zulfikar.ahmed@gmail.com <zulfikar.ahmed@gmail.com>

Attachments12:56 PM (1 minute ago)

to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
Ladies and Gentlemen,
I am pleased to announce that a highly nontrivial mathematical conjecture was checked by physical intuition only.  It is that unlike MINIMAL hypersurfaces of spheres where it is a hard problem of continuous research interest for many many decades top mathematicians like Jim Simons and Shing-Tung Yau and Shing-Sheng Chern and Manfredo do Carmo and their collaborators and students had contributed, and all the theorems that try to understand the length of the second fundamental form restrictions to get rigid answers for minimal hypersurfaces close to the equator and they all have to stop to collect CLIFFORD TORI when the length squared of the second fundamental form  of the hypersurfaces exceeds a certain value which are precisely described by Alencar and do Carmo for constant curvature hypersurfaces.   The point here is that for some natural GEOMETRIC questions like minimal hypersurfaces one is considering critical points of functionals that are are similar to the Einstein-Hilbert action but they have critical points that are arbitrarily complicated but these arbitrary complicated critical points should not arise for the Einstein-Hilbert action for the same sort of physical reasoning that were used by Hilbert and Einstein.  So here we are employing physical intuition to make a mathematical conjecture in a setting where deep mathematics had been employed to try to sort out the rigidity of the solution that comes from mathematical intuition and I checked that the first level of ‘pathology’, the Clifford tori solutions, are NOT critical points of the Einstein-Hilbert action.  This could be a good mathematics paper if someone worked through proving the conjecture in full detail but I am overjoyed because this is the first case of truly surprising geometric result that occurs from a physical model that goes against the most established models of this age, i.e quantum field theory and general relativity buttressing my confidence that S4 model is the correct model.

PHYSICAL INTUITION — and this is unique in the PLANET today until S4 is accepted physics because no one else can claim to have physical intuition about S4 hypersurfaces — the Einstein-Hilbert action does NOT have critical points at these Clifford tori.  This is physical intuition for S4 that is verified by simple computation of the first variation.  But the conjecture is that NO critical point of Einstein-Hilbert action on S4 should exist that is not the equator BECAUSE that should be empty space without matter.


Attachments area

Let (X,g) be a smooth closed oriented four-manifold.  Let \mu\in\Omega^2(X) with *\mu=\mu.  Let $s\in S_X$ be a Spin^c structure and let S_+,S_-, L be the corresponding vector bundles.  If A is a connection on L and \psi is a section of S_{+} then (A,\psi) satisfy the Seiberg-Witten equations when

F(A,\psi) = ( D_A\psi , F_A^{+} - q(\psi) - i\mu)

vanishes.  The space of solutions (A,\psi) do not depend on g or \mu.  This information constitutes the Seiberg-Witten invariants of X.  Let \mathcal{M} = \{ (A,\psi): F(A,\psi)=0\}/\mathcal{G}

where \mathcal{G} is the group of gauge transformations.  Then a Theorem says \mathcal{M} is always compact.  Fixing g for generic \mu the space is a smooth finite dimensional manifold with a smooth circle action.  The dimension is \dim(\mathcal{M}) = b^1 - 1 + b^{2+} + \frac{c_1^2-\tau}{4} where \tau = b^{2+}-b^{2-} is the signature of X.  Here c_1=c_1(L).

The Seiberg-Witten invariant SW_X is defined as follows on spin-c structures over X.  First for topological reasons the dimension b^1-1+b^{2+}+\frac{c_1^2-\tau}{4} is even call it 2d.

(a)  If b^{2+} - b_1 is even SW_X=0.

(b)  If d=0  then \mathcal{M} is a finite set of points then SW_X(s) = \sum_{\mathcal{M}} \pm 1.

