Compact models of the universe
Zulf Ahmed
March 16, 2010
1
Summary
The discovery in 1984 of crystals with rotational symmetry of order 5 and there-after others of orders 8, 10 and 12 provide evidence of a fourth macroscopic dimension. Isotropic cosmic background radiation was discovered in the 1960s whose intensity distribution can be fitted by a thermal Planck form with excel-lent precision. We show how these two pieces of data can be used to show that the universe must be a compact four-dimensional manifold.
An immediate issue is whether our approach can be fruitful given the widespread acceptance of the theory that the universe is expanding. The spherical models of the universe rely on a static universe if the quantum of energy is to be constant. But the expanding universe theory relies crucially on the linear relationship observed between galaxy distances and their redshift. An unorthodox interpre-tation of the redshift is that it is a result of the shape of the CBR intensity dis-tribution, which, as a function of frequencyνis of the formCν3exp(¯hν/kT−1)
wherek is the Boltzmann constant andT is the temperature. This is nonzero but small for starlight frequencies forT ∼2.72K. However, the skew at higher frequencies is toward lower frequencies. This skew could explain the approx-imately linear relation between distance of a galaxy and its redshift, as the photons spend more time in the CBR.
The natural Laplacian and the Dirac operator on spinor fields have discrete spectra on compact manifolds since the spectral theory of self-adjoint operators applies to the Laplacian since it has a compact resolvent. The Dirac operator squared is a Laplacian on spinors and also has discrete spectrum. In particular a round four-sphere of radius 1/¯his able to describe quantum mechanical results in terms of the geometry of the universe.
SupposeM is an embedded three-dimensional submanifold ofS=S4(1/¯h). The Ricci tensorsRM
i jandRSij forM andS for vector fields tangent toM by
RMij =RSij−4HBij−Cij (1)
where Bij is the second fundamental form and Cij is the third fundamental
form for the embedding. Thus ifλ= ¯h2/12 thenRS
ij−λgij = 0 implies
If we set 8πGTij = −4HBij−Cij we recover formally the gravitational field
equations of Einstein:
RMij −λgij= 8πGTij. (3)
Recall that if A(X) = −∇XN then B(X, Y) = g(A(X), Y) and C(X, Y) =
g(A(X), A(Y)), both of which are symmetric tensors.
These are promising observations to support S4(1/¯h) as model of a static
universe where electromagnetism can described by classical Hamiltonian me-chanics and a three-dimensional subspaceMtthat evolves in time describes the
evolution of the physical universe. Since the Einstein gravitational field equa-tions occur as a description of the embedding (via the second fundamental form) there are no apparent conflict between electromagnetism and gravitation. More-over, electromagnetism naturally hasSU(2) gauge invariance in four dimensions and unlike in flat three dimensional space, a classical model of an electron or-biting a nucleus need not lose electromagnetic energy – i.e., stability of matter is not an issue even with classical mechanics.
2
Sketch of arguments
On complete riemannian manifold with a lower bound on Ricci curvature the heat kernel of operators of type ∆ +V where ∆ denotes the positive Laplacian have Gaussian upper bounds under a wide class of potentials V. If the back-ground radiation has occurred as a result of heat diffusion, the observed uniform lower bound implies that the universe cannot be non-compact or open.
Since the CBR intensity distribution has exponential decay past the peak intensity, it is nonzero near starlight frequencies. This produces a skew towards lower frequency at starlight frequencies. The effect of light detected from distant galaxies should be linear in the distance since CBR is isotropic. This observation could perhaps explain the remarkable regularity of Hubble’s relation between galaxy distance and redshift without resorting to an explanation in terms of expansion of the universe.
