Why is the vacuum flat, i.e., Euclidean?

nomadreid

Deviations from the vacuum energy bring about deviations from a Euclidean spatial geometry. Fine; I am not questioning this principle. I am wondering why a Euclidean metric is the base from which everything deviates? An answer that it is the limit of more general metrics only begs the question. A deeper answer going back to first principles would be appreciated. (If the question is not clear, let me know, and I shall try again.)

Driftwood1

Not sure if you can get any more closer to first principles than Euclidean axioms

nomadreid

A system of Euclidean axioms are, to a mathematician, just one of an infinite number of possible axiom systems. If a mathematician-god were to create a new universe, he/she/it might arbitrarily start with the vacuum energy having a metric that was non-Euclidean, and create the rest of the physical laws on that basis. However, he/she/it would not be able, short of continuous miracles, to make the universe come out to be the one we observe around us. So, starting from the other end: if this mathematician-god started from other physical laws known to be valid in our universe, he/she/it would end up being forced to make the vacuum have, in the absence of mass, a Euclidean metric. Put another way: he/she/it would be forced to make the Pythagorean Theorem hold. But what would be the mechanism that would force him/her/it to do so? So, the first principles I am talking about are physical laws that do not presuppose the Euclidean metric, but that imply that metric's necessity.

Driftwood1

I was unaware that mathematicians could select from an INFINITE number of axiom systems

Seems rather a lot to me - anyway!

You make the assumption that "everything" is known about the current Universe we live in and that ALL natural processess are fully understood and mathematically described and proven.

This cannot be further from the truth.

What number is approached when you divide the amount of truth Humans have accumulated by the total absolute truth that exists in the Universe?

nomadreid

Yes; check any mathematician working in the field of Model Theory (in Mathematical Logic): as a trivial example, the number of possible large cardinal axioms that can be assumed is infinite.


No, I was just hoping that enough of them were to be able to answer my question. I accept the possibility that there is not yet enough known to be able to answer my question, even though I am not yet convinced that this is the case.

First, you would have to come up with a quantification of "truth", and second, you cannot divide by zero.

Driftwood1

so you are now claiming that absolute truth is zero?

dividing by zero is a mathematical and semantical limitation - a nuerotic twitch by the cleansing mathematician - afraid of the approximate - the mathematician walks on a paper thin skin with bare feet knowing all too well that beneath the paper is riddled with acid glass splinters

The Problem with Mathematics is that it isnt a Science - more of an Abstract Philosophy

It thrives on proofs and faith - a faith in a lack of errors in the axiom set assumed.

Assuming that the axiom sets available are in fact infinite in number is good exmaple of this neurotic affliction

I always suggest a good bottle of single malt whiskey to the few mathematician firends I know.

The last discussion I had with a mathematician was about 6 months ago - we were arguing as to why MORE than one infinity is needed in the field.

Imagine that - they can't even define infinity - but they seem to be happy with many versions of it

interesting field.........

nomadreid

Let's say that it is a sparse set in the field of statements, so that its measure is zero. Of course, you could try to duplicate Kant's mistake of thinking that there are a priori synthetic truths, but you are going to come up against the likes of Lobachevsky and Gödel.

Well, yes, in the same way that "0 = 1" is a mathematical limitation, and in the same way that ";lkjsdao" in English is a semantic limitation.

That's not its problem, it is its saving grace. Physicists are tied down to the humdrum physical reality; mathematicians can let their imaginations run wild, with the only limitation of trying to avoid contradiction. Mathematics is an art form.

In physics or a court of law, the accused is assumed innocent until proven guilty, but a (good) mathematician is always aware of how far the particular axiom system he/she is working on at the moment (and you can work on several competing ones at once) has been proven consistent, or (mostly) equiconsistent with arithmetic, or whatever.... mathematics is a game that only those in applied mathematics (physicists, etc) take seriously. That's why mathematicians, like blondes, have more fun.

No one ever claimed that mathematicians were sane. But look at the sane people around you -- do you really want to be one of them?

I would guess that the discussion went something like this:
"Oh, you need a lot. The first three are obvious: you need the cardinalities of the sets of: the natural numbers (aleph-null), of the continuum (two to the aleph-null), and the functions from the continuum to the continuum. (two to the two to the aleph-null). But then you should need the first inaccessible cardinal (in order to have a model for the usual mathematics) and of course an uncountable measurable cardinal (in order to make all functions Lebesgue integrable, a must in quantum physics)."
Something like that?

If your mathematician friends can't define infinity, go tell them that they are not looking in the right books. The books crowding my bookcase are full of precise definitions not only of infinity in general but also many, many different kinds of infinity.

I like it. Of course, I don't make any claims to sanity......

Chalnoth

Edit: Bah, misread. This isn't quite the case, though. Euclidean geometry holding has to do with the relationship between energy density and expansion rate. You can have Euclidean spatial geometry with all sorts of stuff in the universe, not just vacuum energy. It's just that vacuum energy tends to make a non-flat universe more and more flat (in actuality, this will happen with any sort of energy density with [itex]w < -1/3[/itex]).

nomadreid

Thanks very much, Chalnoth. This helps a lot, but I need to ask you what w stands for here. (Sorry, not being a physicist I need to search for specific terms, and I did not find this in my search.)

