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Einstein's argument for a finite universe

  1. Dec 27, 2014 #1
    After reading some of Einstein's writings on relativity I am confused as to why he finds it necessary that the universe is finite.

    In his "Relativity", Einstein explains that the ultimate Newtonian picture of the universe (matter in Euclidean space) would be one in which all mass were concentrated in a single area in an otherwise vast ocean of empty space- a universe which would grow increasingly "impoverished". He cites this as unsatisfactory.

    Einstein later references the question of the mass density of the universe as a whole, indicating that if space is infinite, that is, if space is Euclidean or Quasi-Euclidean, then the density of the universe would be 0. However, he indicates that if we admit even a small average mass density for the universe as a whole then we necessarily admit that space is finite.

    I don't understand why he says that any density value greater than zero for the universe would imply finite space. Yes it is true that if matter is finite and space is infinite then the density is 0, and that if space is finite and matter is finite then the density is greater than 0, but what if we were to admit infinite space and infinite matter? Would we not then admit a average density greater than 0 without resorting to the concept of finite space?

    Later in "Geometry and experience" Einstein seems to indicate that there is another reason to believe that space is finite based on reasons other than his arguments using the concept of average density. There he says that "the latest results of relativity" indicate that the universe is probably finite and spherical. I am confused by this because Einstein seemed to say in "Relativity" that the General Theory of Relativity could imply am infinite "Quasi-Euclidean" space of the universe or a finite universe- thus he resorts to the density argument. What is Einstein citing here with respect to "the latest results"? Are there consequences of the theory of relativity which directly imply a finite universe? Is positive curvature of the universe implied by the EFEs for example?

    As a note: the lecture "geometry and experience" was given by Einstein in 1921, while "Relativity" was published in the 40's I believe. Perhaps he thought some results had been arrived at only to latter find that they were unsatisfactory before publishing "Relativity".
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  3. Dec 27, 2014 #2


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    IIRC his argument for this (which may not be fully explicit in the book) is that an infinite space with nonzero matter density would imply an infinite amount of matter, and he did not think an infinite amount of matter was physically possible.

    I think he was referring here to the Einstein static universe, with a nonzero cosmological constant:


    What he apparently failed to consider was that this model is unstable, like a pencil balanced on its point: any small perturbation will cause the universe to either collapse or expand.
  4. Dec 27, 2014 #3


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    It would be easier to discuss this if you could point us to the exact passages you're talking about. Anything Einstein wrote before 1923 should be available online, e.g., at project gutenberg or archive.org.
  5. Dec 27, 2014 #4
    Do you know why Einstein thought that an infinite amount of matter was physically impossible?
  6. Dec 27, 2014 #5
    From Relativity:

    "Considerations on the Universe as a Whole

    XXX. Cosmological Difficulties of Newton’s Theory

    APART from the difficulty discussed in Section XXI, there is a second fundamental difficulty attending classical celestial mechanics, which, to the best of my knowledge, was first discussed in detail by the astronomer Seeliger. If we ponder over the question as to how the universe, considered as a whole, is to be regarded, the first answer that suggests itself to us is surely this: As regards space (and time) the universe is infinite. There are stars everywhere, so that the density of matter, although very variable in detail, is nevertheless on the average everywhere the same. In other words: However far we might travel through space, we should find everywhere an attenuated swarm of fixed stars of approximately the same kind and density. 1
    This view is not in harmony with the theory of Newton. The latter theory rather requires that the universe should have a kind of centre in which the density of the stars is a maximum, and that as we proceed outwards from this centre the group-density of the stars should diminish, until finally, at great distances, it is succeeded by an infinite region of emptiness. The stellar universe ought to be a finite island in the infinite ocean of space.1 2
    This conception is in itself not very satisfactory. It is still less satisfactory because it leads to the result that the light emitted by the stars and also individual stars of the stellar system are perpetually passing out into infinite space, never to return, and without ever again coming into interaction with other objects of nature. Such a finite material universe would be destined to become gradually but systematically impoverished."

