Curvature of Space and some views on motions in macrocosm
The subject curvature has been treated at greater length
in files
within Physics.
1. The geometry of cosmos is a much discussed question.
Is Universe as a whole positively or negatively curved?
Interpreters of Einstein's relativity theory have
meant that he has replaced gravitation as a force with the
concept of curvature of space. Einstein's view on Universe
has been compared with the surface of a horsesaddle. When it's said that the concept of gravitation
as a force my be replaced by the curvature of space, it
seems that something like the gravitational force nevertheless
slips in through the back door through tensor concepts in
Einstein's equations? And apart from that, it remains to
explain why space are curved around big masses, why
the geometrical properties of space is different around
these than in interstellar mediate space between them.
From the viewpoints of our dimension model here it seems
reasonable to state that Universe must be both positively
and antipositively* curved, at least in ddegree step 4→3
(and more like a horse than a horsesaddle). *(Product of
+R and R becomes negative, R for radius of curvature.)
The questions about curvature have been connected with
the question if Universe is finite in size or infinite.
Presumably too Universe is neither finite
or infinite in a quantitative sense (additional), but
both in a qualitative  this with the redefinition of
"infinity" as anticenter in the model. Size
as a property is created through Big Bang and the further
development. It concerns relations  relations between
parts of the whole. For Universe, per definition including
everything, there is nothing "outside" to compare
with, the concept size loses all sense.
To regard cosmic Space as such, as some kind of
abstract network with own existence, an entity in itself,
becomes nonsense, both in our model and according to a lot
of scientists. Its reality lies in relations and is depending
on its complementary part, the masses. The polarity is expressed
in the formula +/ E=mc2. Here the
negative energy E (as Dirac's holes for positrons) has to
be interpreted as not just lack of energy but an alternative
form, that means as antimatter
on a fundamental level.
Hence, with Space and Mass interpreted
as the complementary poles (in ddegree 3 ), the masses
with positive curvature get the "matrix" or counterpart
in an antipositive curvature of Space.
It's also an accepted apprehension of
later years that masses as galaxies are carried away by
expanding Space (not flying themselves).
Choice between concepts optional?
May we regard interpretations in terms of curvature
and in terms of forces as equally
valid? The views and definition of forces in the model here
are ultimately geometrical With the assumptions
in the model that a dimension chain also corresponds to
a series of angle steps, curved lines are in fact
postulated. (Curvature could in that case in reality be
quantified.)
The concepts of curvature seem bound to
a view in dimension degree (ddegree) 3, a concept which
may lose its sense if we could apply a 4dimensional aspect?
Curvature and elliptic versus hyperbolic
geometries could be applied to the forces as vector fields
as such in lower ddegrees, using formulations in terms
of Acceleration outwards (the FA
force) and Gravitation (FG
) as acceleration inwards. (About the forces here.)
In our dimension model one degree of structure gets debranched
into external motion in each dimension step 5 →
4 → 3→... And the ddegree of analysis is optional.
With forces per definition closely
related motions a such, this implies that aspects on
curvature in macrocosm could concern these motions. For
instance the "expanding" of Universe. The
apparently high escape velocity of distant galaxies could
be the expression in ddegree 3 of a static  and "straight"
outward/inward geometry in ddegree 4, when this is transformed
to lower degree and Space to antipositive curvature?
Corresponding "motions" for
masses becomes the 2dimensional rotation as an elliptic
geometrical form.
What should create an antipositive (R) curvature
of Space?
With the simple illustration of a cloth that is pursed
up in the middle, a piece of the radii in the centre shortened,
one gets accelerating breadth of the surface as "wave
tunnels" outwards. Pursing the circumference instead
leads to the football geometry.
With this illustration, the centre as "drawn"
underground, should be the first singularity at Big Bang,
the first 0pole, coming to represent an underlying level
in relation to the first materialized macrocosm.
