Extrasolar scale change in Newton’s Law
from 5D ‘plain’ R2-gravity (on ‘very thick brane’)
I L Zhogin (SSRC, Novosibirsk)∗
Abstract
Galactic rotation curves and lack of direct observations of Dark Matter may indicate that
General Relativity is not valid (on galactic scale) and should be replaced with another theory.
There is the only variant of Absolute Parallelism which solutions are free of arising sin-
gularities, if D=5 (there is no room for changes). This variant does not have a Lagrangian,
nor match GR: an equation of ‘plain’ R2-gravity (ie without R-term) is in sight instead.
Arranging an expanding O4-symmetrical solution as the basis of 5D cosmological model,
and probing a universal function of mass distribution (along very-very long the extra dimen-
sion) to place into bi-Laplace equation (R2 gravity), one can derive the Law of Gravitation:
transforms to 1
with distance (not with acceleration).
1 Introduction
Being a ‘close relative’ of General Relativity (GR), Absolute Parallelism (AP) has many interesting
features: larger symmetry group of equations; field irreducibility with respect to this group; vast
list of compatible second order equations (discovered by Einstein and Mayer [1]) not restricted to
Lagrangian ones.
There is the only variant of Absolute Parallelism which solutions are free of arising singularities,
if D=5 (there is no room for changes; this variant of AP does not have a Lagrangian, nor match
GR); in this case AP has topological features of nonlinear sigma-model.
In order to give clear presentation and full picture of the theory’ scope, many items should be
sketched: instability of trivial solution and expanding O4-symmetrical ones; tensor Tµν (positive
energy, but only three polarizations of 15 carry (and angular) momentum; how to quantize such a
stuff ?) and PN-effects; topological classification of symmetric 5D field configurations (alighting on
evident parallels with Standard Model’ particle combinatorics) and ‘quantum phenomenology on
expanding classical background’ (coexistence); ‘plain’ R2-gravity on very thick brane and change
in the Newton’s Law: 1
goes to 1
with distance (not with acceleration – as it is in MOND [2]).
At last, an experiment with single photon interference is discussed as the other way to observe
very-very long (and very undeveloped) the extra dimension.
2 Unique 5D equation of AP (free of singularities in solutions)
There is one unique variant of AP (non-Lagrangian, with the unique D; D=5) which solutions of
general position seem to be free of arising singularities. The formal integrability test [3] can be
∗E-mail: zhogin at inp.nsk.su; http://zhogin.narod.ru
http://arxiv.org/abs/0704.0857v1
http://zhogin.narod.ru
extended to the cases of degeneration of either co-frame matrix, haµ, (co-singularities) or contra-
variant frame (or contra-frame density of some weight), serving as the local and covariant (no
coordinate choice) test for singularities of solutions. In AP this test singles out the next equation
(and D=5, see [4]; ηab = diag(−1, 1, . . . , 1), then h = det haµ =
Eaµ = Laµν;ν − 13(faµ + LaµνΦν) = 0 , (1)
where (see [4] for more detailed introduction to AP and explanation of notations used)
Laµν = La[µν] = Λaµν − Saµν − 23ha[µΦν],
Λaµν = 2ha[µ,ν], Sµνλ = 3Λ[µνλ], Φµ = Λaaµ, fµν = 2Φ[µ,ν] = 2Φ[µ;ν]. (2)
Coma ”,” and semicolon ”;” denote partial derivative and usual covariant differentiation with
symmetric Levi-Civita connection, respectively.
One should retain the identities [which follow from the definitions (2)]:
Λa[µν;λ] ≡ 0 , haλΛabc;λ ≡ fcb (= fµνhµchνb ), f[µν;λ] ≡ 0. (3)
The equation Eaµ;µ = 0 gives ‘Maxwell-like equation’ (we prefer to omit g
µν (ηab) in contrac-
tions that not to keep redundant information – when covariant differentiation is in use only):
(faµ + LaµνΦν);µ = 0, or fµν;ν = (SµνλΦλ);ν (= −12Sµνλfνλ, see below) . (4)
Actually the Eq. (4) follows from the symmetric part of equation, E(ab), because skewsymmetric
one gives just the identity:
2E[νµ] = Sµνλ;λ = 0, E[µν];ν ≡ 0;
note also that the trace part becomes irregular (the principal derivatives vanish) if D = 4 (this
number of dimension is forbidden, and the next number, D = 5, is the most preferable):
Eµµ = Eaµh
ab = 4−D
Φµ;µ + (Λ
2) = 0.
The system (1) remains compatible under adding fµν = 0, see (4); this is not the case for
another covariant, S,Φ, or (some irreducible part of the) Riemannian curvature, which relates to
Λ as usually:
Raµνλ = 2haµ;[ν;λ]; haµhaν;λ =
Sµνλ − Λλµν .
3 Tensor Tµν (despite Lagrangian absence) and PN-effects
One might rearrange E(µν)=0 that to pick out (into LHS) the Einstein tensor, Gµν =Rµν− 12gµνR,
but the rest terms are not proper energy-momentum tensor: they contain linear terms Φ(µ;ν)
(no positive energy ( !); another presentation of ‘Maxwell equation’ (4) is possible instead – as
divergence of symmetrical tensor).
However, the prolonged equation E(µν);λ;λ = 0 can be written as ‘plain’ (no R-term) R
2-gravity:
(−h−1 δ(hRµνGµν)/δgµν=) Gµν;λ;λ +Gǫτ(2Rǫµτν − 12gµνRǫτ ) = Tµν(Λ
′2, . . .), Tµν;ν = 0; (5)
up to quadratic terms,
Tµν =
2 − fµλfνλ) + Aµǫντ (Λ2);(ǫ;τ) + (Λ2Λ′,Λ4);
tensor A has symmetries of Riemann tensor, so the term A′′ adds nothing to momentum and
angular momentum.
It is worth noting that:
(a) the theory does not match GR, but shows ‘plain’ R2-gravity (sure, (5) does not contain all
the theory);
(b) only f -component (three transverse polarizations in D=5) carries D-momentum and an-
gular momentum (‘powerful’ waves); other 12 polarizations are ‘powerless’, or ‘weightless’ (this is
a very unusual feature – impossible in the Lagrangian tradition; how to quantize ? let us not to
try this, leaving the theory ‘as is’);
(c) f -component feels only metric and S-field (‘contorsion’, not ‘torsion’ Λ – to label somehow),
see (4), but S has effect only on polarization of f : S[µνλ] does not enter eikonal equation, and f
moves along usual Riemannian geodesic (if background has f=0); one may think that all ‘quantum
fields’ (phenomenological quantized fields accounting for topological (quasi)charges and carrying
some ‘power’; see further) inherit this property;
(d) the trace Tµµ =
fµνfµν can be non-zero if f
2 6= 0 and this seemingly depends on S-
component [which enters the current in (4)]; in other words, ‘mass distribution’ is to depend on
distribution of f - and S-component;
(e) it should be stressed and underlined that the f -component is not usual (quantum) EM-
field – just important covariant responsible for energy-momentum (suffice it to say that there is
no gradient invariance for f).
4 Linear domain: instability of trivial solution (with powerless waves)
Another strange feature is the instability of trivial solution: some ‘powerless’ polarizations grow
linearly with time in presence of ‘powerful’ f -polarizations. Really, from the linearized Eq. (1)
and the identity (3) one can write (the following equations should be understood as linearized):
Φa,a = 0 (D 6= 4), 3Λabd,d = Φa,b − 2Φb,a, Λa[bc,d],d ≡ 0 ⇒ 3Λabc,dd = −2fbc,a .
