How Does Differential Geometry Help Calculate the Cosmological Constant?

[marcus]

http://arxiv.org/abs/gr-qc/0609004
Calculation of the Cosmological Constant by Unifying Matter and Dark Energy
Torsten Asselmeyer-Maluga, Helge Rosé
23 pages, submitted to Adv. Theor. Math. Phys

"We show that the differential-geometric description of matter by differential structures of spacetime leads to a unifying model of the three types of energy in the cosmos: matter, dark matter and dark energy. Using this model we are able to calculate the value of the cosmological constant with Lambda = sqrt(14/27) 8 pi G/c^2 rho_obs = 1.4 10^-52 m^-2."

 

[R.X.]

Sounds complete BS to me - quantum corrections will generically add to any unproteced cosmological constant a value of the cutoff, ie the Planck mass, to the power of 4. Moreover, where are the contributions from the QCD phase transition and of the symmetry breaking at the weak scale? Each of them adds corrections to the CC that are wrong by very many orders of magnitude.

To present just one classical number out of the context of a concrete physical model is almost as good as nil. This is a typical example where abstract mathematics without physics input leads to misleading, physically wrong statements.

 

[Helge Rosé]

Sounds complete BS to me - quantum corrections will generically add to any unproteced cosmological constant a value of the cutoff, ie the Planck mass, to the power of 4. Moreover, where are the contributions from the QCD phase transition and of the symmetry breaking at the weak scale? Each of them adds corrections to the CC that are wrong by very many orders of magnitude.

To present just one classical number out of the context of a concrete physical model is almost as good as nil. This is a typical example where abstract mathematics without physics input leads to misleading, physically wrong statements.

I would very appreciate any constructive criticism, but your "statement" is nothing else then unsubstantial polemics. I am not willing to discuss at this level. In case you should have read the paper and there is any well-founded criticism - let me know.

Helge

 

[marcus]

...your "statement" is nothing else then unsubstantial polemics. I am not willing to discuss at this level. In case you should have read the paper and there is any well-founded criticism - let me know.

Helge

You've got it, Helge! In my view at least, you are cordially welcome to ignore purely negative attacks... We get these from time to time, even from what i suspect are highly intelligent qualified people.

I think this forum is usually more sociable and informal than some kind of "thesis defense" ordeal, or a critical "peer-review" jury. We didnt put on our tuxedo suits yet

Some can and will offer constructive crit. But many of us simply like to be exposed to new and various ideas in congenial company.

One should try to see what might be valuable in what the other guy says---that was what Carlo Rovelli advised us, one time he came to visit here.

 

[R.X.]

I would very appreciate any constructive criticism, but your "statement" is nothing else then unsubstantial polemics. I am not willing to discuss at this level. In case you should have read the paper and there is any well-founded criticism - let me know.

Helge

I have read the paper - it is a nice mathematical curiosity, but I severely doubt it to have a physical significance. Because the problem of the cosmological constant is intrinsically tied to quantum corrections - classically it can have any value to start with, but quantum corrections will immediately wipe out any classical value. In perturbative quantum field theory, it is a strongly divergent and not computable quantity; essentially one needs to cut the divergent integral off and this then introduces a cc of the order cutoff^4. Depending on what corrections you consider, the cutoff can be the weak scale or the Planck scale - at any rate this value will be many orders of magnitude too large. Also non-perturbative corrections, such as a vacuum energy stemming from the QCD phase transition, will contribute.

This is at the core of the cc problem, which remains one of the conceptually important open problems today. Without addressing these issues, one cannot claim to have contributed anything to the solution of it.

In susy field theories, the cc is zero by symmetry, but the problem tends to reappear the moment you break susy; say at the weak scale. In (perturbative) string theory the cc is a controlled, in principle computable quantity (in contrast to QFT), but it also requires an extra structure in order to prevent it to acquire a huge value like the Planck mass; some more contrived recent constructions involving fluxes claim to achieve that.

What is often not realized is that the decoupling theorem does not hold for such divergent quantities, which means the cc gets contributions from arbitrarily high mass scales. In other words, it is sensitive to the UV structure of the theory, and without specifying that (eg in the context of a string model), there is no way to meaningfully address this problem.

Actually, string theory is much more clever than QFT, and indeed it can happen (in a 2d toy model) that the 1-loop correction to the cc vanishes despite the theory being not supersymmetry. In a way, the cancellation occurs between whole towers of fermionic and bosonic excitations, but not mass level per mass level (as it is in a susy theory). This particular mechanism is quite impossible to realize in QFT, as it relies on global(modular) properties of the string world sheet. For details look for the paper on "Atkin-Lehner" symmetry by Greg Moore.

 

[marcus]

BTW Helge, you two show remarkable audacity to actually CALCULATE the value of a fundamental parameter.

I probably have some of the deep-seated prejudices as others do, and tend to regard such attempts as "numerology". But I try to see the positive.
In fact two excellent PF members, Hans and Alejandro, have been investigating ways to calculate several basic dimensionless constants---not naively, but as a way of digging up hints of underlying structure.

It is a risky gamble. But I suppose that sometimes if you just happen to be able to calculate a fundamental number, by making certain geometric assumptions, this can inspire you to look at the geometry a different way.

I think you actually scare me on page 3-----you make some assumptions about the geometry (or the differential topology) of the universe which

* seem exotic
* I don't understand, or only slightly understand
* and give rise to a value of Lambda---from the unusual geometry---in a way that seems too good to be true.