(c) If d>0 then SW_X(s) = \int_{\mathcal{M}} e^d where e\in H^2(\mathcal{M},\mathbf{Z}) represents the first Chern class of a bundle \mathcal{M}^0\rightarrow\mathcal{M}.

Consider the K3 surface z_1^4 + z_2^4 + z_3^4 + z_4^4=0.  This four-manifold has b^{2+} =3 and b^{2-}=19.  There is a unique spin-c structure with c_1(s)=0.  For this spin-c structure SW_{K3}=\pm 1 and for all others it is zero.  Freedman’s theorem tells us K3#\bar{\mathbf{CP}^2} is homeomorphic to #_3 \mathbf{CP}^2 #_{20}\bar{\mathbf{CP}^2} but Seiberg-Witten is nonzero for the first and zero for the second.

(These are quick notes from Clifford Taubes’ Park City Lectures to refresh my mind).


The Clifford tori S^m(\sqrt{m/n})\times S^{n-m}(\sqrt{(n-m)/n}) can occur as a constant scalar curvature when the square of the second fundamental form S=n for a hypersurface of S^{n+1}.  The Reilly formula for integral of the scalar curvature R = S_2 gives the first variation

J'(t) = \int_M 2H - 3S_3

We want to check whether these Clifford tori can pop up as critical points.  This is to check for 2H - 3S_3 = 0 for these tori.  The Ricci curvature varies 0 \le Ric \le 3/2.  The mean curvature for these is zero.  These have two distince principal curvatures \lambda_1 for the 2-sphere and \lambda_2 for the circle.  We have relations:

(a) \lambda_2+2\lambda_1 = 0

(b) 2\lambda_1^2 + \lambda_2^2 + 2\lambda_1\lambda_2 = C_0 > 0C

Solving the system gets us \lambda_1 = \sqrt{C_0/2} and \lambda_2=-2\sqrt{C0/2}.  Now just plug into 2H - 3S_3 = -3(C0/2)(-2\sqrt{C0/2}) This is not zero.

This is fabulous.  This tells us that the Clifford tori cannot arise as a critical point of the Einstein-Hilbert action for hypersurfaces of S^4.  Therefore we will have uniqueness of the critical point of Einstein-Hilbert action for the range of squared length of the second fundamental form 0\le S \le 3+\epsilon.  Extending \epsilon beyond 14n(n+4)/(9n+30) approximately 5.16  is a very hard problem according to this paper.  We need to check what this squared length of the second fundamental form translates to in terms of the radius of the universe but this is quite respectable.  We know that for totally umbilical hypersurfaces, i.e. the geodesic spheres in S^4 the first variation will vanish and now we know it will not vanish for these Clifford tori.  It would be nice to prove that there are no other critical points besides these but I am willing to take it on faith for now.

This is actually a very nice geometric result in and of itself because although I basically used the hard work of Alencar and do Carmo, of Cheng and Yau and here Q-M Cheng and S. Ishikawa and did very little work thus far myself, I can now make the bolder conjecture that these are the only critical points for the restricted length squared of second fundamental form.  It’s a little surprising result because if you look at the heavily worked field of minimal (zero mean curvature) and constant mean curvature and constant scalar curvature hypersurfaces it’s quite busy with even a nice Master’s Thesis on these things.  I don’t have intrinsic interest in working on these problems but I think this is fascinating nonetheless because based on physical intuition in this case I obtained the correct answer despite all the results for constant curvature cases that Clifford tori should not be critical points of the Einstein-Hilbert action which is generally not considered in the field.  So I conjecture that empty space should only be the equator of the round four-sphere according to the Einstein-Hilbert action in S4 physics.  In other words I am conjecturing that while it is more difficult to extend the minimal hypersurfaces without pathological examples that these are not going to be a problem for the Einstein-Hilbert action.  I am making this conjecture based on physical intuition rather than mathematical here and this is another theme that I am stealing from the string theorists who have been successful with their brand of physical intuition.  I think this is the first time in history that anyone has made a conjecture based on physical intuition for the round four sphere model of physics, so I am quite proud of this moment.




zulfikar.ahmed@gmail.com <zulfikar.ahmed@gmail.com>

2:35 AM (42 minutes ago)

to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
Ladies and Gentlemen,
The Einstein-Hilbert action is just the integral of the scalar curvature.  For the S4 model I finally have some clarity for what the critical points of the Einstein-Hilbert action can be.  I conjecture that the only critical points will be the equators of the four-sphere but I don’t know how to prove it.

