Bidimensional interferential patterns, in a representation with the Time oriented along the ordinate axe.  Interference is the name of the intersection point of distinct branches


Objects and measurements’ hidden nature 

A few definite steps allow to fully recognise the nature of the Events and the action and function of the devices part of the industrial machinery and of the equipments to count and relate them (e.g., Triggers, microprocessors, PLCs, etc.).   As an initial step, it is necessary to understand that the 4D space-time described by Minkowski and Einstein between 1905 and 1907, still presented as a basic concept to the future Electronics or Electrotechnics Engineers, is no more since long time witheld a theory representing the reality outcoming by the of the experiments.     


 In 1915 Einstein extended Relativity Theory to include also some of the ideas underlined by the figure on side. To every point of a curved differentiable manifold M, can be attached a tangent space, a vector space containing all possible directions along which one can pass through the point. If the parameter t identifying the spatial leaf (or, sheet) is such as to make the tangent vector unitary, then its physical meaning is that of the proper Time t. Reader may figures out the basic meaning of “differentiable manifold” recalling the 2D surface of the Earth or the waves formed in a liquid surface when perturbed by one point source. After 1915 different 3D leaves of the foliation may have different or equal times and no one of the leaves is less real than another.  A multiple-histories conception  



Time-identified Events


A key point regards Time. Time has been in the past considered to be the fundamental phenomena underlying Dynamics. Until 1907 the 3D spatial environment was considered stratified by the universally valid time coordinate t.  

  ”To every point of a curved differentiable manifold M, can be attached a tangent space, a vector space containing all possible directions along which one can pass through the point.   The parameter t identifying the spatial leaf (or, sheet) is such as to make the tangent vector unitary.  Its physical meaning is that of the proper Time t” (  csdn.net/2014) 



Reeb foliation. The visible “leaves”, also named “sheets”, are hypersurfaces whose dimension is 3.   Each Observer, Detector or industrial Machinery occupies a multitude of 3-dimensional leaves of the foliation.  Then, each Detector collects changes in the status of a multitude of detectors densely existing all around.  Each of them sensitive only along an instant of Time (image credit Tambara Institute of Mathematical Sciences, University of Tokio, 2014)

The relativistic point of view about Time and Events conceived in 1907 a 3D flat space labelled by mean of the Time parameter t.    Time continued to allow to order sequentially all the physical Events, as an example, the measures of a Physical property like the spin or its collective counterpart, polarisation. Events identified with 3+1 coordinates.  Still a single-history concept, where a bulk of flat 3D spatial hypersurfaces is crossed by one perpendicular time-like straight line and definite corrections have to be applied to times and distances to account for the relative speedths of the moving bodies.  Single-history concept of 1908 by the laymen still today imagined the environment we are inhabiting.   



 Reeb foliation. The visible “leaves”, also named “sheets”, are 3D hypersurfaces.  Each Observer, Detector or industrial machinery occupies a multitude of 3D leaves of the 4D foliation.  Because of this reason, each Detector (e.g., a pixel in a CMOS-camera in a final inspection system) collects changes in the status of a multitude of detectors densely existing all around. Every one of them sensitive only along an instant of Time (  Tambara Institute of Mathematical Sciences, University of Tokio, 2014)




The ambigous identification of the Events by their timing






In 1915, Einstein in its General Relativity revised deeply Minkowski's idea of 1907, applying new principles (general covariance and equivalence) over topological spaces.   Since then, Time has no more the unambigously defined meaning of Event-identifier it had before. The topological spaces to whom we are referring are the Manifolds, resembling near each one point the known Euclidean space. The neighbourhood of each point of an N-dimensional manifold is homeomorphic to the Euclidean space of dimension N. The simplest 1D manifolds are the curves and the circles, and the 2D manifolds are named surfaces (e.g., plane, sphere). Relativity conceives we, our devices and Machinery be living in a higher dimensional differentiable Manifold M.    