The last ‘D‘Alembert equation’ has the ‘source’ in its right hand side. Some components of Λ
(most symmetrical irreducible parts) do not grow (as well as curvature), because (again, linearized
equations are implied below)
Sabc,dd = 0, Φa,dd = 0, fab,dd = 0, Rabcd,ee = 0,
but the least symmetrical components of the tensor Λ do grow up with time (due to terms ∼ t e−iωt;
three growing polarizations which are ‘imponderable’, or powerless) if the ‘ponderable’ waves (three
f -polarizations) do not vanish (and this should be the case for solutions of ‘general position’).
5 Expanding O4-symmetrical (single wave) solutions and cosmology
The unique symmetry of AP equations gives scope for symmetrical solutions. In contrast to GR,
this variant of AP has non-stationary spherically (O4-) symmetric solutions. The O4-symmetric
frame field can be generally written as follows [4]:
haµ(t, x
a bni
cni eninj + d∆ij
; i, j = (1, 2, 3, 4), ni =
. (6)
Here a, . . . , e are functions of time, t = x0, and radius r, ∆ij = δij −ninj, r2 = xixi. As functions
of radius, b, c are odd, while the others are even; other boundary conditions: e = d at r = 0, and
haµ → δ aµ as r → ∞. Placing in (6) b = 0, e = d (the other interesting choice is b=c=0) and
making integrations one can arrive to the next system (resembling dynamics of Chaplygin gas;
dot and prime denote derivation on time and radius, resp.; A = a/e = e1/2, B = −c/e):
A· = AB′ −BA′ + 3AB/r , B · = AA′ − BB′ − 2B2/r . (7)
This system (does not suffer of gradient catastrophe and) has non-stationary solutions; a single-
wave solution of proper ‘amplitude’ might serve as a suitable cosmological (expanding) background.
The condition fµν=0 is a must for solutions with such a high symmetry (as well as Sµνλ=0); so,
these O4-solutions carry no energy, that is, weight nothing (some lack of gravity ! in this theory
the universe expansion seemingly has little common with gravity, GR and its dark energy [5]).
More realistic cosmological model might look like a single O4-wave (or a sequence of such
waves) moving along the radius and being filled with chaos, or stochastic waves, both powerful
(weak, ∆h≪ 1) and powerless (∆h < 1, but intense enough that to lead to non-linear fluctuations
with ∆h ∼ 1), which form statistical ensemble(s) having a few characteristic parameters (after
‘thermalization’). The development and examination of stability of such a model is an interesting
problem. The metric variation in cosmological O4-wave can serve as a time-dependent ‘shallow
dielectric guide’ for that weak noise waves. The ponderable waves (which slightly ‘decelerate’
the O4-wave) should have wave-vectors almost tangent to the S
3-sphere of wave-front that to be
trapped inside this (‘shallow’) wave-guide; the imponderable waves can grow up, and partly escape
from the wave-guide, and their wave-vectors can be less tangent to the S3-sphere.
The waveguide thickness can be small for an observer in the center of O4-symmetry, but in co-
moving coordinates it can be very large (due to relativistic effect), however still small with respect
to the radius of sphere, L≪ R. It seems that the radial dimension has to be very ‘undeveloped’;
that is, there are no other characteristic scales, smaller than L, along this extra-dimension.
6 Non-linear domain: topological charges and quasi-charges
Let AP-space is of trivial topology: no worm-holes, no compactified space dimensions, no singu-
larities. One can continuously deform frame field h(x) to a field of rotation matrices (metric can
be diagonalized and ‘square-rooted’) haµ(x) → saµ(x) ∈ SO(1, d); m=D−1. Further deformation
can remove boosts too, and so, for any space-like (Cauchy) surface, this gives a (pointed) map,
s : Rm ∪∞ = Sm → SOm; ∞ 7→ 1m ∈ SOm.
The set of such maps consists of homotopy classes forming the group of topological charge, Π(m):
Π(m) = πm(SOm); Π(3) = Z, Π(4) = Z2 + Z2. (8)
Here Z is the infinite cyclic group, and Z2 is the cyclic group of order two.
It is important that deformation to s-field can keep symmetry of field configuration. Definition:
localized field (pointed map) s(x) : Rm → SO(m), s(∞) = 1m, is G-symmetric if, in some
coordinates,
s(σx) = σs(x)σ−1 ∀ σ ∈ G ⊂ O(m) . (9)
The set of such fields C(m)G generally consists of separate, disconnected components – homotopy
classes forming the ‘topological quasi-charge group’ denoted here as Π(G;m) ≡ π0(C(m)G ). These
QC-groups classify symmetrical localized configurations of frame field. Since field equation does
not break symmetry, quasi-charge conserves; if symmetry is not exact (because of distant regions),
quasi-charge is not exactly conserving value, and quasi-particle (of zero topological charge) can
annihilate (or be created) during colliding with another quasi-particle.
The other problem. Let G1 ⊃ G2, such that there is a mapping (embedding) i : C(m)G1 → C
which induces the homomorphism of QC-groups: i∗ : Π(G1;m) → Π(G2;m), so one has to
describe this morphism.
Let us consider the simple (discreet) symmetry group P1 with a plane of reflection symmetry:
P1 = {1, p(1)}, where p(1) = diag(−1, 1, . . . , 1) = p−1(1).
It is necessary to set field s(x) on the half-space 1
Rm = {x1 ≥ 0}, with additional condition
imposed on the surface Rm−1 = {x1 = 0} (stationary points of P1 group) where s has to commute
with the symmetry [see (9)]:
p(1)x = x ⇒ s(x) = p(1)sp(1) ⇒ s ∈ 1× SOm−1.
Hence, accounting for the localization requirement, we have a diad map (relative spheroid; here
Dm is anm-ball and Sm−1 its surface) (Dm;Sm−1) → (SOm;SOm−1), and topological classification
of such maps leads to the relative (or diad) homotopy group ([6]; the last equality below follows
due to fibration SOm/SOm−1 = S
m−1):
Π(P1;m) = πm(SOm;SOm−1) = πm(S
m−1).
Similar considerations (of group orbits and stationary points) lead to the following result:
Π(Ol;m) = πm−l+1(SOm−l+1;SOm−l) = πm−l+1(S
m−l).
If l > 3, there is the equality: Π(SOl;m) = Π(Ol;m), while for l = 2, 3 one can find:
Π(SO3;m) = πm−2(SO2 × SOm−2;SOm−3) = πm−2(S1 × Sm−3),
Π(SO2;m) = πm−1(SOm;SOm−2 × SO2) = πm−1(RG+(m, 2)).
The set of quaternions with absolute value one, H1 = {f, |f| = 1}, forms a group under
quaternion multiplication, H1 ∼= SU2 = S3, and any s ∈ SO4 can be represented as a pair of such
quaternions [6], (f , g) ∈ S3(l) × S3(r), |f | = |g| = 1:
x∗ = sx ⇔ x∗ = f x g−1 = f x ḡ ; |x| = |x∗|.
The pairs (f,g) and (–f, –g) correspond to the same rotation s, that is, SO4 = S
(l) × S3(r)/±.
Note that the symmetry condition (9) also splits into two parts:
f(axb−1) = af(x)a−1, g(axb−1) = bg(x)b−1 ∀(a,b) ∈ G ⊂ SO4. (10)
7 Example of SO2-symmetric quaternion field
Let’s consider an example of SO2{2, 3}−symmetric f–field configuration (g=1), which carries both
charge and SO2-quasi-charge (left, of course), f(x): H = R
4 → H1; f(∞) = 1. The symmetry
condition (10) reads
f(eiφ/2xe−iφ/2) = eiφ/2f(x)e−iφ/2. (11)
We’ll switch to ‘double-axial’ coordinates: x = aeiϕ + beiψj. Let us use imaginary quaternions q
as stereogrphic coordinates on H1, and take symmetrical field q(x) consistent with Eq. (11):
q(x) = x i x̄+ i = −q̄, f(x) = −
1 + q
. (12)
It is easy to find the ‘center of quasi-soliton’ (1-submanifold, S1)
S1 = f−1(−1) = q−1(0) = {a = 0, b = 1} = {x0(ψ) = eiψj}
and the ‘vector equipment’ on this circle:
dx|x0 = da eiϕ + (db+ i dψ)eiψj, 14df
= idb− k ei (ϕ+ψ)da ;
i-vector all time looks along the radius b (parallel translation along the circle S1; this is a ‘trivial‘,
or ‘flavor’-vector). Two others (’phase’-vectors) make 2π−rotation along the circle.
