ELEMENTARY STRING COSMOLOGY ACCORDING TO QUANTUM RELATIVITY
1. The Weyl-Loop as transformed Planck-String-Boson in Quantum Geometry
2. Parameters of the Weyl-String and the Expansion of the Universe as a 'seeded'
Protoverse
3.
1. The Weyl-Loop as transformed Planck-String-Boson in Quantum Geometry
This brief treatment shall attempt to introduce the casual and the expert reader to the
topic of a stringbased universe. The language used shall be mostly devoid of technical mathematics and show a greatly simplified
manner of how to envision a 12-dimensional universe as a simpler and more familiar 3-dimensional one.
The expert reader is invited to rigourise this treatize in coauthorship for possible
publication at appropriate locations.
Exercise-Task 1:
a) I ask the reader to begin in drawing a single circle, say onto a flat piece
of paper and centred on the Nullpoint of an x-axis. This loop is a metrically hitherto undefined Weyl-Loop or Weyl-String.
b) Now define the diameter D=2R=d on this circle, say as coordinates
x=-R and x=+R. Call this partition #1 for an encompassing diameter D identical to 1d. This circle has area of:
A=πR² and a projected volume after rotation about this diameter normal to the x-axis of: V=4πR³/3.
c) Next generalise this Diameter as D=nd for a quantised number count n=1,2,3,4,...,n.
Then for n=2, we have D=2d=4r for R=2r for A=πR² =4πr²
encompassing two circular areas summing to 2πr² and in the ratio 2:1.
We so have half the previous area for the 1-partition 'covered', the other half being
'uncovered'.
In terms of volumes, the 2-partition gives two spherical 'little' volumes 4πr³/3
adding to 8πr³/3, now contained within the encompassing 4πR³/3=32πr³/3 in a ratio of 4:1.
d) For n=3, we find D=3d=6r, A=πR²=9πr² circumscribing
an area of 3πr² in the ratio 3:1 and three 'little' volumes totalling 4πr³
within 36πr³ in a ratio of 9:1.
e) For n=4, we find D=4d=8r, A=πR²=16πr² circumscribing
an area of 4πr² in the ratio 4:1 and four volumes summing to 16πr³/3 within 256πr³/3 in
a ratio of 16:1.
f) For n=n, we find D=nd=2nr, A=πR²=n²πr² circumscribing
an area of nπr² in the ratio n:1 and n volumes summing to 4πnr³/3 within
4πn³r³/3 in a ratio of n² :1.
Most generally for the n-partition then. The areas A diminish in a ratio of n:1
and the volumes V diminish in a ratio of n² :1.
We may now introduce the notion of a 'Radius-Vector-Space' R , which is metrically
utilised to define a 'Sphere-Space' S, the latter enclosing a 'Volume-Space' V.
Using this, one can now define a 'Point-Space' P=V0=1.
The 0-Sphere of S0 resides in Radius-Space R1 and is defined as the Diameter D or the
equipartitioned distance of 2R=V1 between two metric 'points ±V0' aka the locus of S0.
The 1-Sphere of S1 resides in Radius-Space R2 and is defined as the 1D-Manifold or Perimeter
of a Circle, circumscribing an area A=πR²=V2 as the locus of S1.
The 2-Sphere of S2 resides in Radius-Space R3 and is defined as the 2D-Manifold or Surface-Area of
a normally rotated circle, circumscribing a volume V=4πR³/3=V3 as the locus of S2.
The 3-Sphere of S3 resides in Radius-Space R4 and is defined as the 3D-Manifold or Volume
circumscribing a hypervolume H=π² R^4/2=V4 as the locus of S3.
As we have now reached a dimensional transition between manifolds and volumes; allowing
us to describe a 3D-Surface as equivalent to some form of a familiar 3D-Volume; we might hypothesise this mathematical
property to become a natural boundary for a higher dimensional superbrane scenario. This hypersphere or Riemannian
3-Sphere is indeed the one used in contemporary cosmological models, notwithstanding any relationships in regards to C-M-F-Theory.