I am not criticising. I am just letting you know that I find page 3 scary, and a little exhilarating, like looking over a cliff.
Maybe it is just something to get used to.

 

[Helge Rosé]

... I think this forum is usually more sociable and informal than some kind of "thesis defense" ordeal, or a critical "peer-review" jury. We didnt put on our tuxedo suits yet

Some can and will offer constructive crit. But many of us simply like to be exposed to new and various ideas in congenial company.

... One should instead try to see what might be valuable in what the other guy says---that was what Carlo Rovelli advised us, one time he came to visit here.

Thanks Marcus,

it is great to could be here. I very like the discussion and the open mind in the forum. I like even the hard but objective criticism (the careful style) it helps very much (in our last paper some errors could be corrected by this clear mind of the forum). Thats why we post the paper strait to the forum and are curious about the comments.

helge

 

[Careful]

Thanks Marcus,

it is great to could be here. I very like the discussion and the open mind in the forum. I like even the hard but objective criticism (the careful style) it helps very much (in our last paper some errors could be corrected by this clear mind of the forum). Thats why we post the paper strait to the forum and are curious about the comments.

helge

Thanks, you make me blush , but I am too busy at the moment with my own work. I am not going to read something carelessly and then give comments on it.



Careful

 

[Helge Rosé]

I have read the paper - it is a nice mathematical curiosity, but I severely doubt it to have a physical significance. Because the problem of the cosmological constant is intrinsically tied to quantum corrections - classically it can have any value to start with, but quantum corrections will immediately wipe out any classical value. In perturbative quantum field theory, it is a strongly divergent and not computable quantity; essentially one needs to cut the divergent integral off and this then introduces a cc of the order cutoff^4. Depending on what corrections you consider, the cutoff can be the weak scale or the Planck scale - at any rate this value will be many orders of magnitude too large. Also non-perturbative corrections, such as a vacuum energy stemming from the QCD phase transition, will contribute.

Dear R.X.

Nice to see you let your BS-style behind you. Now I very understand the core of your problem with our approach. It is true, QFT assumes the origin of lambda is the vacuum energy - but this is a assumption (like our assumption that the source is geometrically). The vacuum energy assumtion leads to a myriad to large lambda as you and everybody know. At this point you have two alternatives:

1. You can insist in your assumption -> you get the famous "cc problem"
2. You can replace your assumption -> you get the chance for a new explanation

We chose number 2. I guess you will disagree - one can not drop the lambda = vacuum energy assumptions - QFT is a good physical theory and any assumption it suggest must be true.

Here is the point were we disagree. I think, more important than orthodox assumptions is observation and if the experiments says: the is no lambda = E_v then we have to look for a better assumption. Maybe you r way of thinking is an other.

So let me short explain why I don't believe the lambda = E_v. QFT is a great theory and very successful - no doubt. Does this mean any implication of QFT is true - no. Vacuum ergery is supposed as real - e.g. because of the Casimir effect - beause you can measure it. But in the Casimir effect you don't measure the absolut value of E_v - you measure the changes. The changes are real - but the value of E_v, responsible for the lambda = E_v assumption - can not be measured. There is not any effect in QFT which can measure E_v. Thats why lambda = E_v should be dropped. A interesting paper to this questions is astro-ph/0604265

This is at the core of the cc problem, which remains one of the conceptually important open problems today. Without addressing these issues, one cannot claim to have contributed anything to the solution of it.

A new approach must be consistent with the observations - it needs not explain the wrong statements of an other approach (It would be like a question on SRT "But where is the aether?").

I hope I could motivate a little bit my direction of thinking.

helge

 

[Helge Rosé]

BTW Helge, you two show remarkable audacity to actually CALCULATE the value of a fundamental parameter.
I probably have some of the deep-seated prejudices as others do, and tend to regard such attempts as "numerology". But I try to see the positive.
In fact two excellent PF members, Hans and Alejandro, have been investigating ways to calculate several basic dimensionless constants---not naively, but as a way of digging up hints of underlying structure.
It is a risky gamble.

You are absolutely right and we are aware of the risk that a tiny mistake could make the result totally wrong. But we state a hypothesis not a truth. The hypothesis has to output the right values found by observations and it has to survive. This paper is restricted to the global structure and based only on GRT. There are no explicit connections to QFT. The only reason why we could "calculate" the fraction is the fact : global structures are determined by topology and can calculated by topological methods. The relation between curvature and Chern-Simons invariant is a very good idea of Witten. Only by this identification it is possible to calculate some thing. And only a relative quantity like a fraction - the absolute energy densities could only determined by a full dynamical theory. Our current approach is only static.

I think you actually scare me on page 3-----you make some assumptions about the geometry (or the differential topology) of the universe which

* seem exotic
* I don't understand, or only slightly understand
* and give rise to a value of Lambda---from the unusual geometry---in a way that seems too good to be true.

I am not criticising. I am just letting you know that I find page 3 scary, and a little exhilarating, like looking over a cliff.
Maybe it is just something to get used to.

Marcus, this question and honesty is very helpful. I will try to answer it.

* exotic:
To be able to use some mathematical result we have to require some properties of spacetime
compact, closed, differentiable, simply connected.
A physically interpretation could be:
compact - not degenerated infinite any meaningful quantity should have a finite limit
simply connected - there is no more then one spacetime that counts
differentiable - any physical theory needs this
closed - this not evident (the mathematical statements need it - theorems without this requirement could be exist - but are not proved yet)

The differential structure is our basic object (like metric in GRT) - this is an basic assumption.

