The Machian Momentum for the Universe We consider the universe to be a Quantum Universe; that is we sum all fermionic spin-states (N) as Intrinsic Angular Momenta to become the Intrisic Spin Angular Momentum for the Universe as a quantised protoverse. This total momentum we considered mirrored in a dimensional 11-DWittenMirror reflecting 12D-OmniSpace onto 10D-OmniSpace in a complexified within-without topology of a doublesided surface becoming onesided. The modular duality of superbrane theory will prove useful in application in that regard. This rather naturally engages the Heisenbergian Uncertainty Principle in the form of the fermionic (or halfspin) intrinsic angular momentm given in h/4p as the limit for measurement regarding dynamical systems. We engage the absolute polarity cancellation, say as x=v.t=(-v)(-t) in this doubling process in engaging the 4world scenario of George Ryazanov and described in our context in the following analysis. 1. Moments of Inertia and the nested curvatures of dimension mappings The Moment of Inertia of a solid sphere mass M and radius R is (2/5).MR2 and that of a spherical shell of the same dimensions is (2/3).MR2 as determined by integration across the geometries. We consider the universe to be the Hypersphere of Riemann of 4D-Volume [2p2Rmax3], which is also a 3D-surface of toroidal derivative, i.e. a KleinBottle as Moebian extension in 2D. This kind of topology incorporates naturally our doubling of a onesided surface via a dimensional twist of a doublesided surface. This we term the Francom-Moebius Adjacency about a Wolford-Centre or mathematical point singularity. As precisely 8 spheres radius (a) fit into a sphere radius (2a) and volume (4p/3)(8a3)=(32p/3)(a3); we define the curvature factor (3p/2) to relate toroidal volume [(2p2)a3] to spherical volume [(4p/3)a3]. Considering now the topology of such a torus, defined in radii a and 2a; we can observe a positive spherical curvature encompassing a negative hyperbolic curvature in a 3D perspective, the latter which we expand into 4D as the hypersphere of Riemann. The encompassment or enveloping of a lower surface dimension by a volume dimension is however twosided. We must consider the mapping of volumes onto both the inner and outer surfaces in the same dimensional sense. Whilst the inner volume of the earth, say is mapped onto the inner manifold comprising the earth's surface; the outer volume of the earth (the outer space) is mapped onto the outer manifold as the encompassing volume and as given in the 4D hypersphere, which is a 3D surface. This consideration then allows us to extend the 3D/4D scenario in the application of conifolded or enfolded higher dimensional geometries. Especially, there exist toroidal derivatives in 6D, which are called Calabi-Yau manifolds and their extensions into 7D are termed Joycian shaped topologies. We so invoke here the nomenclature of superbrane theory, without its mathematical formalism. In our analysis of the Machian observer, we shall find the principle behind higher dimensional M-Theory of the superbranes. The underlying principle is the Principle of Quantum Relativity (QR) as a natural extension for the Principle of General Relativity (GR), itself a natural consequence for the Principle of Special Relativity (SR). We rename M-Theory in 11 dimensions as FMC-Theory of Quantum Relativity in 12 dimensions for this purpose. The elementary precept of modular duality in M-Theory shall also be the defining foundation for QR. We then consider 3D-surfaces as 4D-volumes to conifold 6D-Calabi-Yau shapes and 7D-Joyce manifolds in the connector dimensions of their enfolding. This process renders the Minkowski-Riemann spacetime metric as a scalerelative enfolding of space within time, thus leading to the 'sharing of dimensions' and any object moving through time with maximum speed 'c', requiring to be necessarily at rest relative to its scale relative spacial dimensions. Any motion of the object relative to the space dimensions, would decompose the Minkowski 4-vector and reduce the speed c via the formalisms of Special Relativity and its postulates. For simplicity, we define the 12 dimensions as a simple superpositioning of