The classical Maxwell’s equations with magnetic charge are:
∇ ×E+ ∂ ∂tB= 0
∇ ×B− ∂
∂tE= 0
∇E=ρ
∇B=µ
Simplifying this system to a toy model of a circle, whereEandB are separable functionsE(x, t) =e(x)m(t) andB(x, t) =b(x)n(t) leads to wave equations for both. For a fixedxthe solutions formandnare both proportional to
exp(
r
bxex
Now consider the Maxwell equations on a sphere with∇× replaced by the Dirac operatorD. Take an inner product with an eigenspinorDϕ=λϕand let e(t) =hE, ϕiand b(t) =hB, ϕiso that the equations reduce to
λe(t) +∂tb(t) = 0
λb(t)−∂te(t) = 0
The solutions aree(t) = C1exp(iλt) andb(t) = C2exp(iλt) where Ci contain
the initial conditions. For the standard round four-sphere,λ=±(2 +k). The solution to the Maxwell equation are obtained by superposition of these:
E(x, t) = ∞
X
k=−∞
C1kϕk(x) exp(i(2 +k)t) (5)
B(x, t) = ∞
X
k=−∞
C2kϕk(x) exp(i(2 +k)t) (6)
The ϕk in turn can be written as combinations of spherical harmonics and
Killing spinors. Since the round four-sphere has constant sectional curvature, it produces a vacuum solution of the gravitational field equations of Einstein with a stress-energy tensor term. Recall that the field equations are:
Rij−
1
2gijR= 8πGTij. (7)
In the absense of matter, these reduce to saying that the Ricci curvature is a constant multiple of the metric, which is true for a sphere.
There is little difficulty in extending this structure to include weak and strong nuclear forces if the gauge group is extended toSU(2)×SU(2)×SU(3). Thus at least formally, a unification presents no difficulties.
However, this theory suggests that the universe is static, and hence leaves open the questions of the origins of the universe. Indeed, if this theory is a correct description of reality, then the universe need not have an origin in the time scales that have been proposed by expanding universe theories. A second issue is this model does not require the existence of a Higgs boson.
3
Gaussian bounds of parabolic kernels
One can apply the estimates established by P. Li and S-T Yau for the Gaussian bounds on parabolic kernel of heat equations in 1986. These estimates are valid for non-compact manifolds with a lower bound on the Ricci curvature. It is known that the CBR is approximately isotropic for at least 300 light years in every direction from us, and hence a uniform lower bound for the density is a natural assumption for the universe.
4
Spectrum of Dirac operator on a sphere
The Dirac operator on Rn is defined to be the square root of the Laplace operator−1
2∆ as follows. LetD=ea∂xawhere repeated indices indicate a sum
and square it:
D2=eaeb∂xa∂xa.
If eaeb +ebea = −2δab where δab is the Kronecker delta, then D2 = −12∆
because the mixed terms cancel. The relation defines an algebra using unit vector basisea of a vector spaceV called the Clifford algebra C(V). It is easy
to see that for first fewnthese are C(R1) =C,C(R2) =H,C(R3) =H⊕H,
andC(R4) =M(2,H). They are associative but obviously non-commutative.
The conjugation defined on unit vectorsea which is−ea extends to the algebra
which includes product terms, as does the transpose operation, and there is a norm defined by multiplication by conjugate transpose, i.e. N(u) =u¯ut. The spin group consists of invertible elements of unit norm, with determinant 1, and composed of even number of reflections.
For a smooth riemannian manifoldM, the spinor bundle is constructed at a pointxin a number of ways. One method is to use the isomorphism between exterior algebra∧TxM and Cl(TxM, g)) where g is the metric at the tangent
space. The spinor group acts as a pointwise endomorphism of this algebra, and smooth sections of this bundle are the spinors. The Lie algebra∫ o(n) acts on the Clifford algebra in such a way that the Lie bracket maps to commutators of spinors.
Locally, using an orthonormal frame (ei) the Dirac operator is defined on
spinors by
Ds=ea∇eas
where∇X is an extension of the Levi-Civita connection to spinors.
Christian Bar has computed the spectrum of the Dirac operators on the round spheres Sn. These are ±(n/2 +k) for k = 0,1,2,3, ... Indeed, let us define Killing spinors with Killing constantµas sectionssof the spinor bundle that satisfy
∇Xs=µXs. (8)
It is known that for µ = ±1
2 the Killing spinors trivialize the spinor bundle.
Thus any spinor has a unique representation in products of eigenfunctionsfiof
the Laplace-Beltrami operator ofSn and a Killing spinor basis Ψ k.