In fact, the field (12) has also symmetry SO2{1, 4}, and this feature restricts possible directions
of ‘flavor’-vector (two ‘flavors’ are possible, ±; the P2{1, 4}−symmetry (this is the π-rotation of
x1, x4) gives the same effect). The other interesting observation is that the equipped circle can be
located also at the stationary points of SO2−symmetry (this increases the number of ‘flavors’).
8 Quasi-charges and their morphisms (in 5D, ie m = 4)
If G ⊂ SO4, the QC-group has two isomorphous parts, left and right: Π(G) = Π(l)(G) + Π(r)(G).
The Table below describes quasi-charge groups for G ⊂ G0 = (O3 × P4) ∩ SO4 (P4 is spatial
inversion, the 4-th coordinate is the extra dimension of G0-symmetric expanding cosmological
background).
Table. QC-groups Π(l)(G) and their morphisms to the preceding group; G ⊂ G0.
G Πl(G) → Πl(G∗) ‘label’
SO{1, 2} Z(e)
e→ Z2 e
SO{1, 2} × P{3, 4} Z(ν) + Z(H)
i,m2→ Z(e) ν0; H0 → e + e
SO{1, 2} × P{2, 3} Z(W )
0→ Z(e) W → e + ν0
SO{1, 2} × P{2, 4} Z(Z)
0→ Z(e) Z0 → e+ e
SO{1, 2} × P{3, 4} × Z(γ)
0→ Z(H) γ0 → H0 +H0
×P{2, 3} 0→ Z(W ) →W +W
‘Quasi-particles’, which symmetry includes P4, seem to be true neutral (neutrinos, Higgs particles,
photon).
One can assume further that an hadron bag is a specific place where G0−symmetry does
not work, and the bag’s symmetry is isomorphous to O4. This assumption can lead to another
classification of quasi-solitons (some doubling the above scheme), where self-dual and anti-self-
dual one-parameter groups take place of SO2−group. The total set of quasi-particle parameters
(parameters of equipped 1-manifold (loop) plus parameters of group) for (anti)self-dual groups,
G(4, 2)×RP 2, is larger than the analogous set for groups SO2 ⊂ G0, which is just O3×G(3, 1) =
RP 2 . If the number of ‘flavor’-parameters (which are not degenerate and have some preferable
particular values; this should be sensitive to discreet part of G – at least photons have the same
flavor) is the same as in the case of ‘white’ quasi-particles, the remaining parameters (degenerate,
or ‘phase’) can give room for ‘color’ (in addition to spin). So, perhaps one might think about
‘color neutrinos’ (in the context of pomeron, and baryon spin puzzle), ‘color W, Z, and Higgs’
(another context – B-mesons), and so on.
Note that in this picture the very notion of quasi-particle depends on the background symmetry
(also to note: there are no ’quanta of torsion’ per se). On the other hand, large clusters of
quasi-particles (matter) can disturb the background, and waves of such small disturbances (with
wavelength larger than the thickness L, perhaps) can be generated as well (but these waves do
not carry (quasi)charges, that is, are not quantized).
9 Coexistence: phenomenological ‘quantum fields’ on classical back-
ground
The non-linear, particle-like field configurations with quasi-charges (quasi-particles) should be very
elongated along the extra-dimension (all of the same size L), while being small sized along usual
dimensions, λ≪ L. The motion of such a spaghetti-like quasi-particle should be very complicated
and stochastic due to ‘strong’ imponderable noise, such that different parts of spaghetti are coming
their own paths. At the same time, quasi-particle can acquire ‘its own’ energy–momentum – due to
scattering of ponderable waves (which wave-vectors are almost tangent to usual 3D (sub)space);
so, it seems that scattering amplitudes1 of those spaghetti’s parts which have the same 3D–
coordinates can be summarized providing an auxiliary, secondary field.
So, the imponderable waves provides stochasticity (of motion of spaghetti’s parts), while the
ponderable waves ensure superposition (with secondary fields). Phenomenology of secondary fields
could be of Lagrangian type, with positive energy acquired by quasi-particles, – that to ensure the
stability (of all the waveguide with its infill – with respect to quasi-particle production; the least
action principle has deep concerns with Lyapunov stability and is deducible, in principle, from the
path integral approach).
10 ‘Plain’ R2 gravity on very thick brane
and change in the Newton’s Law of Gravitation
Let us start with 4d (from 5D) bi-Laplace equation with a δ-source [as weak field, non-relativistic
(stationary) approximation (it is assumed that ‘mass is possible’) for R2-gravity (5)] and its
solution (R is 4d distance, radius):
∆2ϕ = − a
δ(R); ϕ(R2) =
lnR2 − b
(+ c , but c does not matter); (13)
the attracting force between two point masses is Fpoint =
, a, b should be proportional to
both masses.
Now let us suppose that all masses are distributed along the extra dimension with a ‘universal
function’, µ(p),
µ(p) dp = 1. Then the attracting (gravitation) force takes the next form [see
1 These amplitudes can depend on additional vector-parameters (‘equipment vectors’) relating to differential of
field mapping at a ‘quasi-particle center’ – where quasi-charge density is largest (if it has covariant sense).
0 1 2 3 4 5 6
Fig. 1. Deviation δF = F − 1/r2 for different µ(p), see Eq. (14) and text below.
(13); r is usual 3d distance]:
F (r) =
ϕ(r2 + (p− q)2)µ(p)µ(q) dp dq =
V − b V ′, V (r) =
µ(p)µ(q) dp dq
r2 + (p− q)2
. (14)
(Note that V (r) can be restored if F (r) is measured.)
Taking µ1(p) = π
−1/(1 + p2) (typical scale along the extra dimension is taken as unit, L = 1;
it seems that L should be greater than ten AU), one can find rV1(r) = 1/(2 + r) and
F (r) =
8 + 4r
2b(1 + r)
r2(2 + r)2
; or (now L 6= 1) F (r) =
2L(2L+ r)2
, where a = b = 2/L2.
Fig. 1, curve (a) shows δF = F − 1/r2 (deviation from the Newton’s Law; a/b is chosen that
δF (0)=0); two other curves, (b) & (c), correspond to µ2 = 2π
−1/(1 + p2)2, µ3 = 2π
−1p2/(1 + p2)2
(also δF (0)=0; residues help to find rV2 = (10 + 6r + r
2)/(2 + r)3, rV3 = (2 + 2r + r
2)/(2 + r)3).
We see that in principle this theory can explain galaxy rotation curves, v2(r)∝ rF r→∞−→ const,
without need for Dark Matter (or MOND [2]; about rotation curves and DM see [7]; they are
looking for DM in Solar system too, [8]).
Q: Can the ‘coherence of mass’ along the extra dimension be disturbed ? (the flyby anomaly,
the Pioneer anomaly [9]); can µ(p) be negative in some domains of p ?
11 How to register ‘powerless’ waves
This section is added perhaps for some funny recreation (or still not ? who knows). We have
learnt that S-waves do not carry momentum and angular momentum, so they can not perform
any work or spin flip.
