no cosmological scale yields homogeneity. The discoveries of clusters, superclusters, and sub-nuclear particles warn us that we should not assume that there are ultimate largest or smallest objects. Attractive though those assumptions may be and despite the simplifications they might bring to theories, there is no scientific or philosophical justification for such assumptions.

Finally we can conclude that because the curved space theory’s basic postulate, the continuity of matter, does not correspond to reality, neither does the theory. The notion of curved space is not realistic. Although we can endow imaginary space with curvature or any other property we wish, real space is not curved.

Visit

http://www.theuniverse.andmuchmore.com

to read about the greatest postulate failure of all time.

That's right. The world is flat.

(Airlines only use "Great Circle" routes for purposes of avoiding R'lyeh and the opening to the Underearth.)

Silas

Mr. Hanak,

Out of curiosity, which of the following statements do you agree with and which do you disagree with:

1.) The trajectory of a freely falling body is independent of its internal structure and composition.

2.) The outcome of any local non-gravitational experiment is independent of the velocity of the freely-falling reference frame in which it is performed.

3.) The outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed.


Thanks for your indulgence.



<font size=-1>[ This Message was edited by: Wiley on 2002-06-20 17:34 ]</font>

RJH: So what is meant by a continuous distribution of matter? A continuous distribution implies homogeneity in some sense.

Incorrect. In this case the word "continuous" refers to a mathematical property, not a physical property. The equation that describes the distribution of matter must be continuous (i.e., must have no embedded infinities) in order to be integrable. As an example, the equation y=x^2 is continuous, because all finite values of "x" produce finite values of "y". However, the equation y = 1/(x-1)^2 is not continuous, because x=1 provides an undefined 1/0 (infinity) for "y". The word "continuous" means only that the distribution of matter must meet this mathematical definition of continuity.

<font size=-1>[ This Message was edited by: Tim Thompson on 2002-06-21 17:24 ]</font>

I think RJH is not going to enter into conversation with us...

My favorite non-continuous function is the function: y=1 when x is rational, and y=0 when x is irrational. It's charming, since the limit at any point is both 1 and 0!

Silas

Well, some people see empty space in there. My glasses aren't good enough, true, but then I have trouble reading anyway.

It seems that your argument boils down to, "space and matter is not uniformly homogeneous, so space cannot be curved." That doesn't seem to make much sense. A homgeneous universe is a special case, and only a first approximation to our own circumstance, not a requirement of the theory.

Wiley:

I appreciate your question and respect you for your obvious knowledge, thoughtfulness, and sincerity. So please do not think that my answer implies any form of disparagement. For me the three statements are meaningless because:

1.) I think that there can be no such a thing as a freely falling body.

2.) I think that there can be no such a thing as a freely falling reference frame.

3.) I think that there can be no such a thing as the universe.

However, if I correctly understand the thrust of your inquiry what follows is a more relevant response.

There is no question that many of the predictions of relativity theory have been verified, and some of them many times and in multiple ways. However, proof of the consequent is never proof of the antecedent. Karl Popper taught us that falsifiability is the hallmark of science. From him we learned that no matter how much supporting evidence there is for a theory, all that is required is one verifiable contradiction of the theory to completely discredit it.

If a theory is false in some way but nonetheless makes subsequently verified predictions, it means that the predictions are based not so much on the theory as on ideas or logic imported into the theory. For example, the Fitzgerald-Lorentz contraction theory and the Lorentz transformations (a response to the Michelson-Morley experiment) predated Einstein’s work by many years. Einstein used that theory and those transformations in his Special Theory of Relativity.

I believe that the contraction theory and Lorentz transformations will eventually be applied to a non-relativistic theory that will make those same predictions and more. A more detailed explanation, beyond the scope of a bulletin board, can be found in my books. The second book, A Journey Beyond The Universe, describes some surprising fundamental aspects of such a non-relativistic theory. Excerpts from both can be read at:

http://www.theuniverse.andmuchmore.com

They are now also available as inexpensive eBooks at xlibris.com.