four worlds superimposed upon one another, say as: LineSpace (1-2-3)HyperSpace(4-5-6)QuantumSpace(7-8-9)OmniSpace(10-11-12)(13=1-2-3)... Each of the four worlds is the Riemann hypersphere with 3 spacial dimensions and a connector time dimension, which forms however a spacial dimension for the adjacent superimposed world. So LineSpace connects to HyperSpace via a convenient, but ultimately imaginary timedimension and as say expressed in the Minkowsi 4-vector. We could also extend our 4 worlds in ascribing Translation to LineSpace, Rotation to Hyperspace, Vibration to QuantumSpace and Quantization to OmniSpace, the decisive constant being the speed of light in 'free space', denoted as 'c'. 2. The Machian observer and the principles of relativity and the Weyl nullification The rotational frames of references of HyperSpace will become elucidated in the Machian observer. Consider the rotation of a dynamical system about its axis of rotation, say a solid circular disk radius R with angular velocity w=v/R=2pf; f the frequency of rotation and v the tangential velocity linked to angular centripetal acceleration ac =v2/R. The postulates of SR lead to Time-Dilation and Lorentz-Contraction and invoke the notion of no absolute motion being measurable between two oberservers moving uniformely or at rest with respect to one another. This and the invariance of 'c', irrespective as to the dynamical motion of the observer, forms the Principle of Special Relativity. The Principle of General Relativity states, that a dynamical system of SR, that is in uniform motion of an inertial frame of reference and which is experiencing a gravitational field; is indistinguishable from a non-inertial system subject to accelerations, which is not experiencing such a gravitational field. This is known as the Principle of Equivalence. Our rotating disk above so experiences both inertial and non-inertial frames of references, so applying both SR and GR in the one dynamical system. In particular the radius vector must remain invariant, being normal to the central force, causing the centripetal acceleration. The tangent vector experiences Lorentz-Contraction relative to a Bird's Eye Observer (BEO), which remains stationary relative to the rotating frame of reference (FOR). Using SR and allowing an observer to rotate with the FOR must produce the invariance of 'c'. Now any 'observation point' for the corotating FOR must be somewhere on the disk, resolved in the radius vector and the tangent vector. We 'shrink' the radius vector using GR until we encounter the Weyl-Limit. The Riemann Tensor of GR's field equations consists of two parts; the Weyl-Tidal-Tensor and the Ricci-Density-Tensor with the g-metric. The Weyl-Tensor preserves volume, but deforms the volume, whilst the Ricci-Tensor reduces volume in accelerating the particles comprising the warping surface inwards. But our radius vector must remain invariant and when the limit for measurement is reached in the 'point-limit', then there can be no deformation by the Weyl-Tensor, because this would violate the invariance of the radius vector. Subsequently, the Weyl-Curvature must be 0 and the 'point limit' must be a dewarped singularity, which we now rename as the wormhole perimeter lps=rps/2p.rps as the metrically defined geodesic. As mensuration limit, no further Lorentz-Contraction below the wormhole perimeter is possible. This is a consequence of QR in the modular duality definitions for heterotic superstring HE(8x8), which we rename here as the Supermembrane EpsEss. 3. Quantum Relativity definitions for the Mach observer The Identity coupling the macrorealm to the microrealm is given in the inverse properties of SourceBoson Eps and SinkBoson Ess, the subscripts denoting ps=primary sourcesink and ss=secondary sinksource. QR defines the following:......Energy:..........E = moc2 = hf .......................................................................E=hf iff mo=0 .......................................................................E=mo iff f=fss=1/fps=lps/c All elementary particles/wavelets thus carry an intrinsic EigenFrequency or Lightequivalent fss=1/fps=tps, which is the TimeInstanton, the beginning of the universe as