 3-dimensional interferential patterns on a surface, from one point source: an aid to deepen the concept of wave








Readers may figure out what is meant as “differentiable Manifold” recalling the 2D surface of the Earth or the waves formed in a water surface by one point source, like the liquid manifolds before and below.  To grab a more modern understanding of what today is meant as a “measure”, including the containers’ detections of the tracking Triggers, we suggest to focus the multi-branched tree-like structures naturally arising in the figures below.  In this example, a single point source perturbed the state initially flat of a liquid surface. Several deriving outcomes are the effect of that disturbance:

  • all of them equally real also when their measures are visibly different;
  • after a perturbation, they are never visible completely identical outcomes; 
  • comparing different perturbations, they are never visible completely identical outcomes.  
3D interferential patterns hint the relation existing between Waves, Manifolds and Triggers. Going deeper to the rationale of all Triggers, these are the devices to label with different Information different leaves of the foliation M. Thus, allowing us to eliminate an ambiguity whose nature is cleared by General Relativity.  Here, to every point of a differentiable manifold can be attached a tangent space, say a vector space containing all possible  directions along which one can pass through the point. The ambiguity derives by the fact that an infinity of time-like curves γu with tangent vector, can be identified as world line of an object.  If the parameter t on γ is such as to make the tangent vector unitary, then its physical meaning is that of the proper time t of an object.  To let the Readers perceive the physical meaning of this, consider the example the ramp of an airport, itself really a portion of a differentiable manifold.  Infinite trajectories are allowed to planes for landing or take-off. Some of them however more probable than others.  What are the trajectory and kinematic of the plane? …all of them are possible, but only some of them are persistent constructive superpositions. Trying to figure out the kinematical properties of a container on a Conveyor or into a rotary Machine, the pulsed Radar spotting a single plane at time intervals, is replaced by the triad Clock-Encoder-Trigger.  Thus making it possible to infer the containers’ kinematics in all Electronic Inspectors and in the Rotary Machines like Fillers, Labellers or Blow-formers.

3D interferential patterns hint the relation existing between Waves, Manifolds and Triggers. Going deeper to the rationale of all Triggers, these are the devices to label with different Information different leaves of the foliation MThus, allowing us to eliminate an ambiguity whose nature is cleared by General Relativity.  Here, to every point of a differentiable manifold can be attached a tangent space, say a vector space containing all possible  directions along which one can pass through the point. The ambiguity derives by the fact that an infinity of time-like curves γu with tangent vector, can be identified as world line of an object.  If the parameter t on γ is such as to make the tangent vector unitary, then its physical meaning is that of the proper time t of an object.  To let the Readers perceive the physical meaning of this, consider the example the ramp of an airport, itself really a portion of a differentiable manifold.  Infinite trajectories are allowed to planes for landing or take-off. Some of them however more probable than others.  What are the trajectory and kinematic of the plane? …all of them are possible, but only some of them are persistent constructive superpositions. Trying to figure out the kinematical properties of a container on a Conveyor or into a rotary Machine, the pulsed Radar spotting a single plane at time intervals, is replaced by the triad Clock-Encoder-Trigger.  Thus making it possible to infer the containers’ kinematics in all Electronic Inspectors and in the Rotary Machines like Fillers, Labellers or Blowformers






And, what about the Time indicated by clocks lying in different outcoming branches of the foliation?  After 1915 different leaves of the foliation M may have different or equal Times and no one of them is less real than another.  Space dimensionality is ≥4, however superior to the three dimensions we are biologically equipped to perceive and mathematically conceive.  Einstein’s equations of General Relativity are invariant under reparametrization of their time coordinate.  In brief, there is no longer any time.