Higher ddegrees underground (both 0 and
00poles in ddegree 4), inside lower ones, as underlying
level, should potentially generate a curvature of both R
and +R. "Externalization" of these higher ddegrees
as lost in the geometrical structure, as debranched and
transformed into external motions  according to fundamental
postulates  becomes a condition for both antipositive
expanding Space and positive curvature as mass aggregations.
Centre displacements:
All curved lines or pathways in cosmos could perhaps
principally be regarded as following from centre displacements
in relation to first primary 0pole and level developments.
The hypothesis is then related to escape
velocity: The further out from a primary centre a pathway
starts, the lower the energy, the more curved becomes the
pathways. (Cf. satellites,)
Centre displacements  escape velocity,
cf. steps 5 → 4→ 3:
Fragmentation of Mass:
An only positively curved surface defines and includes
one centre (1). An antipositively curved surface defines
instead (partly) 2 or more excluded centra. Hence, a Space
with antipositive curvature, should become a factor in
the fragmentation of masses in Universe.
Cf. in last step of our model: "Motions
from each other", outwards, with the root in 0pole
of ddegree 4, defines anticentra, while "motions towards
each other" with the root in first 00pole of ddegree
4, defines new 0poles.
It's observed too on a real macrocosmic
scale in Universe that huge, seemingly empty parts of Space
separate huge "highways" or accumulations of galaxies,
pointing to the possibility that a more fundamental and
elementary geometrical polarization Mass ↔ Space lays behind and before the later fragmentation steps.
Multilayer creation through antipositive curvature
inwards?
The most characteristic feature of life is antipositive
curvature turned inwards, developing inside positive curvature.
This combination is illustrated on the
biological level in embryonic growth, where surface of the
cell grows faster than the square of the radius, resulting
in multilayer creation within first, surrounding "positively
curved" layer.
Are there processes in the inorganic world that could be
understood in similar terms?
"Disturbances" from outside on big hydrogen clouds
in macrocosm or other such concepts which astronomers and
physicists use?
The nuclei of atoms can be analysed
as shells  also according to physicists. Could we here
eventually have a negative curvature inwards in heavier
nuclei in the relation "empty" space / mass, or
energy /+, which makes the nuclei labyrinthianly multistratified?
Cf. that charge is said to be built up inwards in heavier
nuclei.
The concept "isospin"
is perhaps not only a mathematical help concept? Maybe also
an expression for the "nutrient supply" of the
atom, (its dependence on nourishment as "empty space"?
(According to assumptions about light waves in
Physics here.) For energy metabolism of the nucleus?
In that case there should perhaps exist gateways in the
potential wall around nuclei for this import of nourishment!
Is it possible to imagine that such multilayer
structures in macrocosm  as the rings of dust around planets,
planet systems, spiral galaxies, perhaps star clusters in
a first phase, could be regarded as a similar patterns of
invaginations and openings for the incorporation of matter
to "shells" ?
Some added associations:
a) With the antipositive curvature, implying a kind of
potential acceleration of higher degree, regarded as expression
for and result of a level underlying masses, it would be
possible perhaps to interpret this underlying 4dimensional
geometry as cause to events where these masses throw out
surpluses of matter, as cause to star explosions, protuberances
etc. That means to interpret such "radial" occurrences
in stars (or galaxies?) in geometrical terms of the antipositive
curvature of space. An extrapolation of the ordinary geometrical
views on curvature?
b) The negative radius of curvature is usually in ddegree
2 illustrated by triangles with angle sum less than 180°.
(Combinations of such triangles may formally enclose positively
closed, perhaps elliptic, volumes.) In projection the magnetic
field around the earth forms such narrower angles at the
poles. They may be seen too as doubled in the fieldfree
area between two magnetic poles of the same kind, two Npoles
or two Spoles.
One perhaps more relevant "metaphor"
or expression for this geometry is the fibre directions
around germs (or start) of branches in a tree trunk:
c) The double nature of light:
Could perhaps the double nature of light as waves and particles
be interpreted in terms of the concept curvature?
When a ray of light is sent through a
small hole in a screen, it behaves and is detected as a
particle on a detector screen if the hole is bigger than
the wavelength of the light. But if the hole is smaller
than the wavelength the beam may be described as broken
up, behaving like a wave.