Generally, we define the Surface-Area S for the n-partition as: Sn=dVn/dR=nVn/R
to give the boundary conditions for the Riemannianian hypersphere:
S0=dV0/dR=0.Vo/R=0.1/R=0; S1=dV1/dR=1.V1/R=2;
S2=dV2/dR=2.V2/R=2πR; S3=dV3/dR=3.V3/R=4πR² and S4=dV4/dR=4.V4/R=2π²R³
Here then So encompasses a 'Singularity' as a 1-Ball; a Circle as S1 encloses an Area
as a 2-Ball; a 2-Sphere as a 2D-Surface-Area encloses a 3-Ball as a 3D-Volume and a 3-Sphere as a 3D-Manifold encloses
a 4-Ball as a 4D-Volumar, which can also be expressed as a 3dimensional volume in Radius-Space R3 by the nature of embedded
curvatures.
This then leads us to the next exercise, which shall lead to a description of the quantisation
of spacetime in 12 dimensions, the so called Vafa-F-Space in F-theory, which is the natural 3-brane extension for Witten's
11D-M-Space as a 2-brane (or manifold or membrane), which extends the 10D-C-Space as the 1-brane aka the superstring.
So how does a simple Euclidean flat geometry of the circle give us higher dimensions?
The answer is intrinsic curvature and a deviation from the Euclidean flatness described
in the above.
We continue with Exercise-Task #2:
Here I ask not for any mechanical drawings; the previous circles for the 2-partition
will suffice; but I ask for a little thinking on the reader's behalf.
Imagine the two circles inscribed in the bigger circle of area A=πR² =4πr²
to define the cross-section of a simple 2-Torus. This torus will have an inner doughnut radius of r and an outer 'ring'
perimeter of 2πr as the locus of this perimeter travelling between the two centres of the two defining and inscribed circles
say.
The torus, as a 3D-geometrical manifold intersects itself at the center of the encompassing
sphere of volume V=4πR³/3=32πr³/3 in a point defining a 'Tangent-Plane', which is perfectly Euclidean flat
and so of 0 curvature.
This central point experiences however a negative intrinsic curvature in that the 'singularity'
of the tangent-point is 'funnelled' wormhole-like by the volume of the encompassing sphere, which is not part of the volume
of the torus.
The encompassing sphere, say as 'measured' by an external observer located on the outer
surface of this sphere, would however experience a positive curvature in the closure of the spheroid.
Projecting this observation forwards, we may state, that a universal observer within
the closed outer universe, would encounter a dilemma in measurement.
On the one hand the universe appears perfectly flat as the tangent plane of observation,
but on the other hand and in using localised coordinate systems, the universe is measured as open and hyperbolic with
perhaps scenarios of 'missing mass' and 'missing energy'.
We may also term this superimposition of intrinsic curvatures as describing a multiconnected
universe; and as only an observer situated on the surface of the encompassment (or distant from that by a displacement
exceeding R from the centre), would be aware of the overall closure of the geometrical construct (of singular connectedness)
and hence a projected topology for a universe closed in a higher dimension, but open in a dimensionality reduced by 1.
Continuing our simple quantum geometry; we find VSphere=4πR³/3=32πr³/3
for the 'closed universe' of a Weyl-String as a sort of 'envelope' around a circumscribed volume VTorus=2π²r³
in a ratio of VSphere/VTorus=(32πr³/3)/2π²r³=16/3π=8(2/3π) or about a numerical value
of 1.7.
The surface area of the torus is (2πr)²=4π²r² as the product of the two perimeters
defining the 2-Torus.
We may also here allow the 'curvature factor' 3π/2~4.7123.. to represent an upper
bound for the Feigenbaum-Delta of Chaos Theory (4.66..).
Now the area ratio for the 2-partition was 2:1 and the volume ratio was 4:1 previously
for the Weyl-Loop.
Transforming the two spherical volumes into a single torus by say a 'solid of revolution'
integration, changes the volume ratio to a fractional value from 4 to 1.7 and the surface area ratio becomes 16πr²/4π²r²=4/π.
If we now write: (3π/2)VSphere=8VTorus, we find a geometrical statement
which we can use to establish a perfect identity between the multi-connected toroidal universe and a single-connected spherical
universe; the latter becoming subjected to the 'curvature factor' 3π/2~4.7123.. and which physically, say under
the agency of inertia will become evidenced in the Feigenbaum-Delta of Chaos Theory (4.669..).