5
Hypersurfaces of a 4-sphere
IfM is a 3-dimensional submanifold of a 4-manifoldN, let (ej) be a frame so
thatejare tangent toM, and let (ωj) be a dual coframe The structure equations
onN are:
ThenKABCD,KAC=KABCB,K=KABABare the curvature tensor, the Ricci
tensor, and the scalar curvature. For the standard four-sphere Kijkl=δikδjl−δilδjk.
For tensors onM whereω4= 0 andω4i∧ωi=dω4= 0 we have
ω4i=hijωj,
and we can callh=Ph
ijωiωj the second fundamental form, the eigenvaluesλi
of (hij) the principal curvatures, and H =Phii =Pλi the mean curvature.
From
dωi=ωij∧ωj
dωij =ωik∧ωkj−(1/2)Rijklωk∧ωl
we see
Rijkl=Kijkl+hikhjl−hilhjk.
On a sphere, taking the contractionj=l gives Rik= 2δik+Hhik−hijhjk.
Taking another contraction gives the scalar curvature: R= 6 +H2−S whereS=P
i,jh 2 ij.
6
Maxwell’s equations on a four-sphere
For spinors, the Maxwell’s equation can be written as DE− ∂
∂tB= 0 DB+ ∂
∂tE= 0
In strict analogy with the case of the Maxwell’s equations on a circle, one can use separation of variablesE(x, t) =e(x)m(t) andB(x, t) =b(x)n(t) to obtain
m(t) =p(De)−1be−1(Db)n(t)
n(t) =exp(pb−1(De)e−1(Db)t)
Thus for a fixedx, these equations lead to fluctuations that are not inU(1) but in a subgroup ofSpin(4)
7
Hamiltonian mechanics on H
2The unit radius round four-sphere can be identified with the quaternionic pro-jective spaceHP1. This is defined by the equivalence [q1, q2]∼[αq1, αq2] where
for nonzero quaternionα. OnH2 is defined the standard symplectic formω 0of R8. A Hamiltonian function is a function ˆH :H2 →Rfor which the gradient vector field∇Hˆ preserves the symplectic form: if Φt(x) satisfies
∂
∂tΦt(x) =∇ ˆ
HΦt(x), (9)
then Φ∗
tω0=ω0.
If we restrict attention to Hamiltonians ¯H onH2 with condition
¯
H(αp, αq) = ¯H(p, q) (10)
for unit quaternions α then these functions are lifts of functions on HP1 to
H2 whose restriction toS7 is constant on the S3fibers of the Hopf mapS7→
S4. Although S4 is not a symplectic manifold, Hamiltonian mechanics can
thus be defined on it by the symplectic form onH2 for these restricted set of
Hamiltonians.
As simple examples of projective Hamiltonians, one has ¯H(p, q) =ap¯p+bp¯q+ cq¯qfor positive real constants a, b, c >0. Indeed, if one considers homogeneous polynomials ofp, q and ¯p,q¯whose weights are equal in powers of p, qand their barred counterparts then these will all be projective.
Set ¯H(p, q) = |p|2+|q|2 for which the set ¯H = 1 coincides with the unit
seven-sphere. Now supposeM3
0 is a smooth 3-dimensional hypersurface ofS4,
and letM6
0 be its 6-dimensional preimage inS7.
Let Φt(M06) is well-defined because the Hamiltonian flow generates a
one-parameter family of diffeomorphisms of H2. From the view of a viable S4
physics, two interesting questions are: (a) do the projections of Φt(M06) toS4
satisfy formal gravitational field equations in a physically meaningful way, and (b) can classical electrodynamics onH2result in observed quantum mechanical
results onS4.
Specifically of interest is whether the observed three-dimensional stochastic behavior of an electrons in a hydrogen atom can be understood as a determin-istic system being observed from an evolving three-dimensional subspace via an ergodic pheonomenon.
8
Hamiltonian mechanics for
S
4S. Sternberg has shown in ”Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field,” in 1977, how to describe the classical mechanics of a particle with gauge group G by keeping the Hamiltonian the same but changing the symplectic form on the cotangent spaceT∗M.