In this latter case it seems as the hole
got the role of an underlying centre, (as if the beam became
"pursed in the middle"). Would it eventually be
possible to interpret the wavy pattern of divergence in
terms of negative (or antipositive) curvature?
Olber's paradox and the red shift of light in macrocosm:
The expansion of Universe explains this shift towards
the red spectrum in light from more distant stars and galaxies.
The same expansion theory is also part of the explanation
to "Olber's paradox", the fact that the night
sky is mostly dark in spite of the enormous amount of stars
in Universe. The answer is said to be that the stars farthest
away from us have such a high escape velocity that their
light hasn't reach us yet or cannot reach us. (There are
other hypothetical explanations too among scientists.)
The question here is of these same things
could be explained purely by the negative radius of curvature
in "empty" space and its geometry:
If Space grows faster than proportional
to the cube of radius (and mass the reverse), a beam of
light through space have to take longer and longer jumps
for its propagation, equivalent with accelerating wavelengths.
Growing steps, growing lambda through the negative curvature
of space:
And, if so, much light from distant stars
might just pass by our planet in curved pathways,
when empty Space grows towards us  from the point of view
of the emitting star?
A picture of split light which perhaps passes by our Earth:
Both theories, the dynamic one about expansion of Universe,
and this static one about the negative curvature of space,
could be equally valid, only different aspects on the same
thing. In our model we prefer to regard the two aspects
as a stepwise process in time, where one ddegree of structure
(in ddegree 4) gets debranched and transformed into external
motion in next lower ddegree.
One scientists' explanation of Olber's paradox builds on
the assumption of a fractal distribution of mass in macrocosm
(see Wikipedia). This seems to be in agreement with
the model here, said without closer knowledge.
Two other annotations:
One author has in his book about astronomy proposed that
the light only "gets tired", as an explanation
for this displacement towards the red spectrum. Why in that
case?
According to our interpretation of light
propagation through cosmos, this propagation is dependant
on the negative energy of empty space for its continuous
existence. The empty space may not have the same quality
everywhere, it may have been exploited by heavy masses for
instance and represent different degrees of negative energy.
A "tiredness" of light should then depend on the
lack of a qualitative nourishment! (About vacant
space here.)
Another thought: Universe is crisscrossed by myriads of
beams of light, and a plenty of them must have the same
wave length and also meet antiparallelly  perhaps especially
so over big distances.
Why shouldn't they  reasonably  cancel
each other out through interference?
Motions in macrocosm:
Processes or motions in macrocosm are seldom described by
scientists in terms of dimension degrees or changes in dimensions.
Which are the possibilities to interpret the motional patterns
in such terms, as of different ddegrees or changes in ddegree
of structures, in compliance with the dimension model?
Rotation:
Most obvious fact in cosmos is that all celestial bodies
rotate. A tendency to rotation is also observed in the gigantic
Hclouds out of which stars and galaxies are born. It's
disputable if the scientists have given any good explanation
for this rotation. To quote one such explanation:
"All celestial bodies rotate because
of the law of conservation of angular momentum. This is
one of the most fundamental laws of the universe and comes
from the fact that the laws of the universe do not change
dependant on the direction we are looking.
All celestial bodies have at some time
condensed out of large clouds of gas and dust under the
influence of gravity. These clouds were probably moderately
turbulent so most of the angular momentum cancelled out
but there would always be a small residual in one direction
representing the net rotation of the whole cloud...."
The explanation doesn't feel very satisfactory: Note that
angular momentum is not explained, however conservative
it may be, and its origin is just vague suppositions. Remember
too that also elementary particles of microcosm rotate,
hardly explained in that way?
We could associate the quotation above
with the many studies of the problematic turbulence
as such and how "radial" inflows get translated
to whirls and circular outflows, or how just 23dimensional
circular streams and waves appear in liquids.
Could rotation be interpreted in terms of how the centrifugal
force (out of FA )
manifests itself in ddegree 3 as answer to  or when combined
with  the centripetal force (out of FG)?