The Higher Dimensions I:
Generally, we have now a hypersphere of volume S4=dV4/dR=4.V4/R=2π²R³ ,
which also represents a 2-Torus which selfintersects itself at the centre of its geometry.
Defining the Weyl-Loop as such a quantum geometry; we can scale the metric Radius-Space
R3 as being quantised by spacetime quanta of volume Vspq=2π²Rmin³ and relative to an encompassing Radius-Space
R4, which is rendered R3 by topological connectivity in a Moebian-Twist or dimensional extension.
This, as a 1D-Manifold or 1brane results in a 2D-Manifold or 2brane in that a 1D-linesegment
becomes extended as a 2D-Moebian strip; which can then become subjected to the 'twisting', requiring an additional orthogonal
dimension.
Historically, this underpins and simplifies the idea of the 'second string revolution'
of March 1994, when Edward Witten proposed the superstring class IIA to occupy a membrane dimension as 2D object in 11D
in extension of the superstrings residing in 10D as 1D objects.
If we now extend the topology of the Moebius-Strip to allow the apparent enclosure of
yet an added dimension, we obtain the Klein-Bottle as a 2D-Manifold, yet appearing to encompass a say spherical volume in
3 dimensions.
This topologically Moebionised R4 space then becomes bounded in the toroidal 'Hubble-Volume'
2π²Rmax³ and with the topological consequence, that the 'outside' of this manifold is Moebian connected to its 'inside'.
In terms of information, the 'data' of the 'open universe' becomes 'mapped' onto the
'inner half' of the 'Cosmic Klein-Bottle-Dragon' (CKBD) (the author's literary licence in nomenclature here applies).
As this 'data' is however continuous in the Moebian connectivity of the 'outer half',
the data is also mapped onto the 'outside' of the CKBD and a distiction between the dimensions can be made in principle as
say the Omni-Space of C-M-F comprising dimensions 10-11-12; yet also can be avoided in its unitisation as a 10th dimension
of a string dynamic in 1D, that is our 0-Sphere in metric R1-Radius-Space from before.'
We now consider R3 to comprise our Weyl-Loop quantisation as some finite number
count N, metrically lower bounded in the spacetime quantum in Vspq=2π²Rmin³ and upper bounded (in the first
definition) by VCKBD=2π²Rmax³ =N.Vspq.
The quantum cosmology so begins with Vspq=2π²Rmin³ as an R4-Space around a Vo singularity
and being bounded by R3-Space as metric limit for spacetime quantisation of the Weyl-Loop-Unit.
We can define a 'linearised' 3D Line-Space in an 'opening' of the
Weyl-Loop into a Weyl-String, which then becomes 'attached' as such a string as say one end to the primordial spacetimequantum
Vspq=2π²Rmin³ and which is the Brane-Space in 3D.
An important interlude of required parameters:
We are now required to postulate the existence of relational operators, which in some
fundamental manner allow definition of such abstract symbolisations as used hitherto for labels like space and time and radius
and dimension and so on.
The historical evolvement about this relates to the question and origin of so called
'Natural Laws' for such operators as energy, momentum and position and its derivatives like velocity, acceleration and so
on. Quantum Relativity (QR) postulates, that the notion of the 'Laws of Nature' derives from a realm prexistent to that of
any spacetime metrically defined.
QR suggests, that the elementary nature of the cosmos is that of a form of 'Source-Energy';
an energy not derivable from inertial parameters defining mass; however definable by such via inductive relationships and
logical principles.
QR also postulates, that this 'Source-Energy' is fundamentally related to the notion
of 'Consciousness', which can be defined in labels of 'Self-Awareness' being the timedifferential of frequency as inverse
number count and therefore as the physically and mathematically applicable principle of angular acceleration action upon the
spacetime quanta recursively and iteratively.
This is detailed elsewhere and it suffices here to state, that the 'Source-Energy' aka
the 'Universal Space-Consciousness' can be modelled on a binary algorithm, which is based on the Weyl-Loop closing and opening
as the qubit [0,1].
This algorithm selfiteratively defines itself following principalities coupled to antiprincipalities
in a one-to-one correspondence with dimensional counters and integer relations, which, under guidance of those (10) principles
emerge a number of integer relations, which can be used to calibrate mensuration systems in terms of the 'Source-Energy' aka
the 'Selfawareness' of Spacetime itself.
