Limits of mathematics in cosmology

Lev,

when you say "The concept of expanding Riemannian space", I think at Friedmann-Lemaitre-Robertson-Walker models. This is because expansion is change of distances, hence change of metric. I don't see (yet?) the problem with this model.

"What is a prototype basic concept we will be generalizing when introducing evolving topology?"

Lev,

I was afraid you might ask a poignant question such as this. Your are, without question, infinitely more qualified than I am to discuss the topics at hand. I'm just a retired guy collaborating with another retired guy on a model of dark matter. I'm reluctant to say too much about our half-baked model because of my lack of formal training and the fact that I don't have answers to many questions. Let me just suggest that dark matter particles establish a continuum that occupies and defines the extent of the universe. In other words, we would redefine space to include only the volume contained within the continuum of dark matter, and, of course, including any interstitial gaps among the closely packed DM particles. Further we would stipulate that any volume outside the DM continuum is not part of our universe and is therefore undefined. In our emerging model, DM is, and has been, evolving throughout the expansion of the universe. An important aspect of the evolution of dark matter is that it occupies more and more space as it evolves. It is this property of DM evolution that is driving the accelerating expansion of the universe.

So, when I used the term "evolving topology", I was thinking in terms of a particular model of dark matter. In the context of these remarks, am I misusing the term "evolving topology"?

Regards,

Frank Burdge

Cristi,

I was talking of the situation in mathematics, in which there are no concepts of expanding metric or Riemannian spaces. As you pointed out, physicists do operate with related concepts introduced initially via Friedmann equations, but

[link:en.wikipedia.org/wiki/Metric_expansion_of_space]Perhaps a more complete assessment is that the interpretation of the metric expansion of space continues to provide paradoxes that are still a matter of debate.[2][3][4][5] The prevailing view is that of Chodorowski: "unlike the expansion of the cosmic substratum, the expansion of space is unobservable".[6][/link]

However, my main point is this. The concept of expanding space is so different from the classical concept of space (as the 'founding fathers' of physics envisioned it), and the very process of universe expansion is so unlike any of the classical/dynamical processes, that such process warrants its fuller explication by a radically new, structural model. Structural (as opposed to just equations) in the sense that it should elucidate the nature of the processes that drive the expansion of space, i.e. it appears that the expanding space is the RESULT of such processes. (By the way,the conventional depiction of the expansion by means of the expanding balloon makes a mockery of such a fundamental process.)

"In the context of these remarks, am I misusing the term "evolving topology"?"

I believe so Frank, and although I appreciate your generosity, still you should not assume that I'm "without question, infinitely more qualified than ... [you are] to discuss the topics at hand."

;-)

Lev,

The truth is that I (think I) understand how semi-Riemannian geometry can account for expansion of the Universe, and I don't see where the problem is. Maybe you can point out clearly a phenomenon, an 'element of reality' or what else would you consider, which is inherent to the expansion of our universe but not compatible with the expansion as viewed in general relativity. This may help me to see where the problem is.

Dear Cristi,

If you are happy with "semi-Riemannian geometry" as the model of the inflation/expansion, I don't feel I should be trying to make you unhappy about it. ;-)

Happy or not with semi-Riemannian geometry, this is not the point. Not that I don't see any problem with general relativity, but I thought that knowing precisely what particular problem we are debating would be useful. At least for me. I confess that for me is not that obvious as is expected.

Regards,

Cristi

Cristi,

Again, my reason for opening the discussion is that if we wish to accept the reality of the Big Bang, we should realize that the conventional mathematical and physical concepts of space are due for a radical rethinking and that--in contrast to the previous scientific developments--there is absolutely nothing on the mathematical shelf that can be remolded for this purpose. We need to start practically from the beginning (and that is VERY VERY hard).

However, before proceeding with the more interesting and difficult part related to which way to go, we need to see to which extent there is an understanding of the unprecedented situation we find ourselves in.

I just discovered this site; please excuse my naive question. (You can find me using MathSciNet (or Web of Science) or look at the Sept., 2010 issue of the Pacific Journal of Math for a new article.) :-)

It seems to me that Ricci flows, mean curvature flows, etc. include the ideas of "evolving topology" (after pinching off, the topology could change), "expanding metrics" and expanding Riemannian manifolds. Ricci flow is about changing the metric with "time." Geometric measure theory allows stranger objects (e.g. varifolds, currents.)