But let us conceive that these waves can effect a flip-flop of two neighbor spins. So, a ‘detector’
could be a media with two sorts of spins, A and B. Let sA = sB = 1/2 but gA 6= gB, and let
the initial state is prepared as follows: {<sAz >,<sBz >}(0) = {1/2,−1/2}. Then the process of
spin relaxation starts; turning on appropriate magnetic field Hz (and alternating fields of proper
frequencies) one can measure the detector’s state and find the time of spin relaxation.
The next step. Skilled experimenters try to generate S-waves and to register an effect of these
waves on spin relaxation. The generation of intense ‘coherent’ S-waves could be proceeded perhaps
with a similar spin system subjected to alternating polarization.
12 Single photon experiment (that to feel huge extra dimension),
and Conclusion
Today, many laboratories have sources of single (heralded) photons, or entangled bi-photons (say,
for Bell-type experiments [10]); some students can perform laboratory works with single photons,
having convinced on their own experience that light is quantized (the Grangier experiment)[11].
It is being suggested a minor modification of the single (polarized) photon interference exper-
iment, say, in a Mach-Zehnder fiber interferometer with ‘long’ (the fibers may be rolled) enough
arms. The only new element is a fast-acting shutter placed at the beginning of (one of) the inter-
ferometer’s arms (the closing-opening time of the shutter should be smaller than the flight time
in the arms). For example, a fast electro-optical modulator in combination with polarizer (or a
number of such combinations) can be used with polarized photons.
Both Quantum mechanics (no particle’s ontology) and Bohmian mechanics (wave-particle dou-
ble ontology)[12] exclude any change in the interference figure as a result of separating activity
of such a fast shutter (while the photon’s ‘halves’ are making their ways to the place of a meet-
ing). However, if a photon has non-local spaghetti-like ontology (along the extra dimension) and
fragments of this spaghetti are moving along both arms at once, then the shutter should tear up
this spaghetti (mainly without photon absorption), tear out its fragments (which will dissolve in
‘zero-point oscillations’). Hence, if the absorption factor of the shutter (the extinction ratio of
polarizer) is large enough, the 50/50-proportion (between the photon’s amplitudes in the arms)
will be changed and a significant decrease of the interference visibility should be observed.
QM is everywhere (where we can see, of course), and, so, non-linear 5D-field fluctuations,
looking like spaghetti-anti-spaghetti loops, should exist everywhere. (This omnipresence can be
related to the universality of ‘low-level heat death’, restricted by the presence of topological quasi-
solitons – some as the 2D computer experiment by Fermi, Pasta, and Ulam, where the process of
thermalization was restricted by the existence of solitons. See also the sections 5–8 (and [4]) for
arguments in favor of phenomenological (quantized) ‘secondary fields’ accounting for topological
(quasi)charges and obeying superposition, path integral and so on.)
AP, at least at the level of its symmetry, seems to be able to cure the gap between the
two branches of physics – General Relativity (with coordinate diffeomorphisms) and Quantum
Mechanics (with Lorentz invariance).2 Most people give all the rights of fundamentality to quanta,
and so, they are trying to quantize gravity, and the very space-time (probing loops, and strings,
and branes; see also the warning polemic by Schroer [14]). The other possibility is that quanta
have the specific phenomenological origin relating to topological (quasi)charges.
2Rovelli writes[13]: In spite of their empirical success, GR and QM offer a schizophrenic and confused under-
standing of the physical world.
References
[1] A. Einstein and W. Mayer, Sitzungsber. preuss. Akad. Wiss. Kl 257–265 (1931).
[2] M. Milgrom, The modified dynamics – a status review, arXiv: astro-ph/9810302.
[3] J. F. Pommaret, Systems of Partial Differentiation Equations and Lie Pseudogroups (Math.
and its Applications, Vol. 14, New York 1978).
[4] I. L. Zhogin, Topological charges and quasi-charges in AP, arXiv: gr-qc/0610076; spherical
symmetry: gr-qc/0412130; 3-linear equations (contra-singularities): gr-qc/0203008.
[5] S.M. Carroll, Why is the Universe Accelerating ? arXiv: astro-ph/0310342
[6] B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry – Methods and Applica-
tions, Springer-Verlag, 1984.
[7] M.E. Peskin, Dark Matter: What is it ? Where is it ? Can we make it in the lab ?
http://www.slac.stanford.edu/grp/th/mpeskin/Yale1.pdf; M. Battaglia, M.E. Peskin,
The Role of the ILC in the Study of Cosmic Dark Matter, hep-ph/0509135
[8] L. Iorio, Solar System planetary orbital motions and dark matter, arXiv: gr-qc/0602095;
I.B. Khriplovich, Density of dark matter in Solar system and perihelion precession of planets,
astro-ph/0702260.
[9] C. Lämmerzahl, O. Preuss, and H. Dittus, Is the physics within the Solar system really
understood ? arXiv: gr-qc/0604052; A. Unzicker, Why do we Still Believe in Newton’s Law ?
Facts, Myths and Methods in Gravitational Physics, gr-qc/0702009.
[10] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 81, 5039
(1998); quant-ph/9810080; W. Tittel, G. Weihs, Photonic Entanglement for Fundamental
Tests and Quantum Communication, quant-ph/0107156.
[11] See the next links: departments.colgate.edu/physics/research/Photon/root/ ,
marcus.whitman.edu/ beckmk/QM .
[12] H. Nikolić, Quantum mechanics: Myths and facts, arXiv: quant-ph/0609163 .
[13] C. Rovelli, Unfinished revolution, gr-qc/0604045 .
[14] B. Schroer, String theory and the crisis in particle physics (a Samisdat on particle physics),
arXiv: physics/0603112;
the other sources of contra-string polemic are seemingly the books: P. Woit, Not even wrong;
L. Smolin, The Trouble with Physics (and the blog math.columbia.edu/∼woit/wordpress).
http://arxiv.org/abs/astro-ph/9810302
http://arxiv.org/astro-ph/9810302
http://arxiv.org/abs/gr-qc/0610076
http://arXiv.org/gr-qc/0610076
http://arXiv.org/gr-qc/0412130
http://arXiv.org/gr-qc/0203008
http://arxiv.org/abs/astro-ph/0310342
http://arxiv.org/astro-ph/0310342
http://www.slac.stanford.edu/grp/th/mpeskin/Yale1.pdf
http://arxiv.org/hep-ph/0509135
http://arxiv.org/abs/gr-qc/0602095
http://arxiv.org/gr-qc/0602095
http://arxiv.org/astro-ph/0702260
http://arxiv.org/abs/gr-qc/0604052
http://arxiv.org/gr-qc/0604052
http://arxiv.org/gr-qc/0702009
http://arXiv.org/quant-ph/9810080
http://arXiv.org/quant-ph/0107156
http://departments.colgate.edu/%physics/research/Photon/root/photon_quantum_mechanics.htm
http://marcus.whitman.edu/~beckmk/QM/
http://arxiv.org/abs/quant-ph/0609163
http://arXiv.org/gr-qc/0604045
http://arxiv.org/abs/physics/0603112
http://arXiv.org/physics/0603112
http://www.math.columbia.edu/~woit/wordpress/
	Introduction
	Unique 5D equation of AP (free of singularities in solutions)
	Tensor T (despite Lagrangian absence) and PN-effects
	Linear domain: instability of trivial solution (with powerless waves) 
	Expanding O4-symmetrical (single wave) solutions and cosmology 
	Non-linear domain: topological charges and quasi-charges 
	Example of SO2-symmetric quaternion field
	Quasi-charges and their morphisms (in 5D, ie m=4) 
	Coexistence: phenomenological `quantum fields' on classical background 
	`Plain' R2 gravity on very thick brane  and change in the Newton's Law of Gravitation
	How to register `powerless' waves 
	Single photon experiment (that to feel huge extra dimension), and Conclusion

  Galactic rotation curves and lack of direct observations of Dark Matter may
indicate that General Relativity is not valid (on galactic scale) and should be
replaced with another theory. There is the only variant of Absolute Parallelism
which solutions are free of arising singularities, if D=5 (there is no room for
changes). This variant does not have a Lagrangian, nor match GR: an equation of
`plain' R^2-gravity (ie without R-term) is in sight instead. Arranging an
expanding O_4-symmetrical solution as the basis of 5D cosmological model, and
probing a universal_function of mass distribution (along very-very long the
extra dimension) to place into bi-Laplace equation (R^2 gravity), one can
derive the Law of Gravitation: 1/r^2 transforms to 1/r with distance (not with
acceleration).