In particular the mathematical constants k, h and c² crystallise in connection (and in
that order) with the selftransformation of 'Consciousness' as the 'Source-Energy'.
The integer triplets (15,10,32) and (9,10,16) become adjacent to a 'pure' integer in
11 and an energy (E) proportionality forms as the ratios Frequency F=E/h and Mass M=E/c² with an
'equality' {11=} connecting them. Associating h=(15,10,32) and c²=(9,10,16) with the selfdefining algorithm, then allows identification
of a cosmic unitary mensuration system (denoted as *) for the proportionality constants known as frequency and mass in
Planck's Law E=hf and in Einstein's E=mc².
In other words there exists a calibration for the meter, the second and the kilogram
in SI, where c=3x10^8 (m/s)* and h=1/(15x10^32) (Js)* precisely.
A technologically advanced civilisation, someplacetime in the CKBD could so measure h
and c in some unitary mensuration system and then calibrate its measurements with the integer based 'cosmic' one.
The Higher Dimensions II:
If we now imagine three rotational freedom degrees, each for say cartesian coordinate
axes x, y and z; then we can introduce a colocal 3D Hyper-Space. This allows the open Weyl-Strings to curl
up again in closure in an overall Twistor-Space in 7 dimensions, comprised of 6D-Space plus a time dimension.
Because the freedom degrees are bidirectional in both the translational axes of the Line-Space
as plus/minus say; and also chirally bidirectional as clockwise and anticlockwise say in the Hyper-Space; we can also envison
a 3D Quantum-Space of Vibration about a say mean positional coordinate.
Adding our 4th time-dimension as a connector dimension then forms a dimensional relationship
between Line-Space and Hyper-Space as a real 4th dimension in the linear unfoldment, which is rendered imaginary however
in the angular selfencompassment of the Omni-Space in 12D.
Corollarily, the 7th dimension is the connector time dimension between Hyper-Space and
Quantum-Space and the 10th dimension is the time connector between Quantum-Space and Omni-Space.
The Omni-Space Moebius-Connects back to itself via the 1st dimension, being identical
to a 13th dimension in the circular continuum of integer dimensions.
We so have generated a 12-dimensional universe, perfectly able to be modelled in a Minkowskian
4-vector-cosmology of Euclidean flatness (as the tangent plane at the centre of the Weylian spacetime quantum).
2. Parameters of the Weyl-String and the Expansion of the Universe
as a 'seeded' Protoverse
The most elementary parameter of the Weyl-String is its Light-Path x=ct. This becomes
its wavelength as the perimeter λmin=2πRmin and which describes the now metrically defined quantum loop
around the singularity volume Vo=1 as the mathematical 'Point-Space' and forming a Weyl-Geodesic around this 'singularity'.
From this the c-invariance for generalised wave velocity crystallises immediately in
the minmax boundary/initial conditions: c=x/t=wavelengthxfrequency=λmin.fmax=2πRmin.fmax=λmax.fmin.=Rmax.Ho.
We so find the most elementary ratio defining a dimensionless (Now) time n=Ho.t
and Ho=dn/dt as the meaning behind our time-connector dimensions, given as boundary n0= λmin/Rmax=Ho/fmax.
This is the initial condition for the Weyl-Loop to define a metric cosmology. Two displacement
units cancel in unison with two inverse time units to give a dimensionless 'Now-Time' also known as an instanton. Herte then
Ho represents some constant, call it nodal frequency on the largest cosmological displacement scale and which defines so a
nodal Hubble-Law in Ho=c/Rmax.
The 'nodality' becomes the mass-independent lightspeed expansion of the universe in 11
dimensions; translating to the 3-Sphere forming the nodal boundary of maximum extent for the minimum extent of the Weylian
spacetimequantum to 'expand' into.
This so indicates a 'Steady State Hoylian' universe in a first initial state of a 'Seedling
Cosmology' (based physically on the physics and metric definition of Black Holes and prototypical ylemic neutron stars).
This 'Steady-State' is described in a Hubble-Oscillation, which is a 'bouncing' between
the odd and even nodes.