What I can do now is note the simple observation that the equator is in fact a critical point of the Einstein-Hilbert action.  This follows from the fact that after some algebra in the first variation formula from Robert Reilly which is

\int_{M_t} 2H – 3S_3
where S_3 is the third symmetric polynomial of the principal curvatures and since the hypersurface is three dimensional it’s just the product of the three principal curvatures.  In the case of the equator using the fact that the scalar curvature is exactly 6 for the unit 3-sphere you can get the integrand to vanish.

There are theorems for the characterization of what can happen for constant MEAN curvatures by Alencar and do Carmo and what happes for constant SCALAR curvatures by Cheng and Yau for hypersurfaces of spheres.  They say that these can either be totally umbilic which just means all the principal curvatures are the same or tori of type S^2(r_1)xS^1(r_2).  I don’t think these tori satisfy 2H-3S_3=0 but I have not checked this yet.  More generally I would conjecture that there are no other critical points at all beyond the constant mean and scalar curature ones.  I would leave that as a conjecture.

The outcome of this conjecture would be extremely interesting for it would say that the only empty space in S4 from the Einstein-Hilbert action perspective is the equator and a uniqueness theorem would be very nice for empty space by Einstein-Hilbert action and could get us closer to seeing that Einstein’s equations minus special relativity plus extra dimension (absolutely MANDATED by Bohm-Aharanov actually).  This is a point that the entire field of physics is ignoring.  Bohn-Aharanov says that U(1) invariance of electromagnetism is FALSE.  FALSE so QED is falsified by it.  Quantum field theory is wrong as a description of Nature because it does not explain Lambda>0 which falsifies SR and it does not explain an extra dimension holding on to U(1) invariance of electromagnetism when Bohm-Aharanov shows that there cannot be a U(1) invariance.  These issues are not present in the S4 geometry.  This is why I insist that regardless of the successes of QFT and GR they are WRONG descriptions of Nature. And the right description of Nature is my S4 model.


My conjecture at the moment is that the Einstein-Hilbert action on S^4 has only one solution up to rotations of S^4 which is the equator S^3.  The problem is that when I look at the first variation of the scalar curvature integral, which I extracted from Robert Reilly’s computations in the 1970s, I have

J'(t) = \int_{M_t} 2H-3S_3

I looked around to find some geometers do some computations solving this equation and what I realized is that I was ignorant of the intuition developed by geometers that r-minimal submanifolds cannot exist on spheres.  Jim Simons proved that minimal hypersurfaces cannot exist (so H can not vanish identically) for any hypersurface of S^4.  But constant mean curvature and constant scalar curvature can both occur.  Now maybe the issue is that the above functional should have an exact cancellation for the equator for the integrand for the case of the equator and for no other hypersurface.  We know that the equator has both constant H and constant scalar curvature S_2.  Let’s try to prove that for this particular case the integrand vanishes:

H_0 = 3\lambda

S_2 = 3 \lambda^2

S_3 = \lambda^3

In this case the integrand above is (6-S_2)\lambda so this will vanish if S_2=6 on the unit sphere which is fine.  It vanishes.  I will check more later.  I am just relieved because I was worried that the integrand would not vanish at all.  I should check the constants carefully again.

So then the question is whether it is true that the equator will be the only hypersurface with constant mean and constant scalar curvature and I suspect the answer here is yes and it is already known.  So I should do some research to ensure that this is absolutely true.