What am I missing here? (Sorry if this is a dumb question.)

Hi Kirk,

No the question is not really dumb, but I take the word "evolution" more seriously.

First, the short answer, again, is: I don't think there is any basic math. text in which such concepts as, for example, EVOLVING topology is introduced. Probably because we take the term "evolution" seriously.

The longer answer is this. Yes, indeed, in general, one can introduce, a parametric family of metrics (dependent even on several continuous parameters). But--in our context, i.e. evolution of the universe--can one safely assume that such parametric family capture such evolution?

My answer is NO: it appears that the underlying processes driving the evolution are not continuous (QM), and hence, I suggest, the resulting space cannot evolve in a ANY continuous manner. And so the big question I would actually like to discuss is: How does the space evolves?

Kirk,

First, I'm not trying to "convince" anyone, simply because at this stage in our understanding of the evolution of the universe this would be a foolish undertaking.

Second, as I mentioned above, my reason for opening the discussion is this: it stand to reason that the reality of the Big Bang, in addition to many other reasons I'm planning to give, including those outlined centuries ago by Leibniz, strongly suggest that "the conventional mathematical and physical concepts of space are due for a radical rethinking and that--in contrast to the previous scientific developments--there is absolutely nothing on the mathematical shelf that can be remolded for this purpose"(with my apologies for the repetition). We need to start practically from the beginning (and hence, quite understandably, the tremendous reluctance).

Your line of thinking appears to me as follows: whatever the nature throws at us we can handle it within the conventional formal (continuous) paradigm.

Now back to the "reality" of the Big Bang:

How, within the conventional/continuous paradigm, do you 'grow' the space when all kinds of new "particles" and other entities are emerging and interacting? I don't believe, continuous models, can offer compelling models for the ongoing emergence and interaction of qualitatively different kinds of entities. (I intend to elaborate on this later.)

By the way, even well known physicists, e.g. Frank Wilczek (slide 5), wish to see "more meat to be put on inflation" (i.e. "structure" and "mechanism").

Kirk, I forgot to answer your question about what evolution is. To simplify, I believe it is about the ongoing generation of structurally novel entities.

Lev: It is difficult to argue for or against your statement "the conventional mathematical and physical concepts of space are due for a radical rethinking." Look at the richness Dirac added to mathematics (with the help of L. Schwartz); distribution theory might not exist otherwise. On the other hand, I can't tell if you understand the mathematics that exists now; your claim might be wrong.

Perhaps you could help by explaining your knowledge of mathematics, especially basic (e.g. real analysis) and more advanced (e.g. geometric measure theory, differential geometry) mathematics. For myself, my knowledge of general relativity and QM is limited; I talk to colleagues in physics who do astronomy or QM, study some differential geometry, etc.

To veer wildly off topic, I have become interested in harmonic coordinates; my search for information led me to this site. I believe they were introduced by Einstein and some really nice math papers have appeared starting in 1981 (DeTurck & Kazdan); now Michael Taylor and others use them. Are people here familiar with harmonic coordinates?

Dear Kirk,

A lot of work in harmonic coordinates was done by Vladimir Fock, presented in his book The Theory of Space, Time and Gravitation.

Best regards,

Cristi

Lev,

I'm glad that you reopened this disccusion, because as you might remember from our Email exchanges, I think a lot was left unsaid in the essay contest forum.

For one thing, although it is clear in your essay and other works, that the role you assign to time is identical to that of physical information, you often seem to downplay that aspect in favor of talking about the limitations of mathematics.

Physical information has a mathematical model, however. Could you explain why you don't think that model can be extended enough to overcome your objection to using mathematics in the ontology of physics? You know I am a fan of your "structs" concept -- and that construction is time dependent, is it not? It seems to me that it already has a constructive property suitable to mathematical modeling and computation.

I want to go further, but I'll leave it at that pending your response.

Thanks.

Tom

Kirk,

About my interests and background see the above essay (in my first post) and my web page given in the essay.