The even node is the Weyl-Geodesic, defined in the wormhole radius Rmin=λmin/2π
and the odd node is the Hubble event horizon Rmax.
For a n-time of n=1, Ho=1/t and the linear chronological time will be t1=1/Ho.
QR calculates n(present) to be about 1.1324.., rendering the universe 19.11 Billion years
old in 11D with a nodal half-oscillation taking 16.9 billion years in chronological time.
Exceeding n=1 in a one-directional 'arrow of time', so must indicate an 'electromagnetic
return' of the lightpath parameter c=x/t in regards to the encompassed R4 Radius-Space, described as 10D-Omni-Space; which
is however simultaneous to a second principled 'Expansion-State' for the universe in the modelled 12D-Omni-Space across
the 'Mirror-Dimension'of 11D-Omni-Space.
In simple language, one might say, that the nodal mirror is semi-transparent and laser-like
allows both refracted information into 12D and reflected information into 10D.
Should now the universal observer at the centre of the universe, which is coincident
with the centre of the primordial Weylian spacetime quantum; measure the light parameters obtained, say 'from the distant
stars'; then the electromagnetic return will instigate a dimensional intersection between M-Space and C-Space and depending
on formulations relating the postulates of Special Relativity (SR) to the c-invariance.
The Hubble oscillation will transit its journey from n=1 to n=2 in a state of superposition
to the asymptotic expansion of lightspeed retardation under gravitation in the inertia defined cosmology and so, again depending
on certain limiting redshift scenarios, an accelerating universe will be erroneously assumed to exist; should the universe
be defined as a singular entity expanding into nonexistant prespace.
The inflationary universe defines the 11-dimensional boundary 'instantaneously' and 'creates'
the boundary conditions for the 10-dimensional cosmology to asymptotically approach this boundary condition, which is also
an initial and a final condition.
The 11D-Universe is photonic, whilst the 10D-Universe is inertial. The former carries
constancy in expansion speed c, but the latter decelerates under the auspices of the mass parametric universe defined in the
transformation of the primordial 'Source-Energy' first into kinetic bosonic energy (E=kT) from certain 'Seedlings' and then
into the energy equivalence derived in the below.
As a next Weylian parameter we have the maximum frequency as an angular velocity ωmax=2πfmax=
for c=λmin.ωmax/2π. and which now defines the Energy Eigenstate for the Weyl-Loop as E=hωmax/2π=hfmax.
We can see straight away, that the most elementary parameter for a metric universe
becomes the angular momentum quantum operator h/2π. This operator is independent from the linear radial extension of
say linearly propagated light or electromagnetic photons. The angular velocity ωmax=2πfmax=v/R, where now the linear
tangent velocity v cancels the radius R of the Weyl-Loop as a wormhole perimeter.
Using our primordial algorithmic definitions of the energy equivalence for a (to be defined)
massless and an inertial universe in E=hf=mc²; we then can define this relationship in the following manner.
E=hf iff m0=0 for a massless universe, characterised
by frequency distributions and especially a maximum frequency fmax and a minimum frequency f0 close to but not equal
to 0;
E=mc² iff f0=1/fmax =fmin for an inertial universe, characterised
by mass-frequency equivalents E=hf=mc²=hc/λ and leading to Compton/de Broglie wave matter definitions with de Broglie
wave length λdB=h/mdB.vdBgroup. The minimum frequency so can be defined as a measurement of chronological time in fmin=t0.
and as the instanton in Quantum Relativistic Cosmology.
We have introduced a concept of modular duality, which renders the inverse of frequency
as a number identifiable as a time measurement also equal to a number in the unitisation fmax.fmin=1=λmin.λmax.
This is known as S-Duality in string theory relating vibratory high frequency eigenstates
for energy and momentum to a radius R and describing a physics which is equivalent to relating such winded low frequency eigenstates
to a radius defined as 1/R. There is a corresponding T-Duality relating strong- (or perturbative, convergence <1) and
weak (non perturbative divergence >1) coupling constants between strings, which also uses this identity in (strong
coupling constant).(weak coupling constant)=1. Both can be considered a form of M-Duality or Mirror-Duality as described by
QR.