Assuming the above, we have an actually interesting result: the Einstein-Hilbert action on S4 produces a unique solution that is the equator of the round-four sphere up to rotations.  That is a very powerful result for description of Nature, and it gives me some faith that in fact maybe the Einstein gravitational field equations will be the outcome of our analysis, but without special relativity and by dynamics as evolution of the hypersurface in the normal directions driven by electromagnetism.

For both constant mean curvature and constant scalar curvature hypersurfaces of space forms with positive curvature there are essentially the same restrictions.  Alencar and do Carmo have a theorem that is sharp for constant mean curvature and Cheng and Yau have a theorem for constant scalar curvature.

Theorem of Alencar and do Carmo.  Let M^n be compact and orientable and let M^n\rightarrow S^{n+1}(c) have constant mean curvature.  Choose and orientation so H\ge 0.  Let \phi be the second fundamental form.  For each H set

P_H(x) = x^2 + \frac{n(n-2)}{\sqrt{n(n-1)}} H x - n(H^2+1)

Let B_H be the square of the positive root of P_H=0 when H=0 then B_0=n.  Assume $\latex |\phi|^2\le B_H$  Then either

(i) |\phi|^2=0 and M is totally umbilical or |\phi|^2=B_H


(ii) |\phi|^2=B_H if and only if

(a)  H=0 and $M=S^p(r_1)\times S^q(r_2)$ with p+q=n

(b) H \not= 0 and M^n is an H(r) torus with r^2 < \frac{n-1}{n} and n\ge 3

(c) H\not= 0 and n=2 and M is H(r) torus r \not= \frac{n-1}{n}

Theorem of Cheng-Yau.  Let M be a compact hypersurface with non-negative sectional curvature c>0.  Suppose the normalized curvature of M is constant and greater than c.  Then M is either totally umbilical or a riemannian product of two totally umbilical constantly curved manifolds.

The papers are AlencarDoCarmoConstantMeanCurvature and Cheng-Yau.  Now we use their theorems to get vanishing of the integrand 2H-3S_3 without worry for whether there might be other more complicated solutions and make that a conjecture.  Clearly on the unit S^4(1) we have a formula for the scalar curvature n(n-1) so for 3-sphere it’s 6 and this makes the integrand vanish at the equator of S^4.

If we stay within the bounds set by the Alencar-do Carmo theorem we can only have Clifford tori so we want to get rid of these by computation.  I will leave this problem still open.





zulfikar.ahmed@gmail.com <zulfikar.ahmed@gmail.com>

10:05 PM (9 minutes ago)

to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
Ladies and Gentlemen,
I have absolutely no compunctions, shame, humility, about proclaiming my claim that the connection between Cosmological Constant Lambda = 1.11×10^{-52}m^{-2} and the Planck Constant by setting curvature of a round four sphere so that it’s curvature matches Lambda to be the best scientific model of the past 118 years.
The reason for this is that this is a fundamental model and the claim is that Nature is perfectly, not approximately but perfectly and EXACTLY described by a round four-sphere model.  Then there is a separate issue of how a scientific model about observational phenomena can ever be exact, which I am aware of quite well because I have worked as Scientist II at a biotech and have studied philosophy of science from Princeton where I took courses on philosophy of Quantum Mechanics and had studied Paul Feyerabend and Karl Popper since.  I am aware of the Empiricist tradition and I respect empiricism but on fundamental models  our tradition had to be Platonic.  Special Relativity tested to 10^{-8} precision is impressive but Special Relativity is a Platonic model.  It is based on an invariance of laws of Nature to Lorentz transformations on the flat Minkowski space.  This is in conflict logically with an observed cosmological constant of Lambda>0.  This model is actually false.  This is a matter of simple logic.  Regardless of how much we like this Platonic model we must abandon it when in it’s domain it faces a serious empirical problem such as Lambda>0.  My model is to replace Special Relativity with a round four sphere.  This may not seem like a very sharp and highly educated thing to do but in fact it is the best thing to do.  One cannot confuse the issues on the domain of phenomena that Special Relativity was meant to address.  Lambda >0 cleanly falls into that domain.  The natural instincts of physicists is to DUCT TAPE the problem or to put it aside because of the 10^{-8} precision in observations.  But this is a serious error.  The difference is that between a flat world and a curved world and by this I mean curvature without presence of matter.  This model can completely resolve all issues of gravity and quantum divide comparatively effortlessly.  If I were not confident about this, then I would not be wasting my time ensuring that I make efforts to make sure that my discovery of this feature of Nature is completely public and recognized despite some journals refusing to publish it.
Any effort made in this direction will lead to a complete replacement of general relativity and quantum fields for a beautifully simple physics.