described by GR and the metricated scenarios of mensuration, observation and scalerelative measurements. This gives the inverse proportionality for lightspeed 'c' to the coupling between the vibratory Eps of high frequency/small radius source-part, modular dual to the winded Ess of low frequency/big radius sink-part. Gravitational Radiation is the modular inverse of Electromagnetic Radiation, differing in the scale-independent and algorithmically QR-derived supersonstant c2. A derived minimum/maximum coupling relates the frequency of the 11D-oscillating universe with nodally fixed Hubble Constant Ho=58.04 (km/Mpc.s) and maximum curvature radius Rmax=c/Ho to the wormhole in the constancy for c in: c = Ho.Rmax = fps.lps Also, the dimensionless TauTime(t) in GR, applied to the curvature Rc=c.dt/dt becomes the cycletime n in our descriptions as dn/dt=Ho, n=Ho.t. The wormhole perimeter is directly quantised in terms of the classical electron radius Re=e2/(4peo.mec2), which is proportional to the Compton Radius of quantum-atomic-structure in RCompton=Re/Alpha. Alpha is the electromagnetic finestructure given as Alpha=2pke2/hc=e2/2eohc and as Alpha=60pe2/h in QR, denoting the interaction probability between matter and light as about 1/137. (QR derives a superconductive form for the Action Law: Action = Charge2, electrical conductance quantum Go=2e2/h, the Klitzing constant RK=h/e2 for the Quantum Hall effect and the Josephson Constant in 2e/h). There are exactly 1010 lps/360 wormhole perimeters in one Re; relating to the finestructure of Planck's Constant h as: h= lps/[2Rec3]=lps/ce* in dimensionless Action-Units (relating to the monopole superstring class IIB). {The 360 count is not arbitrary, because the QR derivation for transcendental p engages trigonometric series which lead to the cipher 180 as a limit in Archimedean polygon approximations, also related to QR definitions for magnetic permeability and electric permittivity}. The nodal and 11D oscillating universe grows under agency of 'c' in electromagnetic parameters, encompassing the asymptotically expanding, but decelerating 10D hyperbolically curved 4D-volumed hypersphere as its inner surface mapping in say the 5D-Anti DeSitter hyperspace. The volume as 4D-information becomes the 3D-surface of the inner LineSpace mapping the 5D-surface of the outer HyperSpace. The volume is nested between the surfaces of two worlds, an ekpyrotic universe of LineSpace blending with the Hyperspace, encompassed in the cyclicity of the 10th dimension, reflecting off the M-Space of the Witten-Mirror. Thus the connector-time dimensions are 4-7-10-13=1 and the four worlds are defined in perfect supersymmetry. Maxwell's equations are introduced to the magnetic monopole(as selfdual superstring class IIB) and 12 magnetopolic junction points within a fourfolded propagation of the wormhole toroidal spacequantum define the Unified Field of Quantum Relativity (UFoQR). And yet, this fourfoldedness is just the one world of our Machian Observer. The singularity is no more; it has become the selfrelative BEO at the centre of rotation as the Machian Observer. It is irrelevant if the BEO rotates with the Perimeter FOR or is stationary relative to it. As Machian BEO, shehe is located literally at the centre of the universe, with everything else rotating or moving relatively to himher. The uniform motion of SR implies invariance of c, should the perimeter FOR rotate with the FOR. It is only at the now demetricated singularity, that this corotation becomes arbitrary. Demetricating the field equations of GR blends QuantumTheory with Classical Mechanics, culminating in General Relativity, itself emerged from Special Relativity and the physics of Newton and Galileo. The nullification of the Weyl-Tensor hence dewarps Minkowskian spacetime into the Schwarzschild solutions for the overall cosmic evolution, from the wormhole singularity as limit for the g-metric onto the Sarkar-architecture of the fractal universe in 236.5 million lightyears, then onwards towards the asymptotic boundary defined by the Spherical Standing Wave defined by the Hubble-Oscillation and c-invariance. Meanwhile so the oscillating cosmos has returned in its cyclicity to intersect the dimensions 2.2 Billion years ago, superposing its blueshift onto the cosmological redshift and giving the 'appearance' of an accelerating cosmic expansion, linked to a 'dip' in the Alpha-Electromagnetic Finestructure Constant to about 80 parts per million in a redshift interval from 1.08 to 1.84, mapped again around an Arpian present cycle value of 0.2505. But the finite cosmos is enfolded within the infinite one, creating space and time as it moves ever onwards under invariance of 'c'. 