The inflationary instanton so is defined in a de Broglie phase-speed: vdBphase=λdB.fdB={h/vdBgroupmdB}.{c²mdB/h}=c²/vdBgroup>c
for all vdBgroup<c; and is Rmax.fmax in the inflation of space and the instanton of time.
Setting the minimum frequency t0=f0=fmin=1/fmax introduces the inversion property of
the linearised time t in context with the c-invariance in the 'cycle-time' n=Hot via initial condition n0= λmin/Rmax=Ho/fmax.
Because Ho is a nodal constant as a minimum cosmic frequency; it must change relative
to the oscillation parameter n in terms of the scale factor R(n), which describes the expansion of the asymptotic and mass
parametric universe of deceleration.
Rmax is a boundary curvature radius for example defining the closure of the hypersphere
in the critical density (as 3Ho²/8πGo) in General Relativity (GR) and as the Schwarzschild Radius Rmax=2GoMcritical/c².
The 'true' Hubble-Constant so varies as a function of cycletime n between the nodes in a minimum frequency Ho and a maximum
frequency fmax.
Our minimum frequency fmin=1/fmax describes the time instanton as a linearised minimum
for chronological time t0 and associates with it a quantisation of mass in terms of a characteristic eigenfrequency
for inertia.
The expanding universe so becomes an asymptotic expansion in 10D-C-Space, which is decelerating
under its own gravity in hyperbolic curvature with certain 'missing energy' scenarios instituted by the enveloping 11D-M-Space,
which both oscillates as a 'Hubble-Standing Wave' in between its Hubble nodes of the Quntum Big Bang Singularity of the Weyl-spacetime
quantum and its inflationary set Hubble Event Horizon; and continues its expansion into 12D-F-Space in the dimensional continuity
between the 'outside' and the 'inside' of the Cosmic-Klein-Bottle-Dragon, which describes the Omniverse.
This 12-D expansion so allows a secondary definition for the Omniverse relative
to its 11D-bound of the 11D-Membrane, which is truly the 3D-Manifold of the Riemannian hypersphere as a 3-Sphere.
The 10D-Universe is described with an expansion parameter n/(n+1), which behaves
mathematically like the expansion parameter 'a' in GR.
We so define R(n,t)=Rmax{n/(n+1)} with n=Hot=t.dn/dt for a general cosmic velocity
V(n,t)=c/(n+1)² and for a cosmic deceleration A=-2cHo/(n+1)³.
The problem of the 'missing energy' in terms of acceleration and the 'cosmological constant'
(A mapping Λ say as quintessence) so solves itself immediately.
The deceleration A of the curvature radius R(n,t) is intrinsic to the cosmic evolution.
The 'missing mass' is a consequence of a 'Baryon Seed as Mo' defined in string parameters being less than the critical
mass given by the 11D Hubble-Bubble as a say Mother-Black Hole.
The deceleration parameter in GR is precisely this ratio as qo= Mo/2Mcritical=½Ω
and as the 'Omega' of the ratio for the actual to critical densities in terms of the baryonic seed.
But this ratio is also the ratio between the inertial, say Newtonian mass content of
the universe in the form of m=F/a=GoMo/λmin² and the de Broglie phasal 'hyper acceleration' Rmax.fmax².
So we find: Ω={2GoMo/λmin²}/{Rmax.fmax²}={2GoMo/c²}/Rmax and which
is the Schwarzschild metric applied to the entire universe as a Black-Hole-Mirror with the critical mass Mcritical modulated
by the baryon seedling Mo.
This btw results in a Sarkar-Black-Hole, which as a 'Strominger Black-Hole' is extremal
as a boundary condition for the scaling of superclusters of extent, calculated by QR of diameter so 4.4x10^24 meters. It is
so this difference of mass between an 11D-Mother Black Hole and a 10D-Daughter Black Hole (as the Sarkar supercluster limit
of so 236.5 Million lightyears), which underpins the 'dark matter' and the 'dark energy'.
I'll leave it for now and continue after appropriate responses.
Dear Pedro Gonsalves and wohever else it may be of concern!
I am sending this directly to your address, hoping that the mathematical symbols
copy in your browser.
It is a nuisance, that the yahoo groups will not copy unicode correctly.
I also copy this post on my website, where the symbols copy.
Thanks for anyones considerations.
All the best.
Tony B.