The DS determines the submanifold A and it boundary determines a 3-manifold Sigma. I.e. there is a map DS -> Sigma. The only global and important 3-MF is space itself - that's why we assume sigma = cosmos.

For a moment assume that all. You have a 3-MF with a curvature. Now you take GTR: curvature = engery and Witten: curvature = Chern-Simons. If you want to calculate the energy fraction you can now do that by a fraction of CS invariants. If you know the topological form the the 3-MF you can do that. The DS determines this form. Now assume the simplest form of the DS and try the calculation of the CS. If the fraction is ok - you have a new approach, if not - drop everything.

helge

 

[CarlB]

Vacuum ergery is supposed as real - e.g. because of the Casimir effect - beause you can measure it. But in the Casimir effect you don't measure the absolut value of E_v - you measure the changes. The changes are real - but the value of E_v, responsible for the lambda = E_v assumption - can not be measured. There is not any effect in QFT which can measure E_v. Thats why lambda = E_v should be dropped. A interesting paper to this questions is astro-ph/0604265

As I'm sure you know, Julian Schwinger was not fond of the QFT vacuum and the Casimir effect. He founded source theory to avoid it, and there are a bunch of calculations that compute Casimir effect without a vacuum energy. I'd link to a 2006 paper that goes into this, but arxiv is down.

Schwinger's motivation was to avoid the need to cancel infinities, but it also had the effect of suggesting that the QFT vacuum was a mathematical artifact. This is in analogy with the situation in QM with regard to density matrices:

With density matrices, you can rewrite QM in Clifford algebra notation and completely avoid using spinors. In doing this, you can also derive spinors from the density matrix formalism. Spinors do nasty things like get factors of -1 when rotated through 2 pi. This reeks of an arbitrary "mathematics" effect rather than "physics", and it comes from a sort of inability to consistently define the square root of a vector without introducing an arbitrary complex phase.

Now when you translate the above from QM into QFT, you end up getting rid of the vacuum state. To put it into the language of creation and annihilation operators, in eliminating the vacuum one is recognizing that nature never annihilates something without creating something else. The QM density matrix formalism way of saying the same thing is that nature never allows you to detect a quantum state, but instead restricts you to only make measurements -- and a measurement consists of an initial state and a final state. That is, in QM, the states are unphysical, only the combination of initial and final make physical sense, and in QFT, the creation and annihilation operators are unphysical, only operators consisting of combinations of them as seen in nature make physical sense.

I started a thread to discuss the vacuum:
https://www.physicsforums.com/showthread.php?t=130683
but it hasn't had any significant response yet. I'd love to hear cogent arguments for and against why the QFT vacuum is required in quantum mechanics. I think that Schwinger's arguments were ignored rather than answered; and that because the mainstream way of making calculations was somewhat easier. If you have any arguments for or against the vacuum, I'd like to hear them. Perhaps it belongs here in "beyond the standard model" rather than the QM forums.

A thread that discusses how one gets from density matrices to spinors is on PF here: https://www.physicsforums.com/showthread.php?t=124904

Carl

 

[Helge Rosé]

As I'm sure you know, Julian Schwinger was not fond of the QFT vacuum and the Casimir effect. He founded source theory to avoid it, and there are a bunch of calculations that compute Casimir effect without a vacuum energy. I'd link to a 2006 paper that goes into this, but arxiv is down.

Thanks for this interesting info - could you post the arxiv link.

To put it into the language of creation and annihilation operators, in eliminating the vacuum one is recognizing that nature never annihilates something without creating something else.

I'm not engaged in the E_v debate , I think it is a inner-QFT problem and it could be disappear if we get a better understanding of the other great QFT-problems (singularities, notion of spacetime point, relation to GR).

But your statement is interesting: there is never nothing - only changes. It seems that the characterization of differential structure leads to a similar statement.

DS is a global state of spacetime. Spacetime don't change - a 4-dim set in 4 dimensions can not vary (it would be 5-dim). I.e. also DS is globally fixed. Does it mean that there is no DS dynamics. Torsten and I discuss this a long time.

Now we think: The global state of DS is constant (invariant) - this is an expression of the conservation law (or call it gauge invariance). But DS is locally represented by a decomposition of 3-dim supports Sigma_i (of singular connections). Now, if you change one Sigma_i an other Sigma_j has to changed too because the total DS state has to be invariant. The Sigma_i are the fermions - if you annihilate one an other one has to be created because of the conservation law.
I do not say this explains in any way that there is no E_v. It is an alternative represention (like yours) which do not use the problematic notion of a vacuum state.

Perhaps it belongs here in "beyond the standard model" rather than the QM forums.
Carl

Maybe you start a own thread here or we move the discussion to your QM thread.

helge

 

[Mike2]

I'm not engaged in the E_v debate , I think it is a inner-QFT problem and it could be disappear if we get a better understanding of the other great QFT-problems (singularities, notion of spacetime point, relation to GR).

Forgive me if I've missed a key argument. But why can't it be true that the calculated value of the cosmological constant and the globally measured value both be correct? Could it not be that the vacuum energy is affected by gravity so that it is actually stronger inside the gravity wells of galaxies? Or would there be other consequences contradicted by observation if the vacuum energy were different by 120 OOM? Thanks.