Introduction
Being a ‘close relative’ of General Relativity (GR), Absolute Parallelism (AP) has many interesting
features: larger symmetry group of equations; field irreducibility with respect to this group; vast
list of compatible second order equations (discovered by Einstein and Mayer [1]) not restricted to
Lagrangian ones.
There is the only variant of Absolute Parallelism which solutions are free of arising singularities,
if D=5 (there is no room for changes; this variant of AP does not have a Lagrangian, nor match
GR); in this case AP has topological features of nonlinear sigma-model.
In order to give clear presentation and full picture of the theory’ scope, many items should be
sketched: instability of trivial solution and expanding O4-symmetrical ones; tensor Tµν (positive
energy, but only three polarizations of 15 carry (and angular) momentum; how to quantize such a
stuff ?) and PN-effects; topological classification of symmetric 5D field configurations (alighting on
evident parallels with Standard Model’ particle combinatorics) and ‘quantum phenomenology on
expanding classical background’ (coexistence); ‘plain’ R2-gravity on very thick brane and change
in the Newton’s Law: 1
goes to 1
with distance (not with acceleration – as it is in MOND [2]).
At last, an experiment with single photon interference is discussed as the other way to observe
very-very long (and very undeveloped) the extra dimension.
2 Unique 5D equation of AP (free of singularities in solutions)
There is one unique variant of AP (non-Lagrangian, with the unique D; D=5) which solutions of
general position seem to be free of arising singularities. The formal integrability test [3] can be
∗E-mail: zhogin at inp.nsk.su; http://zhogin.narod.ru
http://arxiv.org/abs/0704.0857v1
http://zhogin.narod.ru
extended to the cases of degeneration of either co-frame matrix, haµ, (co-singularities) or contra-
variant frame (or contra-frame density of some weight), serving as the local and covariant (no
coordinate choice) test for singularities of solutions. In AP this test singles out the next equation
(and D=5, see [4]; ηab = diag(−1, 1, . . . , 1), then h = det haµ =
Eaµ = Laµν;ν − 13(faµ + LaµνΦν) = 0 , (1)
where (see [4] for more detailed introduction to AP and explanation of notations used)
Laµν = La[µν] = Λaµν − Saµν − 23ha[µΦν],
Λaµν = 2ha[µ,ν], Sµνλ = 3Λ[µνλ], Φµ = Λaaµ, fµν = 2Φ[µ,ν] = 2Φ[µ;ν]. (2)
Coma ”,” and semicolon ”;” denote partial derivative and usual covariant differentiation with
symmetric Levi-Civita connection, respectively.
One should retain the identities [which follow from the definitions (2)]:
Λa[µν;λ] ≡ 0 , haλΛabc;λ ≡ fcb (= fµνhµchνb ), f[µν;λ] ≡ 0. (3)
The equation Eaµ;µ = 0 gives ‘Maxwell-like equation’ (we prefer to omit g
µν (ηab) in contrac-
tions that not to keep redundant information – when covariant differentiation is in use only):
(faµ + LaµνΦν);µ = 0, or fµν;ν = (SµνλΦλ);ν (= −12Sµνλfνλ, see below) . (4)
Actually the Eq. (4) follows from the symmetric part of equation, E(ab), because skewsymmetric
one gives just the identity:
2E[νµ] = Sµνλ;λ = 0, E[µν];ν ≡ 0;
note also that the trace part becomes irregular (the principal derivatives vanish) if D = 4 (this
number of dimension is forbidden, and the next number, D = 5, is the most preferable):
Eµµ = Eaµh
ab = 4−D
Φµ;µ + (Λ
2) = 0.
The system (1) remains compatible under adding fµν = 0, see (4); this is not the case for
another covariant, S,Φ, or (some irreducible part of the) Riemannian curvature, which relates to
Λ as usually:
Raµνλ = 2haµ;[ν;λ]; haµhaν;λ =
Sµνλ − Λλµν .
3 Tensor Tµν (despite Lagrangian absence) and PN-effects
One might rearrange E(µν)=0 that to pick out (into LHS) the Einstein tensor, Gµν =Rµν− 12gµνR,
but the rest terms are not proper energy-momentum tensor: they contain linear terms Φ(µ;ν)
(no positive energy ( !); another presentation of ‘Maxwell equation’ (4) is possible instead – as
divergence of symmetrical tensor).
However, the prolonged equation E(µν);λ;λ = 0 can be written as ‘plain’ (no R-term) R
2-gravity:
(−h−1 δ(hRµνGµν)/δgµν=) Gµν;λ;λ +Gǫτ(2Rǫµτν − 12gµνRǫτ ) = Tµν(Λ
′2, . . .), Tµν;ν = 0; (5)
up to quadratic terms,
Tµν =
2 − fµλfνλ) + Aµǫντ (Λ2);(ǫ;τ) + (Λ2Λ′,Λ4);
tensor A has symmetries of Riemann tensor, so the term A′′ adds nothing to momentum and
angular momentum.
It is worth noting that:
(a) the theory does not match GR, but shows ‘plain’ R2-gravity (sure, (5) does not contain all
the theory);
(b) only f -component (three transverse polarizations in D=5) carries D-momentum and an-
gular momentum (‘powerful’ waves); other 12 polarizations are ‘powerless’, or ‘weightless’ (this is
a very unusual feature – impossible in the Lagrangian tradition; how to quantize ? let us not to
try this, leaving the theory ‘as is’);
(c) f -component feels only metric and S-field (‘contorsion’, not ‘torsion’ Λ – to label somehow),
see (4), but S has effect only on polarization of f : S[µνλ] does not enter eikonal equation, and f
moves along usual Riemannian geodesic (if background has f=0); one may think that all ‘quantum
fields’ (phenomenological quantized fields accounting for topological (quasi)charges and carrying
some ‘power’; see further) inherit this property;
(d) the trace Tµµ =
fµνfµν can be non-zero if f
2 6= 0 and this seemingly depends on S-
component [which enters the current in (4)]; in other words, ‘mass distribution’ is to depend on
distribution of f - and S-component;
(e) it should be stressed and underlined that the f -component is not usual (quantum) EM-
field – just important covariant responsible for energy-momentum (suffice it to say that there is
no gradient invariance for f).
4 Linear domain: instability of trivial solution (with powerless waves)
Another strange feature is the instability of trivial solution: some ‘powerless’ polarizations grow
linearly with time in presence of ‘powerful’ f -polarizations. Really, from the linearized Eq. (1)
and the identity (3) one can write (the following equations should be understood as linearized):
Φa,a = 0 (D 6= 4), 3Λabd,d = Φa,b − 2Φb,a, Λa[bc,d],d ≡ 0 ⇒ 3Λabc,dd = −2fbc,a .