Classical Hamiltonian mechanics is a system with a Hamiltonian function H : T∗M → R on the cotangent space. IfdA is the electromagnetic field, a 2-form on the manifoldM, it can be thought of as a 2-form on T∗M where a new symplectic form isω0+edA. Sternberg shows how Hamiltonian flow with
this new symplectic form describes that of H modified by the electron gauge field and the symplectic flow is with respect toω0.
In particular, at least for minimal coupling, Sternberg’s procedure shows how to retain the Hamiltonian and change the symplectic form to describe classical mechanics for any particle in the gauge groupSU(2)×SU(2)×SU(3).
9
Magnetic charges
In four dimensions, the differential 2-forms on a manifold decomposes by eigen-values of the Hodge *-operator into self-dual and anti-self-dual 2 forms. A connection A on a principal SU(2) bundle has curvature FA which lies in a
Lie algebra valued 2-forms, and the anti-self-dual connections are those with FA+ = 0. These are the minima of the Yang-Mills functional on the space of connections modulo gauge transformations. For S4 anti-self-dual connections
are called instantons. Forc2 =k these form a space of dimension 8k−3 by
the Atiyah-Drinfeld-Hitchin-Manin construction, which is based on a one-to-one correspondence between instantons and certain line bundles onCP3 which oc-cur as subbundles of a trivial complex vector bundle. In particular, instantons of unit charge are in one-to-one correspondence with pairs of points onS4 modulo simultaneous rotations byS3.
10
Ergodic theorem
The basic ergodic theorem theorem says that if T is a measure-preserving bounded operator that is ergodic in the sense thatT f =fonly for constant func-tions, then the Cesaro means (1/N)PN−1
n=0 T
nf converges inL2tof∗ satisfying T f∗ =f∗. The problem of interest is to determine if the uniform distribution of a hydrogen electron on a two-sphere could be explained by a deterministic movement of electron on a three-sphere in four dimensions such that observed from a 3 dimensional submanifold, one observes a uniform distribution.
11
Magnetic charges as light or dark monopoles
I identify positive and negative magnetic charges to a ”light” and a ”dark” pole in order to match observation of material composed of magnetic monopoles. It is a natural question how a macroscopic fourth spatial dimension is observed, for it is not observed with the usual senses. In dreams, visions, and hallucinations, we observe objects external to us. The chemical pathways of the nervous system are unidirectional, and nothing can be seen as an image in the mind unless either there is an activation of retinal neurotransmitters or (speculatively) there is an
activation of the pineal gland or the ”third eye”. Extensive experience with visions, hallucinations, and lucid dreams were the primary direct experience of a macroscopic fourth dimension that led to the S4 model. While accepted scientific theories have explanations of such phenomena as diseases of the mind, I could not see how these could be generated within the mind and then projected onto the visual system. That the material composing visions and hallucinations are magnetic in nature is clear from my own experiences. Quantitative studies of this phenomena are virtually non-existent, however.
In my theory, the four dimensional spirit of the human being connects with the three-dimensional body in seven chakras: crown, third eye, throat, heart, solar plexus, sacral, and root. The four dimensional spirit of the universe leaves it’s trace on the three dimensional physical subspace by the ubiquitous CBR. A key reason why these speculative identifications may be important is because the universe is actually four dimensional and not three, and hence a precise understanding of the human spirit requires a four-dimensional physical theory. Two problems of current interest are perhaps amenable for applications of the model sketched here: the problem of consciousness, and the protein folding problem, which may be enormously helped by the additional symmetry provided bySU(2) rather thanU(1).
Metaphysical issues have traditionally been separated from the domain of science, but just as in ancient Egypt, knowledge was considered unified for art, science, and spirit, perhaps it can be possible if a sufficiently simple four dimensional physics is developed.
Interesting consequences of the model proposed here include: there need not have been an earlier time when the universe was hot and dense. In particular, there may not have been an origin of the universe. The currently accepted range for the age of the universe is between 13-16 billion years and is based on an expanding model of the universe. If the doubts raised for the interpretation of the redshift are taken seriously, there is no reason not to assume that the universe is eternal.