Time 

A physically meaningless arbitrary parameter




“The time coordinate t, characterizing a foliation, is in the modern General Relativity theory reduced to just one of the four arbitrary and physically meaningless spacetime coordinates”




Since decades the symbol M conventionally identifying a manifold, has been replaced by another named S.  S derives by the Latin language prefix of Superspace and a still more actual denomination makes frequently use of a Greek language prefix hyperspace.  The time coordinate t characterizing a foliation, is in the modern General Relativity theory reduced to just one of the four arbitrary and physically meaningless spacetime coordinates.  On the date when these notes have been compiled, a promising version of the hyperspace S  is considered to have at least 5 dimensions, when some other versions of the theories adopt 26 dimensions. With reference to the example in the figure below, the hyperspace S is the manifold each of whose points A, B, C, …represents all of the infinite points in a 3D space:  

  • smooth lines numbered 19, 20, 21, 22 are wave crests of probability amplitude function ΨE (x, t), where E is the Energy, x a spatial coordinate and t the time parameter; 
  • dashed lines, same as before for an energy E + ΔE;
  • S(G) and S’(G) are the Hamilton-Jacobi dynamical phases due to different waves.  Their elliptic intersection points, magnified by the lens, regions of constructive interference.    


 Imagine to have a lens powerful enough to magnify details <10-30 cm.   In this realm, it becomes possible to see the true nature of the Space.  Hyperspace S (also named, superspace) is the manifold each of whose points A, B, C, … represents the infinite amount of points existing in a 3D space.  Smooth lines numbered 19, 20, 21, 22  are wave crests of probability amplitude function Ψ(x, ), where E is the Energy, x is a spatial coordinate and t is the time parameter.   Dashed lines, the same for an energy E + ΔE.   S(G) and S’(G) are the Hamilton-Jacobi dynamical phases due to different waves, their elliptic intersection points being regions of constructive interference, what we normally name particles.  The submanifold H of S is the classical history experienced by the geometry of that 3D space, when space has been started off under some particular set of dynamical initial conditions (  Wheeler, J. A., et al./1973)















 

What the figure before is showing is a dynamical geometric object (geometro-dynamics is a synonimous of general relativistic), describing the dynamics of the 3D spatial geometry.   Namely, a propagation of the intrinsic curvature on space-like hypersurfaces with respect to a time coordinate t that labels a foliation of the spacetime. Spacetime arising dynamically in this way by the succession of leaves. 



3-Geometry contains physical Time

The extrinsic curvature describing the embedding of the 3D geometries into 4D spacetime, is represented by the corresponding momenta. Therefore, the three dimensional-geometry, representing the dynamical state of general relativity, is the carrier of information about physical time.  It contains physical time and not depend any more on it. Below, four examples of the non-trivial consequences of the new point of view: 

  1. all time-like distances from an ‘initial’ place;   
  2. given yesterday’s geometry, today’s geometry could not be the one of tomorrow’s; 
  3. a mechanical clock is allowed to go ‘wrong’.  Then, considering the example of a planet, one would have to know the initial angle and the initial rotation velocity in order to derive time from motion;
  4. a speed of ‘speed of time’ is a tautology.

Waves and manifolds are commonly associated. This also at the macroscopic scale of our industrial Machinery, equipments, inspection and detection devices. A full understanding of this last decisive concept is visually aided by the figure here at left side.   Here, two gaussian wave pulses with a minimal amplitude difference, are travelling on a nondispersive string in opposite directions.   Visibly, they pass through each other and superimpose without any reciprocal disturbance.   


  Constructive interference showing the far reaching implications of the Principle of Superposition.  Two gaussian wave pulses with a minimal amplitude difference, travelling in opposite directions on a nondispersive string.  Visibly, they pass through each other without being disturbed.  The net displacement is the sum of the two individual displacements  














The net displacement is the sum of the two individual displacements. Now, remember that a measurement instrument, whatever its technology, features a defined sensitivity, the stimulus' minimum amount, whose existence it can detect. As an example, set the sensitivity be equal to what in the graphics on side has the value 1, an we’ll have the net result of:

  • what dot-like objects shall be observed (measured), each one of them associated to an Information (named Signal):
  • where the superposition reaches amplitudes >1;
  • when the gating system of the measurement instrument (Trigger, detector or a complete electronic inspector) lies in one of those time intervals (the inspection window) allowing further processing.