The last ‘D‘Alembert equation’ has the ‘source’ in its right hand side. Some components of Λ
(most symmetrical irreducible parts) do not grow (as well as curvature), because (again, linearized
equations are implied below)
Sabc,dd = 0, Φa,dd = 0, fab,dd = 0, Rabcd,ee = 0,
but the least symmetrical components of the tensor Λ do grow up with time (due to terms ∼ t e−iωt;
three growing polarizations which are ‘imponderable’, or powerless) if the ‘ponderable’ waves (three
f -polarizations) do not vanish (and this should be the case for solutions of ‘general position’).
5 Expanding O4-symmetrical (single wave) solutions and cosmology
The unique symmetry of AP equations gives scope for symmetrical solutions. In contrast to GR,
this variant of AP has non-stationary spherically (O4-) symmetric solutions. The O4-symmetric
frame field can be generally written as follows [4]:
haµ(t, x
a bni
cni eninj + d∆ij
; i, j = (1, 2, 3, 4), ni =
. (6)
Here a, . . . , e are functions of time, t = x0, and radius r, ∆ij = δij −ninj, r2 = xixi. As functions
of radius, b, c are odd, while the others are even; other boundary conditions: e = d at r = 0, and
haµ → δ aµ as r → ∞. Placing in (6) b = 0, e = d (the other interesting choice is b=c=0) and
making integrations one can arrive to the next system (resembling dynamics of Chaplygin gas;
dot and prime denote derivation on time and radius, resp.; A = a/e = e1/2, B = −c/e):
A· = AB′ −BA′ + 3AB/r , B · = AA′ − BB′ − 2B2/r . (7)
This system (does not suffer of gradient catastrophe and) has non-stationary solutions; a single-
wave solution of proper ‘amplitude’ might serve as a suitable cosmological (expanding) background.
The condition fµν=0 is a must for solutions with such a high symmetry (as well as Sµνλ=0); so,
these O4-solutions carry no energy, that is, weight nothing (some lack of gravity ! in this theory
the universe expansion seemingly has little common with gravity, GR and its dark energy [5]).
More realistic cosmological model might look like a single O4-wave (or a sequence of such
waves) moving along the radius and being filled with chaos, or stochastic waves, both powerful
(weak, ∆h≪ 1) and powerless (∆h < 1, but intense enough that to lead to non-linear fluctuations
with ∆h ∼ 1), which form statistical ensemble(s) having a few characteristic parameters (after
‘thermalization’). The development and examination of stability of such a model is an interesting
problem. The metric variation in cosmological O4-wave can serve as a time-dependent ‘shallow
dielectric guide’ for that weak noise waves. The ponderable waves (which slightly ‘decelerate’
the O4-wave) should have wave-vectors almost tangent to the S
3-sphere of wave-front that to be
trapped inside this (‘shallow’) wave-guide; the imponderable waves can grow up, and partly escape
from the wave-guide, and their wave-vectors can be less tangent to the S3-sphere.
The waveguide thickness can be small for an observer in the center of O4-symmetry, but in co-
moving coordinates it can be very large (due to relativistic effect), however still small with respect
to the radius of sphere, L≪ R. It seems that the radial dimension has to be very ‘undeveloped’;
that is, there are no other characteristic scales, smaller than L, along this extra-dimension.
6 Non-linear domain: topological charges and quasi-charges
Let AP-space is of trivial topology: no worm-holes, no compactified space dimensions, no singu-
larities. One can continuously deform frame field h(x) to a field of rotation matrices (metric can
be diagonalized and ‘square-rooted’) haµ(x) → saµ(x) ∈ SO(1, d); m=D−1. Further deformation
can remove boosts too, and so, for any space-like (Cauchy) surface, this gives a (pointed) map,
s : Rm ∪∞ = Sm → SOm; ∞ 7→ 1m ∈ SOm.
The set of such maps consists of homotopy classes forming the group of topological charge, Π(m):
Π(m) = πm(SOm); Π(3) = Z, Π(4) = Z2 + Z2. (8)
Here Z is the infinite cyclic group, and Z2 is the cyclic group of order two.
It is important that deformation to s-field can keep symmetry of field configuration. Definition:
localized field (pointed map) s(x) : Rm → SO(m), s(∞) = 1m, is G-symmetric if, in some
coordinates,
s(σx) = σs(x)σ−1 ∀ σ ∈ G ⊂ O(m) . (9)
The set of such fields C(m)G generally consists of separate, disconnected components – homotopy
classes forming the ‘topological quasi-charge group’ denoted here as Π(G;m) ≡ π0(C(m)G ). These
QC-groups classify symmetrical localized configurations of frame field. Since field equation does
not break symmetry, quasi-charge conserves; if symmetry is not exact (because of distant regions),
quasi-charge is not exactly conserving value, and quasi-particle (of zero topological charge) can
annihilate (or be created) during colliding with another quasi-particle.
The other problem. Let G1 ⊃ G2, such that there is a mapping (embedding) i : C(m)G1 → C
which induces the homomorphism of QC-groups: i∗ : Π(G1;m) → Π(G2;m), so one has to
describe this morphism.
Let us consider the simple (discreet) symmetry group P1 with a plane of reflection symmetry:
P1 = {1, p(1)}, where p(1) = diag(−1, 1, . . . , 1) = p−1(1).
It is necessary to set field s(x) on the half-space 1
Rm = {x1 ≥ 0}, with additional condition
imposed on the surface Rm−1 = {x1 = 0} (stationary points of P1 group) where s has to commute
with the symmetry [see (9)]:
p(1)x = x ⇒ s(x) = p(1)sp(1) ⇒ s ∈ 1× SOm−1.
Hence, accounting for the localization requirement, we have a diad map (relative spheroid; here
Dm is anm-ball and Sm−1 its surface) (Dm;Sm−1) → (SOm;SOm−1), and topological classification
of such maps leads to the relative (or diad) homotopy group ([6]; the last equality below follows
due to fibration SOm/SOm−1 = S
m−1):
Π(P1;m) = πm(SOm;SOm−1) = πm(S
m−1).
Similar considerations (of group orbits and stationary points) lead to the following result:
Π(Ol;m) = πm−l+1(SOm−l+1;SOm−l) = πm−l+1(S
m−l).
If l > 3, there is the equality: Π(SOl;m) = Π(Ol;m), while for l = 2, 3 one can find:
Π(SO3;m) = πm−2(SO2 × SOm−2;SOm−3) = πm−2(S1 × Sm−3),
Π(SO2;m) = πm−1(SOm;SOm−2 × SO2) = πm−1(RG+(m, 2)).
The set of quaternions with absolute value one, H1 = {f, |f| = 1}, forms a group under
quaternion multiplication, H1 ∼= SU2 = S3, and any s ∈ SO4 can be represented as a pair of such
quaternions [6], (f , g) ∈ S3(l) × S3(r), |f | = |g| = 1:
x∗ = sx ⇔ x∗ = f x g−1 = f x ḡ ; |x| = |x∗|.
The pairs (f,g) and (–f, –g) correspond to the same rotation s, that is, SO4 = S
(l) × S3(r)/±.
Note that the symmetry condition (9) also splits into two parts:
f(axb−1) = af(x)a−1, g(axb−1) = bg(x)b−1 ∀(a,b) ∈ G ⊂ SO4. (10)
7 Example of SO2-symmetric quaternion field
Let’s consider an example of SO2{2, 3}−symmetric f–field configuration (g=1), which carries both
charge and SO2-quasi-charge (left, of course), f(x): H = R
4 → H1; f(∞) = 1. The symmetry
condition (10) reads
f(eiφ/2xe−iφ/2) = eiφ/2f(x)e−iφ/2. (11)
We’ll switch to ‘double-axial’ coordinates: x = aeiϕ + beiψj. Let us use imaginary quaternions q
as stereogrphic coordinates on H1, and take symmetrical field q(x) consistent with Eq. (11):
q(x) = x i x̄+ i = −q̄, f(x) = −
1 + q
. (12)
It is easy to find the ‘center of quasi-soliton’ (1-submanifold, S1)
S1 = f−1(−1) = q−1(0) = {a = 0, b = 1} = {x0(ψ) = eiψj}
and the ‘vector equipment’ on this circle:
dx|x0 = da eiϕ + (db+ i dψ)eiψj, 14df
= idb− k ei (ϕ+ψ)da ;
i-vector all time looks along the radius b (parallel translation along the circle S1; this is a ‘trivial‘,
or ‘flavor’-vector). Two others (’phase’-vectors) make 2π−rotation along the circle.