Or, in other (?) terms: Could rotation and first angular
momentum have its origin in an angle between the magnetic
axis of a body and what comes to be the "rotational
axes"? An angle decided by inherent ddegree steps
between poles, e.g. the forces FA
 FG , then FE
 FM ? Two coordinate
axes, in a smaller angle of decreased ddegree, which in
itself give the basis for the concept angular momentum
? (M and Efields as polarized from the EMforce born in
ddegree step 32 in a dimension chain of forces, according
to suggestions here.
Angle steps hypothetically assumed as halvings: ...180°→
90°→ 45° →22,5°. (Generally the steps
in our dimension model is hypothetically assumed as also
connected with angle steps, in step D4 →
D3 180°→ 90°, implying
a change from radial to tangential (perpendicular) relation.)
Relations between rotation and other phenomena:
a) There is a formula for the relation between the
time it takes for a blast wave to propagate from the inner
of a mass to the whole mass and the shortest possible rotation
time, called "dynamic time scale":
It points to a translation or transformation
process (or jump) from linear Lwaves of ddegree 4 in our
model to ddegree 3 and rotation speed. A dimension step.
It's said about the Crab nebula that the
rotation energy of the central pulsar gets "translated
to" the whole nebula. (The source does not tell how.)
Do we here have a small step towards rotation as a solid
body, as friction, revealing the polar combination
of radial and circular forces?
b) There is also the relation between shortest possible
rotation time (Rt) and density:
The time is inversely proportional to the square root of
mean density:
Rt ~1 /√density.
I. e.: Higher density implies lower rotation time, = higher
rotational speed.
Density has in our model been assumed
as the only and first concept for the "physical quantity"
in ddegree step 5 → 4. (As
the Mass and Space concepts follow out of next step 4 →
3.). Then we should have two operators in this step, the
square root of x and inversion.
With the loop model of a dimension chain,
steps 5 → 4 → 3 → are connected
with steps (← 2 ← 1 ←
/00 from debranched degrees meeting the other way
around, here as motional patterns:
Rotational speed of 3dimensional bodies:
Inversion as expression for "the other way around"?
c) The rotation velocity has also a relation to
the spectral type of stars: hot O and Bstars rotate about
10 times faster than cooler Gstars, the type of our sun.
Why this is so the astronomers don't know, at least they
didn't a couple of decades ago. It seems to contradict the
formula above, if density of at least the cores in stars
are increasing from O and Bstars towards Gstars
for fusion of heavier elements. (Mostly only Hesynthesis
in OBstars.) See file Stars.
Has perhaps the formula Rt ~ 1/√density
above a limited validity? Could it eventually concern the
vertical line through middle step in the figure above, with
reference to figure
6 and figure
11 in file Stars: decreasing speed of rotation
towards the middle step? It's very high for neutron stars
(still higher for black holes if they rotate at all?). Corresponding
to OBstars at the other end of the chain? If so, the problem
could support the loop model.
The problem seems connected with that of stars where high
surface temperature seems opposed to high temperature and
density in the cores.
There is also such examples as these of the relation between
density and temperature:

Density, u/m3 
Temperature, K 
H2clouds 
^{<910}

^{<1}

H0 clouds

^{<68}

^{<12}

nets of tunnels 
< 105

105

More of structure as bound kinetic energy in colder clouds.
Parenthetically: Could neutrinos and antineutrinos be
connected with the properties Density and Temperature
respectively? In files about forces, the
weak interaction force is hypothetically assumed as
derived through step 5→
4 (Fw1) and 1 → 0/00 (Fw2)
in the dimension chain of forces. Fw1 and Fw2 connected
with neutrinos and antineutrinos. Cf. density, neutron
stars and p + e + antiν → n.
d) Stars as our sun doesn't rotate as a solid body.
The rotation time one turn at the poles is 37 days, at the
equator only 25 days. That's a relation about 3/2.
