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




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