The use of the expansion as opposed to the energy density gives me an intuitive feel how one would find the flatness being Euclidean, but I am not sure how I would do this formally. Do you have any link that shows such a calculation?

The only such proofs I have seen basically assume what they want to prove. That is, they assume that the "default case" is Euclidean, and then show how mass-energy is sparse enough to make space close to the default case. But they never explain why the default case should be Euclidean to begin with. I have a similar problem with the latter half of your quotation: whereas I would also be interested in the details as to why vacuum energy will make a non-flat universe more and more flat (note the implicit request here), I would also like to know why "flat" has to be (?) Euclidean. Or is this an empirical result?

Chalnoth

The parameter [itex]w[/itex] is the relationship between energy density and pressure for a given sort of matter. For photons, for example, [itex]w = 1/3[/itex], as photon pressure is one third their energy density. For normal matter, there is no pressure on cosmological scales, so [itex]w = 0[/itex].

Well, this drops right out of the first Friedmann equation:

[tex]H^2 = {8\pi G \over 3}\rho - {k \over a^2}[/tex]

Here the curvature is represented by [itex]k[/itex]. If we just consider the current energy density and expansion rate (where we define the scale factor so that [itex]a=1[/itex] now), we have:

[tex]k = H_0^2 - {8\pi G \over 3}\rho_0[/tex]

If you want to know why [itex]k=0[/itex] indicates Euclidean space, well, that comes down to the spatial components of the Ricci curvature tensor being zero in this case. You can also see it in the metric, where the spatial components of the FLRW metric reduce to the Euclidean metric times a function of time if [itex]k=0[/itex] (the function of time, the scale factor, doesn't affect the spatial curvature).

This goes back to the first Friedmann equation above:
[tex]H^2 = {8\pi G \over 3}\rho - {k \over a^2}[/tex]

This indicates that the effect of the curvature drops off as [itex]1/a^2[/itex]. So if your energy density dilutes more slowly than that, the spatial curvature gets smaller and smaller with time.

nomadreid

Thank you very much, Chalnoth. That helps a lot. I'll be working through these equations.

Driftwood1

Not in a French Court

And of course in most Western legal Courts (especially in the USA) the presumption of innocence directly depends on the cirminality of your defence council and this in turn depends on how much money and/or political influence you have.

Mathematics as it was sparked off by people like Pythagoras and Euclid is for dreamers and abstract ideologues who fear the centre of cyclones/hurricanes because the pressure is deceptively lower.

Mathematicians are the last remaining philosophical priests that are being dragged into the quasi-reality created by the modern scientific demi-gods

You do realise this dont you?

nomadreid

Another name for a dreamer is a person with imagination and creativity. Now that Physics has ascribed to a version of the link between reality and numbers that Pythagoras espoused, and to the axiomatic method of which Euclid was a pioneer, the division between "realists" and "dreamers" is no longer as clear as many believe.

One must distinguish between the two realms of mathematics, that of pure mathematics and that of applied mathematics. Although the two areas overlap considerably, there is still a lot of mathematics that has no known use. Those who delve into these areas can be said to be immune, for the present, to being "dragged". Whereas I agree that they are philosophers -- in the sense that they think, which is not a bad thing -- if they are priests, then I do not know of their congregation, as most people do not pay them much attention. The lay public now and then sees some popular article about mathematicians or a mathematical result, such as the solution of Fermat's Last Theorem by Singh or of Poincare's Conjecture by Perelman, and are given a tiny glance at applied mathematics in schools or their profession, but otherwise it is the results of physicists and, even more, by engineers that grabs their attention. In any case, I do not object to mathematics being used in the service of science: after all, this allowed me to get a good answer to my question in this thread (see Chalnoth's answer). By the way, if scientists were considered demi-gods, people might pay more attention to them too: but a look at the political landscape indicates that this is, alas, not the case.
Finally, I have given up on the word "reality". It joins the words such as "love", "justice", "liberty" in its ambiguity.

Driftwood1

I hope that you are not taking my random linguistic ramblings seriously?

I must warn you in advance that it may lead you to a lifetime of therapy and expense

what "word" have planted in the place of "reality"?

Chalnoth

This is slightly off-topic, but all of science assumes that there exists some sort of self-consistent reality external to our own perceptions. The expectation is that our perceptions of this reality are imperfect, and that we can't ever fully know its nature. But we can achieve various degrees of confidence as to the truth of certain aspects of said reality using science.

So a decent-enough definition of reality is "that which exists independent of our own perceptions".

Driftwood1

...so you are referring to an absolute reality which we can only approximate at best?

This is fine until you get to the weird world of Quantum mechanics.

If the Qauntum physical world is as described by the theory - ie basically a stochastic world - where is our absolute reality then?

Chalnoth

Nope, quantum mechanics doesn't change this a bit. It just means that the absolute reality is properly described by a wave function, which, as before, we are doomed to know imperfectly.