Could this relation be interpreted as
an expression for ddegree 3, represented by the central
rotational axes, and ddegree 2 as the equator plane? According
to the principle of more kinetic energy in lower ddegrees.
e) Rotation in galaxy clusters seems only partially
developed. This observation agrees with the views regarding
levels of mass
organization in macrocosm, where this highest
level gets its character from ddegree 4 or step 43. Rotation
assumed as the motional pattern developed in ddegree 3,
(figure
here).
f) When rotation around secondary and tertiary,
external centra is added, it may be said to illustrate 3dimensional
pathway motions; a consequence of rotations on different
levels of mass organization when it concerns planets around
the sun or stars around the galaxy. In these cases it could
simply be regarded as a 3dimensional motion in ddegree
2, since planets are born out of more or less 2dimensional
dust rings around the sun and stars in the "plane"
around the galaxy.
This complex winding up of masses have
similarities with the winding up of DNA in several steps
to compact chromosomes.
How look at other types of motional patterns than rotation
in macrocosm, processes illustrating other ddegrees in
compliance with the model here?
Vibration (D1)  pulsation (D4).  Pole exchange (D5),
ddegrees of motions:
We have to regard the 4dimensional vector fields both between
masses and within them, as underlying level in matter.
We have to identify formations of layers
(structures of ddegree 2) within masses besides the formation
of orbital planes separating masses.
And where to look for 1dimensional structures
with 4dimensional motions as pulsation? Difficult to distinguish
from 1dimensional "vibration" in 4dimensional
vector fields, Vdiv and Vconv.
 1dimensional motions as vibration or linear pathway
motions when halved:
Longitudinal Lwaves have in this model
been assumed as the motion in 4dimensional fields. Pressure
waves within masses would be one example
There is the hypothesis too, proposed
by a scientist, about pressure waves upholding the structure
and arms of our spiral galaxy. Perhaps supported by observed
changes between red and blue shift in light from sources
near the galaxy centre?
Another example could be the very expansion
of Vacant Space, derived from the FA
force in our model.
 3dimensional motions could also be identified
within layers of stars as our sun, layers in a macroscale
as 2dimensional: the convection currents as presumably
3dimensional could be the result of the stratification
into layers?
 4dimensional motions as pulsation, expansion/contraction
in "1dimensional structures" in cosmos? It's
of course questionable if there is any sense in talking
about 1dimensional structures. They become rather abstract.
In the case of pulsars for instance, inner
processes leads to emission of electromagnetic beams in
opposite directions from the magnetic poles, said to have
their source in the rotational energy of the pulsar. Here
the emitted beams could be regarded as 1dimensional "structures"
on a macroscale, created by inner processes which seems
multidimensional. (Hence "the other way around",
from motion to structure.)
According to scientists' suggested interpretations,
the origin of these beams are also related to the angle
between the axes of rotation and of the magnetic field,
that's 1dimensional "structures". The inner processes
include variations in rotational velocity and subpulses
within these, as well as assumed couplingdecoupling between
core and crust. (Together surely possible to interpret as
steps towards 4dimensional motions?) (Wikipedia.)
Why the described processes in pulsars should
be limited to neutron stars, if so, is another question.
Other 1dimensional "structures" would
be what is called "field lines", if differentiated,
separated through polarization processes like interference
and diffraction. It feels difficult however to attribute
4dimensional motions to individual field lines!?
Difficult likewise to attribute such motions
to spiral arms of galaxies when possible to regard as 1dimensional
structures on a macroscale  if not as "inhalation
/ exhalation" through the mentioned pressure waves.
 5dimensional motion of 0dimensional structures
as "pole exchange"?
The very meeting point where vectors inwards transforms
to vectors outwards may apply to the birth of Universe as
a whole.
If there are "black holes" where
mass crashes to a point, its reasonable to think that the
energy transforms to another form or reality, possibly as
a suggestion here to the expansion force of Vacant Space,
the opposite to Mass.
Gravitational collapses, implosions and
connected explosions, could be regarded as a secondary manifestation
of such pole exchanges.
*