In fact, the field (12) has also symmetry SO2{1, 4}, and this feature restricts possible directions
of ‘flavor’-vector (two ‘flavors’ are possible, ±; the P2{1, 4}−symmetry (this is the π-rotation of
x1, x4) gives the same effect). The other interesting observation is that the equipped circle can be
located also at the stationary points of SO2−symmetry (this increases the number of ‘flavors’).
8 Quasi-charges and their morphisms (in 5D, ie m = 4)
If G ⊂ SO4, the QC-group has two isomorphous parts, left and right: Π(G) = Π(l)(G) + Π(r)(G).
The Table below describes quasi-charge groups for G ⊂ G0 = (O3 × P4) ∩ SO4 (P4 is spatial
inversion, the 4-th coordinate is the extra dimension of G0-symmetric expanding cosmological
background).
Table. QC-groups Π(l)(G) and their morphisms to the preceding group; G ⊂ G0.
G Πl(G) → Πl(G∗) ‘label’
SO{1, 2} Z(e)
e→ Z2 e
SO{1, 2} × P{3, 4} Z(ν) + Z(H)
i,m2→ Z(e) ν0; H0 → e + e
SO{1, 2} × P{2, 3} Z(W )
0→ Z(e) W → e + ν0
SO{1, 2} × P{2, 4} Z(Z)
0→ Z(e) Z0 → e+ e
SO{1, 2} × P{3, 4} × Z(γ)
0→ Z(H) γ0 → H0 +H0
×P{2, 3} 0→ Z(W ) →W +W
‘Quasi-particles’, which symmetry includes P4, seem to be true neutral (neutrinos, Higgs particles,
photon).
One can assume further that an hadron bag is a specific place where G0−symmetry does
not work, and the bag’s symmetry is isomorphous to O4. This assumption can lead to another
classification of quasi-solitons (some doubling the above scheme), where self-dual and anti-self-
dual one-parameter groups take place of SO2−group. The total set of quasi-particle parameters
(parameters of equipped 1-manifold (loop) plus parameters of group) for (anti)self-dual groups,
G(4, 2)×RP 2, is larger than the analogous set for groups SO2 ⊂ G0, which is just O3×G(3, 1) =
RP 2 . If the number of ‘flavor’-parameters (which are not degenerate and have some preferable
particular values; this should be sensitive to discreet part of G – at least photons have the same
flavor) is the same as in the case of ‘white’ quasi-particles, the remaining parameters (degenerate,
or ‘phase’) can give room for ‘color’ (in addition to spin). So, perhaps one might think about
‘color neutrinos’ (in the context of pomeron, and baryon spin puzzle), ‘color W, Z, and Higgs’
(another context – B-mesons), and so on.
Note that in this picture the very notion of quasi-particle depends on the background symmetry
(also to note: there are no ’quanta of torsion’ per se). On the other hand, large clusters of
quasi-particles (matter) can disturb the background, and waves of such small disturbances (with
wavelength larger than the thickness L, perhaps) can be generated as well (but these waves do
not carry (quasi)charges, that is, are not quantized).
9 Coexistence: phenomenological ‘quantum fields’ on classical back-
ground
The non-linear, particle-like field configurations with quasi-charges (quasi-particles) should be very
elongated along the extra-dimension (all of the same size L), while being small sized along usual
dimensions, λ≪ L. The motion of such a spaghetti-like quasi-particle should be very complicated
and stochastic due to ‘strong’ imponderable noise, such that different parts of spaghetti are coming
their own paths. At the same time, quasi-particle can acquire ‘its own’ energy–momentum – due to
scattering of ponderable waves (which wave-vectors are almost tangent to usual 3D (sub)space);
so, it seems that scattering amplitudes1 of those spaghetti’s parts which have the same 3D–
coordinates can be summarized providing an auxiliary, secondary field.
So, the imponderable waves provides stochasticity (of motion of spaghetti’s parts), while the
ponderable waves ensure superposition (with secondary fields). Phenomenology of secondary fields
could be of Lagrangian type, with positive energy acquired by quasi-particles, – that to ensure the
stability (of all the waveguide with its infill – with respect to quasi-particle production; the least
action principle has deep concerns with Lyapunov stability and is deducible, in principle, from the
path integral approach).
10 ‘Plain’ R2 gravity on very thick brane
and change in the Newton’s Law of Gravitation
Let us start with 4d (from 5D) bi-Laplace equation with a δ-source [as weak field, non-relativistic
(stationary) approximation (it is assumed that ‘mass is possible’) for R2-gravity (5)] and its
solution (R is 4d distance, radius):
∆2ϕ = − a
δ(R); ϕ(R2) =
lnR2 − b
(+ c , but c does not matter); (13)
the attracting force between two point masses is Fpoint =
, a, b should be proportional to
both masses.
Now let us suppose that all masses are distributed along the extra dimension with a ‘universal
function’, µ(p),
µ(p) dp = 1. Then the attracting (gravitation) force takes the next form [see
1 These amplitudes can depend on additional vector-parameters (‘equipment vectors’) relating to differential of
field mapping at a ‘quasi-particle center’ – where quasi-charge density is largest (if it has covariant sense).
0 1 2 3 4 5 6
Fig. 1. Deviation δF = F − 1/r2 for different µ(p), see Eq. (14) and text below.
(13); r is usual 3d distance]:
F (r) =
ϕ(r2 + (p− q)2)µ(p)µ(q) dp dq =
V − b V ′, V (r) =
µ(p)µ(q) dp dq
r2 + (p− q)2
. (14)
(Note that V (r) can be restored if F (r) is measured.)
Taking µ1(p) = π
−1/(1 + p2) (typical scale along the extra dimension is taken as unit, L = 1;
it seems that L should be greater than ten AU), one can find rV1(r) = 1/(2 + r) and
F (r) =
8 + 4r
2b(1 + r)
r2(2 + r)2
; or (now L 6= 1) F (r) =
2L(2L+ r)2
, where a = b = 2/L2.
Fig. 1, curve (a) shows δF = F − 1/r2 (deviation from the Newton’s Law; a/b is chosen that
δF (0)=0); two other curves, (b) & (c), correspond to µ2 = 2π
−1/(1 + p2)2, µ3 = 2π
−1p2/(1 + p2)2
(also δF (0)=0; residues help to find rV2 = (10 + 6r + r
2)/(2 + r)3, rV3 = (2 + 2r + r
2)/(2 + r)3).
We see that in principle this theory can explain galaxy rotation curves, v2(r)∝ rF r→∞−→ const,
without need for Dark Matter (or MOND [2]; about rotation curves and DM see [7]; they are
looking for DM in Solar system too, [8]).
Q: Can the ‘coherence of mass’ along the extra dimension be disturbed ? (the flyby anomaly,
the Pioneer anomaly [9]); can µ(p) be negative in some domains of p ?
11 How to register ‘powerless’ waves
This section is added perhaps for some funny recreation (or still not ? who knows). We have
learnt that S-waves do not carry momentum and angular momentum, so they can not perform
any work or spin flip.
