Limits of mathematics in cosmology

I did ask Brendan to start the forum under a longer name "The Intrinsic Inability of All Conventional Mathematical Structures to Model the Conception of Expanding Space and the Ways Out", but he decided on the shorter name.

I believe the issue is relevant not just to cosmology but to physics too.

So, I would like to discuss the situation in cosmology and physics that has been overlooked. The situation is described in the above longer title, and very briefly the main point is this.

If we are to take seriously the inflationary scenario, we need an appropriate fundamental mathematical structure for modeling it, but the conventional mathematics can offer no such structure (there are really only several basic math structures). Again, I believe--and the history of science supports this belief--that without the help of such basic formal structure we will not be able to gain a satisfactory insight into the nature of such inflationary processes, which might be critical to the development of physics and cosmology.

A possible way out I intend to discuss later (see also my essay).

Lev, I read your linked essay with great interest. The evolving transformations system (ETS) could indeed be a very useful way to describe the evolutionary processes. I see within ETS elements of process flow, such as those incorporated in manufacturing process planning, computer program design, and project planning. Process flow charting is, and has been, a very useful tool. I apologize if my obvious simplification of the ETS is offensive. I do, however, question your inclusion of the parenthetic "irreversible" in Postulate 1, copied here for the convenience of readers:

Postulate 1: the universe is a family of evolving and interactive classes of (irreversible) processes.

If the universe is, as I suspect, cyclic, it will eventually collapse, thereby returning to a hot soup of photons, electrons, quarks, and gluons, or some similar conglomeration. In that event, all of the evolutionary processes that formed the structure of the universe during its expansion phase, will have been reversed. Hence, the processes that constitute the family of evolving and interactive classes of process, may entirely reversible. I also suspect that any gains in understanding cosmology, or physics in general, that result from the application of ETS, will be readily definable in the language of mathematics.

I look forward to reading the comments of those more qualified to comment than I.

Best regards,

Frank Burdge

Frank,

Thank you for your comments!

However can we postpone the discussion of ETS till later? My main objective in starting this forum was to address the basic inadequacy of the conventional mathematical structures to deal with the inflationary scenario.

(By the way, if what you mentioned at the end is true "that any gains in understanding cosmology, or physics in general, that result from the application of ETS, will be readily definable in the language of mathematics" the ETS would be a futile exercise, and although I'm very eager to address all these issues, but first things first. ;-) )

Dear Lev,

could you please develop the title "The Intrinsic Inability of All Conventional Mathematical Structures to Model the Conception of Expanding Space and the Ways Out"?

mainly, I think it would be useful to have a description of "the Conception of Expanding Space", and a justification of "The Intrinsic Inability of All Conventional Mathematical Structures to Model" it.

Do you mean that current mathematics or that mathematics in general cannot resolve this task?

Best regards,

Cristi

Cristi,

You are pointing us exactly in the right direction.

By "the intrinsic inability" I mean that all known math. structures (topological in particular) don't possess the capabilities to model the conception of expanding space, as it relates, for example, to the 3d (or higher dimensional) space of the universe as it expansion.

Again, I'm suggesting that to come to grips with the physical evolution of the universe we need a clearer formal guide as to the nature of the process that is responsible for the inflation/expansion. All the usual cosmological talk of inflation or expansion rely on the NUMERIC calculations and not on some qualitative formal structure. The mathematical concept of space does not give us a reasonable clue as the nature/structure of those processes. Hence, instead of trying to look for handicraft 'interfaces' just to bring us to the familiar concept of space, we need to look for new formal language that may clarify the situation. In this respect, several known forays of quantum gravity researchers into various discrete models that are responsible for 'generating' space are not accidental.

Let me try again, Cristi.

In physics, if we want to explain to a graduate student relevant concepts of space, we can point to a Riemannian (in GR) or Hilbert (in QM) spaces; but what can we point to if we are asked about the concept of an expanding space? (The concept of expanding Riemannian space doesn't exist and for good reasons.)

Lev,

I believe that it is not an "intrinsic inability" of mathematics that limits our ability to describe expanding space. The limitation lies in our lack of understanding of the roll of dark matter (DM) and dark energy (DE) in the evolution of the universe. It is only a matter of time and effort until physicists, theoretical and experimental, will discover the nature and evolution of DM and DE. Once a successful model of DM and DE is established, the topological structure of space will be clearly seen, and the mathematical modeling of that evolving topology will be relatively straightforward.

Regards,

Frank Burdge

Frank,

"Dark matter" and "dark energy" are simply first signs of the troubles to come.

However, before we jump to the "evolving topology", we should understand the logic of generalization. Topological spaces are the generalization of metric spaces, which are generalization of the Euclidean spaces, etc. What is a prototype basic concept we will be generalizing when introducing evolving topology?

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?

Hi Lev. I don't claim to be an expert in this area but the criticisms you mention don't seem convincing.

1. If you use GMT flows (e.g. level set method), you don't get "continuity" in some sense. I would have to dig around for references or examples if I wanted to be more specific but "Brakke flows" can exhibit this type of behavior. If you work with BV (or SBV) functions, varifolds or various types of (rectifiable) currents, "discontinuous" behavior can occur.

2. I'm not sure that one can model quantum mechanics very well. If your goal is a "theory of everything (in physics)," then I would start with an easy project like proving the Riemann Hypothesis. :D (What are the existing "good models" of QM? What besides probability measures, functional analysis, PDEs, etc. are involved with these models of QM? This is a vague question. One of my colleagues does QM & QC and has been a "fellow" (?) at KITP; I'll eventually try to formulate my question in a better manner.)

3. Varifolds are Radon measures on the manifold crossed (Cartesian product) with the appropriate Grassmannian. Don't people like Robert Bartnik, Tom Ilmanen, etc. (maybe Gerhard Huisken, Klaus Ecker) use GMT to study general relativity? (Maybe I should throw in Fred Almgren's "multi-valued functions" for fun.)

I don't think "basic math texts" would ever cover this stuff at a serious level. I'm not sure what you mean by "EVOLVING topology" as a concept; it sounds completely trivial. Now "evolution" is not trivial but the idea that topology changes as a parameter (e.g. time) changes is trivial.

What is evolution? Imagine the praise if the evolution of the universe turned out to be the solution of some flow problem and you identified this flow problem. Could QM be included in a "measure-theoretic" flow problem? I have no idea if this even makes sense but it reminds me of "string theory - banes - M theory" and multiverse (meta-universe, metaverse) theory.

I understand that physicists want to understand "reality" and not just some mathematical theory. If you would be so kind as to define "reality" this would help. ;D