4. The Mach momentum for the universe Let us now calculate the Machian Momentum for the Universe and the intrinsic angular momentum for it as a Quantum Universe, a Protouniverse or monosong. The geometry can be modelled on a prolate ellipsoid with major axis invariance under rotation. The focal points would remain mathematical singularities as the 0-brane of Quantum Relativity. Any minor axis rotation would merge the focus points in a point circle and the 1-brane of QR would define an oblate ellipsoid of phase shifted multiverses; integrated as angular function of the complex OmniSpace as the Omniverse. As there is no gravitational field outside the enfolded protouniverse, uniform motion of the protoverse in inertial frames is not equivalent to accelerated motion in noninertial frames by the postulates of GR. Subsequently, rotating the protouniverse in the nongravitational undefined spacetime without does not cause centripetal forces balanced by centrifugal (coriolis) forces within the enfolded and metrically defined spacetimes. The Machian and Einsteinian postulates of 'Relativity of Inertia' hence becomes the BEO -frame within the protoverse as referring to the centre of rotation as the Machian observer, a reference previously known as 'relative to the fixed stars'. It is only in that FOR, that the BEO becomes independent from any other coexisting relative rotational systems in the measurement of the invariance of light, say as in the Sagnac Effect. The Total Momentum for the Universe is given in:..................(2/3)M.Rmax2.w = N.h/4p We consider the Universe's mass M to be the surface mapping of 10D onto the inside of the 11D-manifold and use the moment of inertia of a spherical shell. (This is the 2-brane/2-sphere of LineSpace). We incorporate all fermionic particles, including the (anti)neutrinos, whose number is given in the baryon/photon ratio (Eta h) and a number which increases N by a factor of about 109. QR calculates, for a present cycletime np=1.1324.., the following parameters: Deceleration Parameter qo=0.014015 for Sarkar-Architecture at cosmological redshift z=7.477 or 236.5 Mly. Critical Density rc=3Ho2/8pG=Mmax/Vmax=Rmax.c2/2GVmax ..........................rc=Ho2/4p2G for 11D-curvature) Mo=1.81371262x1051 kg* M∞=6.47061227x1052 kg* Rmax=1.59767545x1026 m*; R10D(n)=Rmax(n/(n+1)) m; R11D(n)=n.Rmax m* No=1.8275x1078 N∞=6.5197x1079 h10D=3.64x10-10 h11D=4.82x10-11 w= 2pf = 2p/T = 3Noh(np+1)2/{8ph10DMo(np.Rmax)2} = 3.07x10-50 1/s for the asymptotic 10D-mass parametric universe for Period T=16p2h10DMo(np.Rmax)2/3Noh(np+1)2=2.05x1050 s or 6.49x1042 Years w = 2pf = 2p/T = 3N∞h/{8ph11DM∞(npRmax)2} = 5.08x10-50 1/s for the oscillating 11D-electromagnetic universe for Period T=16p2h11DM∞(npRmax)2/3N∞h = 1.24x1050 s or 3.91x1042 Years The QR definition for time sets the natural limit for this in the minimum/maximum ratio of lps/Rmax=nps. The Timeinstanton tps=fss=3.333..x10-31 s thus relates nps=1/lss.Rmax=6.26x10-49; mapping the 'real' time tps onto the 'imaginary' time Ho.nps = Ho.lps/Rmax = Ho2.tps = 1.176x10-66 1/s. This analysis has lead us into the realms of the poets and dreamers. Leonardo de Vinci's 'Vitruvian Man' has literally become the Machian Observer, encompassing the entire universe as 'Cosmic Man', the Adam Kadmon of Lurianic-Hebrew Kabbalah or the Purusha of Hindu cosmology. Molecules are mapped as star systems from the 'inner space' and biovital cells become the galaxies of 'outer space'. Every sentient being is mapped holographically as a neuron in all other 'immortalised' non-telomere depleted 'neuronal brains'. The monosongs become tunes, melodies and a symphony of the supermembranes as particle/wavelets of the UFoQR.