 

[Helge Rosé]

Forgive me if I've missed a key argument. But why can't it be true that the calculated value of the cosmological constant and the globally measured value both be correct? Could it not be that the vacuum energy is affected by gravity so that it is actually stronger inside the gravity wells of galaxies? Or would there be other consequences contradicted by observation if the vacuum energy were different by 120 OOM? Thanks.

Hi Mike, nice to read you.

Sorry, I think I don't understand your point. Could you explain:
what is the calculated value - lambda_vacuum ?
what is the globally measured value?

As I understand is lambda a homogenous effect caused by an energy contibution. If your contribution is to big no effect can reduce it again. I mean real energy is extensive.

 

[Mike2]

Hi Mike, nice to read you.

Sorry, I think I don't understand your point. Could you explain:
what is the calculated value - lambda_vacuum ?
what is the globally measured value?

As I understand is lambda a homogenous effect caused by an energy contibution. If your contribution is to big no effect can reduce it again. I mean real energy is extensive.

Well, I'm at work, and I didn't have time to go into a lot of detail. I was assuming everyone was aware of the CC problem - that the calculated value from QFT is 120 orders of magnitude larger than the value derived from GR for the energy density which provides a pressure sufficient to cause the acceleration in the universe's expansion that is observed.

I don't know of any argument that says that the CC cannot be influenced by gravity such that the cc value obtained here on Earth using QFT cannot have a different value in intergalatic space. Here on Earth the CC is calculated using QFT using the parameters (put in QFT by hand) obtained by experiments here on earth. If we tried those same experiments in intergalatic space to determine the parameters, would the QFT using those parameters result in the CC we observe for the universe as a whole? Would that adjusted QFT predict results in contradiction with present observations - That seems unlikely since we cannot measure something in intergalatic space (nothing there to measure by definition) - unless it would cause strange things with the light traveling through it.

 

[CarlB]

Thanks for this interesting info - could you post the arxiv link.

Schwinger's stuff was written back before arXiv, so you can only access echoes of it there now. See the discussion on the top of page 4 of the following:
http://www.arxiv.org/abs/hep-th/9811054
or if you have university access:
http://prola.aps.org/abstract/PRA/v45/i7/p4241_1

I couldn't find a good description of source theory in a convenient spot, but here's K. Milton's historical notes:
http://www.arxiv.org/abs/hep-ph/9505293

The above mentions gauge theories as being the death of source theory. It's likely that my mind has reversed the two ideas as I see an unphysical vacuum as being a part of the unphysical gauge freedoms.

Schwinger's derivation of the spinor structure of QM from measurement principles is titled "Quantum Kinematics and Dyanmics", $14 on Amazon:
https://www.amazon.com/gp/product/080538510X/?tag=pfamazon01-20

His reference to the fictitious vacuum are in section 2, very early in the book. I quoted this, in a very poorly written and confused paper which should otherwise be ignored, see sections IV and V, pages 6-9 of
http://brannenworks.com/GEOPROB.pdf

Eventually I will get around to fixing the above, but I still don't understand "mass", and until I do, I don't see any reason to waste my time writing gibberish about it. (Some will say this hasn't stopped me in the past.) But the above four pages will give an introduction to Schwinger's theory where the vacuum comes in only as an arbitrary mathematical method of splitting a density matrix version of QM into state vectors.

What is said to have killed source theory is gauge theory, but this is a subject that also is very closely tied to the vacuum, as the above notes will make clear. In short, passing to the density matrix form eliminates the U(1) gauge freedom and the other gauge freedoms can be eliminated by an analogous operation. And in terms of Schwinger's measurement algebra, going back to the pure density matrix form eliminates the need for the vacuum.

But your statement is interesting: there is never nothing - only changes. It seems that the characterization of differential structure leads to a similar statement.

DS is a global state of spacetime. Spacetime don't change - a 4-dim set in 4 dimensions can not vary (it would be 5-dim). I.e. also DS is globally fixed. Does it mean that there is no DS dynamics. Torsten and I discuss this a long time.

At best I am only a "first class maverick amateur" physicist (thanks Kea), and you've just gone over my head already. I was only a physics major for two years of graduate school when my main interest was elementary particles. But I spent a bit over a decade studying mathematics, undergraduate and then graduate, and I figure that I can understand this if you point me at a good article written at 1st year or 2nd year grad student level and give me a lot of time. Sometimes the best sources are PhD theses.

Carl

 

[Helge Rosé]

Schwinger's stuff was written back before arXiv, so you can only access echoes of it there now. See the discussion on the top of page 4 of the following:
http://www.arxiv.org/abs/hep-th/9811054
or if you have university access:
http://prola.aps.org/abstract/PRA/v45/i7/p4241_1

...

Thanks for the links.

... a good article written at 1st year or 2nd year grad student level and give me a lot of time. Sometimes the best sources are PhD theses.
Carl

At the moment there only heavy math articles. A much of topology is needed - difficult to understand for many physicists - including me :-) Maybe the new book of Torsten and Carl Brans will help: https://www.amazon.com/dp/981024195X/?tag=pfamazon01-20

best regards

helge

 

[Helge Rosé]

Well, I'm at work, and I didn't have time to go into a lot of detail. I was assuming everyone was aware of the CC problem - that the calculated value from QFT is 120 orders of magnitude larger ...

Of course this is well known.