    "XXXII. The Structure of Space According to the General Theory of Relativity

    ACCORDING to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known. We know from experience that, for a suitably chosen co-ordinate system, the velocities of the stars are small as compared with the velocity of transmission of light. We can thus as a rough approximation arrive at a conclusion as to the nature of the universe as a whole, if we treat the matter as being at rest. 1
    We already know from our previous discussion that the behaviour of measuring-rods and clocks is influenced by gravitational fields, i.e. by the distribution of matter. This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe. But it is conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun. We might imagine that, as regards geometry, our universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake. Such a universe might fittingly be called a quasi-Euclidean universe. As regards its space it would be infinite. But calculation shows that in a quasi-Euclidean universe the average density of matter would necessarily be nil. Thus such a universe could not be inhabited by matter everywhere; it would present to us that unsatisfactory picture which we portrayed in Section XXX. 2
    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. On the contrary, the results of calculation indicate that if matter be distributed uniformly, the universe would necessarily be spherical (or elliptical). Since in reality the detailed distribution of matter is not uniform, the real universe will deviate in individual parts from the spherical, i.e. the universe will be quasi-spherical. But it will be necessarily finite. In fact, the theory supplies us with a simple connection1 between the space-expanse of the universe and the average density of matter in it."

    From Geometry and Experience:

    From the latest results of the theory of relativity it is probable that our three-dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry, if only we consider parts of space which are sufficiently great.
    Now this is the place where the reader's imagination boggles...."
  7. Dec 27, 2014 #6


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    I'm not sure. In fact, I'm now not even sure that that was the thought behind the passage you refer to (Section XXXII.2 in what is quoted in your post #5); the passage itself does not give any argument, it just makes the flat statement that the universe can't be quasi-Euclidean if the average density of matter differs from zero. It's hard to guess what Einstein was thinking here because we now know this statement to be false: we know there is a solution (the critical density FRW solution) to the Einstein Field Equations in which space is quasi-Euclidean (overall zero spatial curvature on average), but in which the average density of matter is positive. AFAIK this passage in the book was written before the FRW solutions were discovered.
  8. Dec 28, 2014 #7
    Indeed all those passages quoted are previous to Einstein acknowledging the existence of non-static solutions of the EFE, and in static universes following the cosmological principle the density of infinite spaces is either zero or undefined. That's why Einstein reasoned that for any defined non-zero positive density the universe couldn't be "quasi-Euclidean", i.e. infinite. This should answer in part the OP inquiry about Einstein preference for a finite universe.
  9. Dec 28, 2014 #8
    Wouldn't an infinite universe and infinite matter be impossible under the "law of conservation of mass" and "conservation of energy"?

    But isn't it said that are universe is expanding?
  10. Dec 28, 2014 #9


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    No. In GR, those laws are local, not global; they tell you that mass and energy can't be created or destroyed in any small piece of spacetime, but they don't say anything about how much mass or energy there is globally.

    Yes, but Einstein didn't know that when he was developing his static universe model; the expansion of the universe wasn't discovered (by Hubble) until more than a decade later. Once that discovery was made, Einstein called his introduction of the cosmological constant into his field equation in order to allow a static solution "the greatest blunder of my life"--if he had stuck to his original equation, which predicted that the universe must be either expanding or contracting, he could have predicted the expansion of the universe a decade before it was discovered.
  11. Dec 28, 2014 #10


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    Was this actually established, or was it just assumed without proof? IIRC it was the latter, but I'm not too familiar with the details of the literature in cosmology from this period.
  12. Dec 29, 2014 #11
    First I should add to my sentence "quasi-euclidean infinite spaces", since the negative curvature static case is infinite but has a density, only it is a negative density and thus hard to consider physically admissible (as first shown by Friedman in his 1924 second paper on GR). Friedmann equations were not available at the time Einstein wrote the above quoted stuff, but he had further motives to dislike infinite universes, as in the static case they demanded to impose physically valid and coherent with relativity boundary conditions at infinity. In the end this was what made him introduce the cosmological constant, to obtain a finite (spherical) universe keeping as well the static condition it is the only possibility allowed by the EFE. Only when he realized non-static solutions were admissible he saw it as a big blunder. With the benefit of hindsight most can see the blunder was ignoring the non-static prospect.
    I think it was assumed heuristically by Einstein, in any case I don't think there were any formal proofs at the time other than the operation x/∞ tends to zero and ∞/∞ is not defined.
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