But let us conceive that these waves can effect a flip-flop of two neighbor spins. So, a ‘detector’
could be a media with two sorts of spins, A and B. Let sA = sB = 1/2 but gA 6= gB, and let
the initial state is prepared as follows: {<sAz >,<sBz >}(0) = {1/2,−1/2}. Then the process of
spin relaxation starts; turning on appropriate magnetic field Hz (and alternating fields of proper
frequencies) one can measure the detector’s state and find the time of spin relaxation.
The next step. Skilled experimenters try to generate S-waves and to register an effect of these
waves on spin relaxation. The generation of intense ‘coherent’ S-waves could be proceeded perhaps
with a similar spin system subjected to alternating polarization.
12 Single photon experiment (that to feel huge extra dimension),
and Conclusion
Today, many laboratories have sources of single (heralded) photons, or entangled bi-photons (say,
for Bell-type experiments [10]); some students can perform laboratory works with single photons,
having convinced on their own experience that light is quantized (the Grangier experiment)[11].
It is being suggested a minor modification of the single (polarized) photon interference exper-
iment, say, in a Mach-Zehnder fiber interferometer with ‘long’ (the fibers may be rolled) enough
arms. The only new element is a fast-acting shutter placed at the beginning of (one of) the inter-
ferometer’s arms (the closing-opening time of the shutter should be smaller than the flight time
in the arms). For example, a fast electro-optical modulator in combination with polarizer (or a
number of such combinations) can be used with polarized photons.
Both Quantum mechanics (no particle’s ontology) and Bohmian mechanics (wave-particle dou-
ble ontology)[12] exclude any change in the interference figure as a result of separating activity
of such a fast shutter (while the photon’s ‘halves’ are making their ways to the place of a meet-
ing). However, if a photon has non-local spaghetti-like ontology (along the extra dimension) and
fragments of this spaghetti are moving along both arms at once, then the shutter should tear up
this spaghetti (mainly without photon absorption), tear out its fragments (which will dissolve in
‘zero-point oscillations’). Hence, if the absorption factor of the shutter (the extinction ratio of
polarizer) is large enough, the 50/50-proportion (between the photon’s amplitudes in the arms)
will be changed and a significant decrease of the interference visibility should be observed.
QM is everywhere (where we can see, of course), and, so, non-linear 5D-field fluctuations,
looking like spaghetti-anti-spaghetti loops, should exist everywhere. (This omnipresence can be
related to the universality of ‘low-level heat death’, restricted by the presence of topological quasi-
solitons – some as the 2D computer experiment by Fermi, Pasta, and Ulam, where the process of
thermalization was restricted by the existence of solitons. See also the sections 5–8 (and [4]) for
arguments in favor of phenomenological (quantized) ‘secondary fields’ accounting for topological
(quasi)charges and obeying superposition, path integral and so on.)
AP, at least at the level of its symmetry, seems to be able to cure the gap between the
two branches of physics – General Relativity (with coordinate diffeomorphisms) and Quantum
Mechanics (with Lorentz invariance).2 Most people give all the rights of fundamentality to quanta,
and so, they are trying to quantize gravity, and the very space-time (probing loops, and strings,
and branes; see also the warning polemic by Schroer [14]). The other possibility is that quanta
have the specific phenomenological origin relating to topological (quasi)charges.
2Rovelli writes[13]: In spite of their empirical success, GR and QM offer a schizophrenic and confused under-
standing of the physical world.
References
[1] A. Einstein and W. Mayer, Sitzungsber. preuss. Akad. Wiss. Kl 257–265 (1931).
[2] M. Milgrom, The modified dynamics – a status review, arXiv: astro-ph/9810302.
[3] J. F. Pommaret, Systems of Partial Differentiation Equations and Lie Pseudogroups (Math.
and its Applications, Vol. 14, New York 1978).
[4] I. L. Zhogin, Topological charges and quasi-charges in AP, arXiv: gr-qc/0610076; spherical
symmetry: gr-qc/0412130; 3-linear equations (contra-singularities): gr-qc/0203008.
[5] S.M. Carroll, Why is the Universe Accelerating ? arXiv: astro-ph/0310342
[6] B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry – Methods and Applica-
tions, Springer-Verlag, 1984.
[7] M.E. Peskin, Dark Matter: What is it ? Where is it ? Can we make it in the lab ?
http://www.slac.stanford.edu/grp/th/mpeskin/Yale1.pdf; M. Battaglia, M.E. Peskin,
The Role of the ILC in the Study of Cosmic Dark Matter, hep-ph/0509135
[8] L. Iorio, Solar System planetary orbital motions and dark matter, arXiv: gr-qc/0602095;
I.B. Khriplovich, Density of dark matter in Solar system and perihelion precession of planets,
astro-ph/0702260.
[9] C. Lämmerzahl, O. Preuss, and H. Dittus, Is the physics within the Solar system really
understood ? arXiv: gr-qc/0604052; A. Unzicker, Why do we Still Believe in Newton’s Law ?
Facts, Myths and Methods in Gravitational Physics, gr-qc/0702009.
[10] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 81, 5039
(1998); quant-ph/9810080; W. Tittel, G. Weihs, Photonic Entanglement for Fundamental
Tests and Quantum Communication, quant-ph/0107156.
[11] See the next links: departments.colgate.edu/physics/research/Photon/root/ ,
marcus.whitman.edu/ beckmk/QM .
[12] H. Nikolić, Quantum mechanics: Myths and facts, arXiv: quant-ph/0609163 .
[13] C. Rovelli, Unfinished revolution, gr-qc/0604045 .
[14] B. Schroer, String theory and the crisis in particle physics (a Samisdat on particle physics),
arXiv: physics/0603112;
the other sources of contra-string polemic are seemingly the books: P. Woit, Not even wrong;
L. Smolin, The Trouble with Physics (and the blog math.columbia.edu/∼woit/wordpress).
http://arxiv.org/abs/astro-ph/9810302
http://arxiv.org/astro-ph/9810302
http://arxiv.org/abs/gr-qc/0610076
http://arXiv.org/gr-qc/0610076
http://arXiv.org/gr-qc/0412130
http://arXiv.org/gr-qc/0203008
http://arxiv.org/abs/astro-ph/0310342
http://arxiv.org/astro-ph/0310342
http://www.slac.stanford.edu/grp/th/mpeskin/Yale1.pdf
http://arxiv.org/hep-ph/0509135
http://arxiv.org/abs/gr-qc/0602095
http://arxiv.org/gr-qc/0602095
http://arxiv.org/astro-ph/0702260
http://arxiv.org/abs/gr-qc/0604052
http://arxiv.org/gr-qc/0604052
http://arxiv.org/gr-qc/0702009
http://arXiv.org/quant-ph/9810080
http://arXiv.org/quant-ph/0107156
http://departments.colgate.edu/%physics/research/Photon/root/photon_quantum_mechanics.htm
http://marcus.whitman.edu/~beckmk/QM/
http://arxiv.org/abs/quant-ph/0609163
http://arXiv.org/gr-qc/0604045
http://arxiv.org/abs/physics/0603112
http://arXiv.org/physics/0603112
http://www.math.columbia.edu/~woit/wordpress/
	Introduction
	Unique 5D equation of AP (free of singularities in solutions)
	Tensor T (despite Lagrangian absence) and PN-effects
	Linear domain: instability of trivial solution (with powerless waves) 
	Expanding O4-symmetrical (single wave) solutions and cosmology 
	Non-linear domain: topological charges and quasi-charges 
	Example of SO2-symmetric quaternion field
	Quasi-charges and their morphisms (in 5D, ie m=4) 
	Coexistence: phenomenological `quantum fields' on classical background 
	`Plain' R2 gravity on very thick brane  and change in the Newton's Law of Gravitation
	How to register `powerless' waves 
	Single photon experiment (that to feel huge extra dimension), and Conclusion