... the cc value obtained here on earth

the measured lambda is homogeneous - i.e. it is not a function of space. The value of lambda is a global quantity - it is e.g. measured by observation of the homogeneous, global cosmic background. There is no reason caused by observation to assume that lambda is changing with space. Sure, Weinbergs model says that a varying vacuum energy could produce the observed lambda - but then QFT has to explain at first why vacuum is spatial distributed in such a very special way. (And again, there is no inner-QFT effect that allow for a measurement of E_vac - i.e. QFT alone would not notice that E_vac is spatial distributed.)

best regards
helge

 

[Mike2]

Of course this is well known.


the measured lambda is homogeneous - i.e. it is not a function of space. The value of lambda is a global quantity - it is e.g. measured by observation of the homogeneous, global cosmic background. There is no reason caused by observation to assume that lambda is changing with space. Sure, Weinbergs model says that a varying vacuum energy could produce the observed lambda - but then QFT has to explain at first why vacuum is spatial distributed in such a very special way. (And again, there is no inner-QFT effect that allow for a measurement of E_vac - i.e. QFT alone would not notice that E_vac is spatial distributed.)

best regards
helge

Let's see... you've mentioned that lambda is measured here on Earth by the interactions that we observe in the accelerators. And you've mentioned how lambda is measured from the CMB (where can I learn more about that?). But the CMB was created when the universe was dense and here on Earth we are in a gravity well. So could it be that lambda is much smaller in intergalatic space?

You mention that the measured lambda is homogenous. But we cannot measure it in intergalatic space where there is nothing to measure. Are you sure that is not just an assumption that it is homogenous?

You also say, "Sure, Weinbergs model says that a varying vacuum energy could produce the observed lambda - but then QFT has to explain at first why vacuum is spatial distributed in such a very special way." QFT does not take into account the affects of gravity. And that's my question - does gravity affect lambda?

 

[Helge Rosé]

Let's see... you've mentioned that lambda is measured here on Earth by the interactions that we observe in the accelerators.

No.

And you've mentioned how lambda is measured from the CMB (where can I learn more about that?).

http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation

...
Are you sure that is not just an assumption that it is homogenous?
...
And that's my question - does gravity affect lambda?

The lambda term in Einsteins equation is a special form (constant*g_mn) of an energy density (energy-stress-tensor). Thus lambda corresponds to an engery contibution. This energy curves the space, i.e. it has a gravitationally cause. In the observations (supernovae, cmb) we see this cause. Gravity don't "affect" lambda - the lambda term has a gravitationally cause.

see: http://en.wikipedia.org/wiki/Cosmological_constant
http://relativity.livingreviews.org/Articles/lrr-2001-1/

 

[Mike2]

No.


http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation



The lambda term in Einsteins equation is a special form (constant*g_mn) of an energy density (energy-stress-tensor). Thus lambda corresponds to an engery contibution. This energy curves the space, i.e. it has a gravitationally cause. In the observations (supernovae, cmb) we see this cause. Gravity don't "affect" lambda - the lambda term has a gravitationally cause.

see: http://en.wikipedia.org/wiki/Cosmological_constant
http://relativity.livingreviews.org/Articles/lrr-2001-1/

Thanks. I will take a closer look at these some time latter. But for now, the change in the acceleration suggests that the cosmological constant is changing, right? IIRC, the supernova data show that before about 4 billion years ago, the universe was still in a decelerating phase of expansion. But afterwards is started to accelerate. Does this not suggest that the cosmological constant is changing? And if it is changing with respect to the size of the universe, then does this not suggest that it has a spatial dependence - or depends on gravity?

 

[torsten]

BTW Helge, you two show remarkable audacity to actually CALCULATE the value of a fundamental parameter.

I probably have some of the deep-seated prejudices as others do, and tend to regard such attempts as "numerology". But I try to see the positive.
In fact two excellent PF members, Hans and Alejandro, have been investigating ways to calculate several basic dimensionless constants---not naively, but as a way of digging up hints of underlying structure.

It is a risky gamble. But I suppose that sometimes if you just happen to be able to calculate a fundamental number, by making certain geometric assumptions, this can inspire you to look at the geometry a different way.

I think you actually scare me on page 3-----you make some assumptions about the geometry (or the differential topology) of the universe which

* seem exotic
* I don't understand, or only slightly understand
* and give rise to a value of Lambda---from the unusual geometry---in a way that seems too good to be true.

I am not criticising. I am just letting you know that I find page 3 scary, and a little exhilarating, like looking over a cliff.
Maybe it is just something to get used to.

Hi Marcus,
I'm back from my vacation.
I understand your doubts. So let me explain it more carefully. Maybe you stop at page 3 but the interesting things come later.