zulfikar.ahmed@gmail.com <zulfikar.ahmed@gmail.com>

8:25 PM (14 minutes ago)

to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
​Ladies and Gentlemen,
If you take a very large distance away from the nitty gritty of technicalities and contemplate the Einstein’s gravitational field equations
Ric_{ij} – Rg_{ij} + Lambda*g_{ij} = T_{ij}

ignoring constants, this is a deterministic equation obviously because i,j are coordinates of spacetime.  But there is a deceptive quality to the determinism here. They are equations that are fundamentally highly ‘thermodynamic’ in nature meaning that the quantities involved in them are gross averages except the metric.  A metric is a sharp quantity precisely putting an inner product on a tangent space.  But these other quantities like Ric_{ij} and R are blunt objects.  It is a work of course of tremendous intellectual genius to model Nature in 1915 by such an equation but I do not believe that this is the right description of Nature at all.  It is too blunt a description to be fundamental.  But it is worthwhile noting that Ric_{ij} and R themselves are AVERAGES of the Riemann curvature tensor.  This is an interpretation of these objects that became clear to me while reading Terry Tao’s expository article on Hamilton and Perelman’s end to Bill Thurston’s Geometrization program.  It’s a nice article:


But with my penchant for simple things above all else I thought about Terry’s interpretation of Ric and R which I am not sure whether he invented or whether its ancient folklore probably the latter because maybe Chavel’s book on Eigenvalues in Riemannian geometry also contains it.  It is that scalar curvature is the average of all sectional curvatures of 2-planes randomly selected in the tangent space at a point, and Ricci curvature Ric(X,X) for a unit vector is the average of sectional curvatures of all 2-planes that pass through X.

Now this is not very deep technically but when you do have a spectacular model such as Einstein’s gravitational equations that claims to describe actual Nature then it is worth thinking about how determinism is going into this equation.  It’s saying that the precise metric of the universe shall evolve according to these average curvatures in this specific way.  This is a very surprising exact model of Nature actually and I think it is not right but to the extent that it is right, it should be more like a PV=nRT type of approximation of more precise exact laws, well I of course think if it is right it should be a consequence of individual atoms and molecules following Dirac equation and Maxwell’s equations on spinors of S4.  Of course the Einstein gravitational field equations are extremely blunt on the force driving evolution of the metric.  Consider all the possibilities opened up by Riemann in his famous paper on the foundations of geometry introducing the Riemann tensor in the presence of Gauss.  Ricci and Scalar curvatures are very blunt objects compared to the full curvature tensor.  Most likely Einstein’s equations are the right sort of idea but I am not yet convinced that it can be exactly right because it is placed on top of special relativity which is falsified by Lambda>0.

I would be most interested in actually showing that the gravitational field equations are still approximately right but the mathematics has to determine that rather than my own prejudices on the matter.
I will return to the problem of probabilistic description of exact laws from the matter and EM equation on S4 but it’s not simple for me to understand what can be exactly right.  Things are not so easy in past the limits of empiricsm after all.