I guess we differ a lot in how we evaluate the really great progress in mathematics at present. For example, the line of development of distributions (or 'generalized functions') I would not call 'great', since I rely on the external historical evaluation of mathematics (as opposed to your, internal): as an arsenal of various useful formal languages for science, I believe the true greatness of mathematics is proportional to the power and beauty of the languages it offers. And my point is that today--for the first time in the history of science--we need new formal languages that we have NEVER had before.

Kirk,

I also forgot to answer your question about what I might call "evolving topology". (By the way, I'm not suggesting that simply evolving topology is enough, but rather that when we get the right formalism it will NATURALLY exhibit something of this nature, and I intend to discuss later such a candidate formalism.)

Let me take one of (ten or so) equivalent ways of generating topology: via the a base (a subfamily of open sets whose unions can generate all open sets). We probably need both: the underlying set of elements should be growing and the base itself should be changing in non-trivial ways.

The reason I'm having a hard time expressing this in the conventional math. language has to do with fact that in the formalism I have in mind (ETS)we are not dealing with 'point' sets but rather each object has a non-trivial 'discrete' structure that also evolves.

Tom,

I'm also glad you are joining the forum!

1. "the role you assign to time is identical to that of physical information, you often seem to downplay that aspect in favor of talking about the limitations of mathematics."

I didn't quite get what you mean, but I will give it a try. According to ETS formalism, there is no conventional concept of time: the struct embodies the new, richer flow of time.

Although I don't quite get what you mean by "physical information", what I have been stressing is that the fundamentally new in mathematics (and science) concept of structural representation--which as I just mentioned contains irreversible temporal information--comes to the fore.

2. "Physical information has a mathematical model, however. Could you explain why you don't think that model can be extended enough to overcome your objection to using mathematics in the ontology of physics?"

Again, Tom, the main break with the conventional math. is the concept of structural representation. As I mentioned in my essay, mathematics has evolved from the very beginning based on the 'point'/spatial representation. So I am convinced that when you switch to structural representation everything, including something equivalent to 'topology' and 'algebra' also change unrecognizably.

Hi Lev,

I am not fully conversant with ETS formalism, though I am willing to learn. Nevertheless, as I read your essay, I can discern no difference between a "temporal stream of interconnected primitives" which form structs, and a bit stream of primitive states.

Relevant to (1). What I mean is, suppose that time gives structure to space:

Then the orientability which defines the stream (indeed, even allows us to give meaning to the word "stream") is a space-time structure _because_ it is oriented. If we find that all of these primitive spacetime structures -- assuming scale invariance and infinite self similarity -- are oriented the same, we have to allow a 2-dimensional object just on the principle of orientability alone.

So primitives that are featureless unless oriented (streaming) implies both time dependency and spatial domain. Because this domain is 2 dimensional and therefore bounds an infinity of points, and because we can further assume the nonlinear evolution that time dependency implies, whatever structures that emerge in d > 2 are already oriented in the same direction. That is, we see that the 2-dimension topology (S^1) on the manifold of a two ball (S^2) is a three dimension object from any point of S^3, the four dimension sphere on which we live. This leaves the 2-ball itself undefined! We know there's correspondence between a point on the 3-sphere manifold and an interior point of the 2-ball, though, because we map an internal plane self similar to the S^2 manifold, of zero curvature. That's not the "real" structureless interior of the 2-ball though, is it? -- we only know that whatever forms may result on the plane are structured by our singular assumption of orientability.

I won't go into Riemannian geometry (every Riemann surface is orientable), but I think the implications are more or less obvious, and interesting.

Anyway, physical information theory follows the same rules as thermodynamics (Shannon). So I think you can make the connection between the Jacobson-Verlinde treatment of gravity as identical to information, and the conclusion that you and I have both reached by different routes: time is identical to information. If gravity is time dependent, then, there exists at least one quantum gravity model in which time is identical to information, because the classical gravity spacetime trajectory is reversible and the quantum information time trajectory is not. That the time metric can orient to all points of the space gives structure to an otherwise formless object. The field of information in the n-dimension stream, n > 1, is ordered by the 1-dimension time metric.

Therefore, time is more primitive than space -- I would have to be convinced that time is more primitive than the integers, before I would give up on a mathematical ontological model.

Regarding (2): I'm not convinced that we can't treat structs in a topological model using scale invariance. More later?

Tom