We start with a 4-manifold having some obvious mathematical properties. The most important property is the existence of a smooth structure (or differential structure). But that property restricts the topology of the 4-manifold and determines a special 3-manifold in the 4-manifold. That means, that all properties of the differential structure are located at a (contractable) subset $$A$$ of the 4-manifold. The boundary of that subset is a 3-manifold having the structure of a so-called homology 3-sphere. An example of such a 3-manifold is the Poincare sphere which is not an exotic manifold. It is rather similar to an ordinary 3-sphere.
Now we look at the splitting of such a homology 3-sphere into simplier pieces and found a surprise: every homology 3-sphere is build from a sum of 3-spheres, Poincare spheres and hyperbolic 3-manifolds. Now our main conjecture is simply: that these 3 pieces are the 3 possible energy components in the universe: dark matter, dark energy and matter. Especially matter are hyperbolic 3-manifolds. It is known from Thurstons work that the most complex 3-manifolds are hyperbolic 3-manifolds. Furthermore these 3-manifolds have a surprising property: One cannot scale its size (Mostow rigidity).
But what is a hyperbolic 3-manifold? There is a generic model for such manifold. Take solid torus $$D^2\times S^1$$ embedded into a 3-sphere and knot this solid torus. Now cut out this solid torus. The resulting manifold (with boundary a torus) is in most cases a hyperbolic 3-manifold. Thus we obtain matter (in our model) from a knot. Now we have keep in mind that a knot is made from a braid which includes also the ribbons. But that is not far from the Bilson-Thompson model which is discussed in one thread.
Now back to the 4-manifold: The differential structure restricts the possible topology. The simplest case is the so-caled K3 surface used in string theory very often. The corresponding 3-manifold is a so-called (2,5,7) Brieskorn sphere which splits into two Poincare spheres, a non-determinable number of 3-spheres and some hyperbolic 3-manifolds. Thus we determine the structure of the whole cosmos and the structure of the dark energy component. Now we can determine the ratio of the whole energy to the dark energy which is calculated to be a topological constant . There is no mystic in that result if one is willing to accept the assumptions.
More difficult is the relation of that result to standard QFT or Loop QG. But we work on that having found some very interesting relations. Part of that can be found in the summary part of the paper.
So, we think that the cosmologica constant is related to quantum theory and we were able to calculate some expectation value to get its value. But we don't have a dynamics.

I hope that clarifies some questions.

Torsten

 

[arivero]

Well, thanks to the Dirac Large Number Thing, cosmology predictions of fundamental parameters have a different status than particle ones. In fact even a lot of crackpot theories are DLNH in disguise or ignorance.

Sounds complete BS to me .

Off toppic: Is BS a kind of translation of French "emmerder"?

 

[selfAdjoint]

[quote-arivero]Off toppic: Is BS a kind of translation of French "emmerder"?[/quote]Off toppic: Is BS a kind of translation of French "emmerder"?[/quote]

The B stands for bull. The S stands for the substance often found on the ground under a bull.

When Harry Truman was president he referred to some Republican idea as "Bull manure". A lady protested to Truman's wife Bess, "That is so vulgar! You should persuade him to speak more politely." Bess replied, "You don't know how hard it was for me to get him to say 'manure' !".

 

[a.k.]

We start with a 4-manifold having some obvious mathematical properties. The most important property is the existence of a smooth structure (or differential structure). But that property restricts the topology of the 4-manifold and determines a special 3-manifold in the 4-manifold. That means, that all properties of the differential structure are located at a (contractable) subset $$A$$ of the 4-manifold. The boundary of that subset is a 3-manifold having the structure of a so-called homology 3-sphere.

ok, it is a bit early this morning (or late this night), nevertheless I have to make some remarks. Your idea is actually quite nice, at least mathematically, but there remain some questions open regarding the consistency of your derivation. You start with the above described setting which constrains some of the topological and differentiable properties of the underlying 4-manifold (call it M) and some embedded homology 3-sphere sigma, called 'universe'. Nevertheless you argue on a GRT ground and GRT requires the introduction of a globally defined metric on M, so my first arising question was if it is actually possible in this formulation to define a metric on M which (e.g. by restriction) gives sigma an interpretation of a 'spacelike slice' of spacetime (in some sense), the compactness of M could make it hard to formulate already a time variable and I wonder if the embedding of sigma in M actually implies that there is at least a neighborhood of sigma that is diffeomorphic to a product sigma \times [0,1], you use this implicitly in formula (5). So ok, assuming this you look at some sort of local Robertson Walker metric with sigma as its spacelike splice ( so [0,1] \times sigma) but in contrary to traditional GRT sigma is a special Brieskorn sphere, i.e. a connected sum of hyperbolic spaces, spheres, poincare spheres. What I do not understand at this point is why you seem to start here with a fixed geometry i.e. a metric on each of these pieces (inducing Levi-Civita-Connections) and later argue about 'choices' hen it comes to fix the value of the Chern simons invariant, you should make it eventually clearer at this point that your already STARTED with a geometric (i.e. levi-civita) connection fixing the Chern simons value, the formulations here seem to me a bit misleading.
Do you know actually if this particular Brieskorn sphere is realized as the link of some polynomial (possibly all Brieskorn spheres are?) and if so, if the Milnor fibre realizing sigma as its boundary could be interpreted as an 'Akbulut cork' and what would this mean physically. Maybe one could speculate about possible physical meaning of a degeneration of this 'Akbulut cork', in the sense that for some parameter going to zero, the cork A as the Milnor fibre of its polynomial degenerates to a singular object, this one could maybe call the 'big bang'. so I think apart from the above mentioned problems which tend to touch the question for dynamic development the paper reflects some quite nice ideas.

 

[torsten]

ok, it is a bit early this morning (or late this night), nevertheless I have to make some remarks. Your idea is actually quite nice, at least mathematically, but there remain some questions open regarding the consistency of your derivation. You start with the above described setting which constrains some of the topological and differentiable properties of the underlying 4-manifold (call it M) and some embedded homology 3-sphere sigma, called 'universe'. Nevertheless you argue on a GRT ground and GRT requires the introduction of a globally defined metric on M, so my first arising question was if it is actually possible in this formulation to define a metric on M which (e.g. by restriction) gives sigma an interpretation of a 'spacelike slice' of spacetime (in some sense), the compactness of M could make it hard to formulate already a time variable and I wonder if the embedding of sigma in M actually implies that there is at least a neighborhood of sigma that is diffeomorphic to a product sigma \times [0,1], you use this implicitly in formula (5).