Well, I'm no scientist, but I will admit to loving the great taste of SPAM(tm)!

T.T. :

My dictionary gives: Uniform: identical from place to place; same. Continuous: being in immediate spatial relationship.

Einstein wrote “There are stars everywhere, so that the density of matter, although very variable in detail, is nevertheless on the average everywhere the same.”(Relativity, Albert Einstein, Crown, 1961, p105)

On the same page he wrote “However far we might travel through space, we should find everywhere an attenuated swarm of fixed stars of approximately the same kind and density.”

He also wrote “If we are to have in the universe an average density of matter which differs from zero, however small may be that difference, then the universe cannot be quasi-Euclidean. (ibid. p114)

He wrote that his original ideas on relativistic structure of space were based on “two hypotheses:
(1) There exists an average density of matter in the whole of space which is everywhere the same and different from zero
(2) …..” (ibid. p133)

By his own words he acknowledged the hypothetical nature of assertion (1) above. In the contexts of his repeated usage of the word ‘same’, the word ‘uniform’ is an accurate synonym. That is why I labeled the idea of ‘the same density of matter throughout space as the uniformity hypothesis elsewhere in this forum. As an aside, in his hypothesis (1) he is implicitly considering things on some kind of cosmological scale.

The notions of average density, and especially average density everywhere, imply some kind of continuity of the physical property ‘density.’ The more point-like the matter contained in a given volume of space, the less continuous its distribution. If the matter were concentrated to a sufficiently small volume relative to that volume of space, that matter would no longer be distributed in that volume. When matter is concentrated to that degree, averaging is neither applicable nor necessary; one has either the density of the matter itself, or the density of that volume of space.

If the matter in a given region of space is more or less uniformly distributed, then local variations of density can be taken into account by mathematical integration. But, that cannot be done for point-like distributions (singularities, you know). Same thing as the above, in fewer words.

Silas :

Indeed charming, but my favorite non-continuous function is the square wave, with its discontinuous amplitude. Why? Because it can (magically?) be represented by a series of continuous trigonometric functions via the Fourier expansion. Thanks, but please, no more.

[b[GrapesOfWrath:[/b]

Homogeneous means ‘everywhere the same.’ In the foundations for the theory variations in homogeneity are permitted, but not its absence. See my reply to T.T. above




.

Absence of homogeneity? I think that is permitted.

Hmm… technically, the limit is neither 1 nor 0. The function is either 1 or 0, but it has no limit at any point.

Nothing magical about it. A series of continuous functions need not be continuous itself.

I concur: SPACE is not curved, or to abridge, SPACE is not. Space mut be that singularly HUMAN invention to symbolize "that medium which contains ie possesses Gravity", as if every dang thing in our world needed something 'greater' to serve as its 'vessel'. No; gravity pervades the cosmos but it doesn't need a well-defined perfectly structured (3D) yet totally imaginary VESSEL in which it 'dwells'. Gravity stands on its own.

Since space is non-existant, then it is neither uniform nor 'curved' (whatever that means). What is indisputable is only that gravity is not uniform.

So the Moon orbiting the Earth is dependent on another force besides gravity? What would this force be? Note "dependent" in this sense means that if we could "magically" remove this unknown force, the Moon would no longer orbit the Earth.

This is purely a semantic argument to sidestep the issue. The question is simply "do you or do you not believe in the principle of relativity for non-gravitional physics?"

Not quite. All modern theories of gravity, not just General Relativity, rest upon the three principles I listed in my previous post. It is fine to take issue with any or all of these postulates, but one needs to defend this position.

To the best of our measurement ablity, these postulates are true. (Between 8-11 significant figures, depending on the postulate.) If these postulates are true, space is curved. It may not curve the way Einstein said it would, but space is curved.