At first, thank you for the first real reaction on our paper. Now to your questions:
In that formulation there is no global foliation but there is always a codimension-1-foliation which looks locally like $$\Sigma\times [0,1]$$. Thus there is no global metric on the 4-manifold but there is always a metric for evry chart which fit together smoothly. Yes, there is metric which can be interpreted that $$\Sigma$$ is space-like.
There is a deep theorem in differential topology which guaranteed the existence of a tubular neighborhood of an embedded $$\Sigma$$.

So ok, assuming this you look at some sort of local Robertson Walker metric with sigma as its spacelike splice ( so [0,1] \times sigma) but in contrary to traditional GRT sigma is a special Brieskorn sphere, i.e. a connected sum of hyperbolic spaces, spheres, poincare spheres. What I do not understand at this point is why you seem to start here with a fixed geometry i.e. a metric on each of these pieces (inducing Levi-Civita-Connections) and later argue about 'choices' hen it comes to fix the value of the Chern simons invariant, you should make it eventually clearer at this point that your already STARTED with a geometric (i.e. levi-civita) connection fixing the Chern simons value, the formulations here seem to me a bit misleading.

Yes you are right that seems a bit misleading. The point is that the Chern-Simons invariante is an invariant of the 3-manifold for a general SU(2) flat connection. But we know that one of these connections induces a Levi-Civita connection. Fintushel and Stern proved that the minimal Chern-Simons invariante is induced from the Levi-Civita connection.

Do you know actually if this particular Brieskorn sphere is realized as the link of some polynomial (possibly all Brieskorn spheres are?) and if so, if the Milnor fibre realizing sigma as its boundary could be interpreted as an 'Akbulut cork' and what would this mean physically. Maybe one could speculate about possible physical meaning of a degeneration of this 'Akbulut cork', in the sense that for some parameter going to zero, the cork A as the Milnor fibre of its polynomial degenerates to a singular object, this one could maybe call the 'big bang'. so I think apart from the above mentioned problems which tend to touch the question for dynamic development the paper reflects some quite nice ideas.

Yes, all Brieskorn spheres are realizable as the link of some polynomial. In case of $$\Sigma(2,5,7)$$ it is $$x^5-y^7$$ leading to a (5,7) torus knot. Yes right one can interprete the Akbulut cork as the Milnor fiber of that singularity. Now to your big bang question. A key point in the whole construction is the fact that the Akbulut cork is contractable. That means we find a homotopy which contract that 4-manifold to a point. But a neighborhood of a point in a 4-manifold is always a 4-ball $$D^4$$ with boundary a 3-sphere. Thus we can state that there was a big bang beginning from a point. In that time the 'universe' had the topology of a 3-sphere. But later that changes to the Brieskorn sphere. Thus from the topological point of view we had a kind of phase transition 3-sphere -> Brieskorn sphere.

Torsten

 

[a.k.]

At first, thank you for the first real reaction on our paper. Now to your questions: In that formulation there is no global foliation but there is always a codimension-1-foliation which looks locally like $$\Sigma\times [0,1]$$. Thus there is no global metric on the 4-manifold but there is always a metric for evry chart which fit together smoothly. Yes, there is metric which can be interpreted that $$\Sigma$$ is space-like.

hm, I do not quite understand, so you mean M (or A?) can be equipped with a lorentz-metric that makes it locally isometric to a product $$\Sigma\times [0,1]$$ equipped with a Robertson-Walker-like warped product and in this sense $$\Sigma$$ would be 'locally' spacelike?

There is a deep theorem in differential topology which guaranteed the existence of a tubular neighborhood of an embedded $$\Sigma$$.

yes I already guessed something like this, the point is that interpreting A as a Milnor fibre one should have something like a canonical foliation, at least locally, given by a family of intersections of the Milnor fibre with appropriately small spheres or ellipsoids, where the leaves of the foliation correspond to different radii of the spheres/ellipsoids, this is what made me think of the 'central fibre' of a quasihomogeneous (Brieskorn) polynomial which is in a natural way a symplectic manifold with contact type boundary and carries a (outside of the singularity) global foliation which collapses the boundary to the singular point for vanishing radii of the intersecting ellipsoids, these ellipsoids are preserved by a natural $$\mathbb{C}^*$$-action on the fibre, whose radial trajectories remind me a bit of matter fields in GRT.

Now to your big bang question. A key point in the whole construction is the fact that the Akbulut cork is contractable. That means we find a homotopy which contract that 4-manifold to a point. But a neighborhood of a point in a 4-manifold is always a 4-ball $$D^4$$ with boundary a 3-sphere. Thus we can state that there was a big bang beginning from a point. In that time the 'universe' had the topology of a 3-sphere. But later that changes to the Brieskorn sphere. Thus from the topological point of view we had a kind of phase transition 3-sphere -> Brieskorn sphere.

hm, I still wonder if there is some kind of 'akbulut cork' for 'singular' spaces and if it could be represented under some conditions (to its boundary) by fibres of Milnor fibrations or complete intersections inducing its differentiable structure, then one could maybe wonder what a singular Akbulut cork would mean physically, so instead of contracting the whole of A one could argue whether one would have locally around the 'big bang' something like a 'cusp' or a 'double point' (or something more complicated) in the four manifold itself, which would mean that spacetime is algebraically singular instead of just having some 'degenerated metric' along the spacelike slices as in GRT; I admit this picture is not quite clear, but to interpret the spacetime as differentiably indurced by a Milnor fibre induces a huge machinery of mathematical strcuture and results, I wonder what 'monodromy' would mean in this picture, or even symplectic monodromy, what does it mean that the Akbulut cork sits in a topologically or symplectically non-trivial fibration. Do you have eventually an idea if the Eta-Invariant on the Milnor bundle over the circle could play a possible role, you mentioned the Eta-invariant on the link, which determines some Chern-Simons values; I calculated some eta-invariants over the Milnor bundle itself, I still wonder about physical applications.

Andreas

 

[torsten]

Dear Andreas,
sorry for the late answer:

hm, I do not quite understand, so you mean M (or A?) can be equipped with a lorentz-metric that makes it locally isometric to a product $$\Sigma\times [0,1]$$ equipped with a Robertson-Walker-like warped product and in this sense $$\Sigma$$ would be 'locally' spacelike?

Yes you understand it right.

yes I already guessed something like this, the point is that interpreting A as a Milnor fibre one should have something like a canonical foliation, at least locally, given by a family of intersections of the Milnor fibre with appropriately small spheres or ellipsoids, where the leaves of the foliation correspond to different radii of the spheres/ellipsoids, this is what made me think of the 'central fibre' of a quasihomogeneous (Brieskorn) polynomial which is in a natural way a symplectic manifold with contact type boundary and carries a (outside of the singularity) global foliation which collapses the boundary to the singular point for vanishing radii of the intersecting ellipsoids, these ellipsoids are preserved by a natural $$\mathbb{C}^*$$-action on the fibre, whose radial trajectories remind me a bit of matter fields in GRT.

Thats interesting what you wrote. I have to learn more about Milnor fibrations. But from my knowledge about foliations I know that a codimension-1 foliation always exists on a manifold with vanishing Euler characteristics. Then the contractability of the Akbulut cork A enfoces them to have such codimension-1 foliation. But I think more is true. I would suppose that there exists also a Milnor fibration but I thought the Milnor fibers are non-trivial manifolds? Maybe I'm wrong. I know the example of the singularity $$x^2+y^3+z^5$$ written as a complex surface $${\mathbb{C}}^2/E_8$$. Then the Milnor firbation produces a 4-amnifold with intersection form $$E_8$$ and with boundary the Poincare sphere. Is it right or do I miss the point?

hm, I still wonder if there is some kind of 'akbulut cork' for 'singular' spaces and if it could be represented under some conditions (to its boundary) by fibres of Milnor fibrations or complete intersections inducing its differentiable structure, then one could maybe wonder what a singular Akbulut cork would mean physically, so instead of contracting the whole of A one could argue whether one would have locally around the 'big bang' something like a 'cusp' or a 'double point' (or something more complicated) in the four manifold itself, which would mean that spacetime is algebraically singular instead of just having some 'degenerated metric' along the spacelike slices as in GRT; I admit this picture is not quite clear, but to interpret the spacetime as differentiably indurced by a Milnor fibre induces a huge machinery of mathematical strcuture and results, I wonder what 'monodromy' would mean in this picture, or even symplectic monodromy, what does it mean that the Akbulut cork sits in a topologically or symplectically non-trivial fibration. Do you have eventually an idea if the Eta-Invariant on the Milnor bundle over the circle could play a possible role, you mentioned the Eta-invariant on the link, which determines some Chern-Simons values; I calculated some eta-invariants over the Milnor bundle itself, I still wonder about physical applications.

Andreas

I agree with you. One should consider the big bang as an intrinsic singularity of the 4-manifold and a cusp is a good way to begin. That remembers me on the work of Gompf about the nucleus of a complex surface. A nucleus is a 4-dim submanifold which can be written as an eliptic fibrations, i.e. the fibers are tori except for a countable number of fibers. A cusp is a singular torus (a cone over the trefoil knot) and Gompf shows that the differential structure of the nucleus changes if one change a torus to a cusp. But that infected the differential structure of the whole 4-manifold. Thus it seems to me that your ideas fit quite well into the general picture.

hm, I do not quite understand <our last sentence. Did you consider something like an equivariant (with respect to the circle) Eta invariant? Or does the wording "eta-invariants over the Milnor bundle itself" means something different?

Best wishes, Torsten

 

[MTd2]

Hmm! Nice to see Asselmeyer and Helge were here!

Besides trying to see in the fundamental level as an analogy with condensed matter, this is another approach that I am fond of. I got Asselmeyer's book and also bought the 2 best books on exotic extructures, that is, "The Wild World of 4 Manifolds" and "4-Manifolds and Kirby Calculus". Everything to study Asselmeyer, Helge and Brans ideas.

So, I've got a couple of questions for you guys:

1. Using Theorem 9.4.10, p. 371, of "4-Manifolds and Kirby Calculus", given that there is no preferencial place to fix (r,t) can I conclude that an exotic 4 manifold leads to a localy source of randomness?

2. In the case of 4-manifolds the case of exocticness is like and "intermadiate" between a complete nonsense, that is, non-differentiability and non-diffeomorphism, and the total boringness, differtiability and diffeomorphism. So, by checking the book "The Wild World of 4-Manifolds", I forgot the page, in which it talks about the 11/3 conjecture, try to find a graphic in which it shows the exclusion of non-differencial manifolds, using different theorems. Some of those little balls are exotic manifolds, I think. So, given that a region of a manifold can be exoctic, could I think of those balls as, in a certain unkown way, linked to the excited states of fundamental particles.

Thanks!