The submanifold H of S is the classical history experienced by the geometry of that 3-dimensional space positive definite, when space has been started off under some particular set of dynamical initial conditions. An entire 3D space, can be imagined as something half-way between a true mathematical point and what a point represents in the hyperspace S.  In particular, a temporary configuration of a: 

  • particle, ...is an Event, a single point in the space-time;  
  • space,   ...is a 3-dimensional geometry, a single point in the hyperspace S.   


“An oscillating drop of water undergoes  fission.  The topology changes.  A point marks the place of separation of the two masses of liquid. That point lacks the full neighborhood of points that characterizes a normal point. Such a critical point is ruled out from any proper manifold by the very definition of the term manifold in mathematics.  Before the division, the surface of the drop constituted a manifold.  After the division, it is again a manifold, consisting of two disparate  pieces.  At the instant of division it is not a manifold.  But little attention does the drop pay to this distinction. It divides, despite all definitions.  No more reason does one see in the definition of manifold against space changing its topology

John Archibald Wheeler, 1967


ja wheeler with tullio regg












 John Archibald Wheeler at Princeton, 1971.  The one with the bald spot in the front row is the relativist Tullio Regge (   R. Matthews/1976)


Signalling, in the detectors and equipments used in the Food and Meverage Machinery is frequently electromagnetic. The most modern design address is toward the transition to fiber optic signalling between all equipments.  Another widespread classic idea needing to be updated to the experimental reality known in the Third Millennium, regards the signalling.  

   3-dimensional interferential patterns on a limited surface, from a point source






The common view assumes that signalling is happening in a common 3D space: transmitter and receiver should be exchanging informations carried by energy moving through the 3D space the laymen think to be inhabiting.   But, an arbitrary metric space A is defined to be positive definite if each of its finite subsets is positive definite with their induced metrics.  3-dimensional spaces have positive definite metric, what implies that physical effects cannot propagate from one point of the 3D geometry to another.       





 The space-time manifold M as seen by the roditor perspective, looks flat and simple. The opposite than the topologies suggested, after comparison of theory and experiments. Then, the obvious question: “Why do we see a different (flat) reality?”  An answer is that we are not biologically equipped, in terms of amount of sensors and neurons, to perceive the fine details corresponding to extremely small time intervals. And, also if we’d be, then we’d not have enough memory where to record such an amount of informations





fine structure med hr

Why?  Because a physical quantity local to a point, and another local to another point, have no reciprocal coupling: they commute. The meaning of the sentence above in bold characters has far reaching consequences. It implies that also the appearances of material objects, a macroscopic case of the measurements (and, triggerings) of the Electronic Inspectors and in the industrial Machinery, are exclusively happening between different 3-geometries where we, our electronic inspection devices and Machinery are existing. The first clear idea of this by Minkowski, dated 1908, was valid only for the flat, ideal, non existing Euclidean spaces. In 1967, these concepts were carried further by J. A. Wheeler, then the greatest living Relativist.  He was capable to establish a firm bridge between the two most successful physical theories, General Relativity and Quantum Mechanics. 


 The YE-1 Compact Muon Solenoid, part of the Large Hadron Collider (LHC), is a complex assemble of measurement instruments. The entire measurement process backed by a supercomputer, equipped with an impressive memory to record the huge flow of data following each collision event (  CERN/ 2006)



Topology-Spinors connection






In other sections and until now, we spoke about the spacetime, looking at its topological properties.  That is not all of the story and there is an additional property to consider: the spin.  There is a criteria to identify a topologic variety as a candidate to be the master plan of the physical space where they happen binary classifications based on measurements.  The same effects we observe on macroscopic objects or that we can logically infer, have to be autonomously rendered by that topology.   Imagine a cube.   Attached, like in the figure below, to four of its corners four elasting strings. 




  Imagine a cube and four elasting strings attached to its corners.  Select any axis running through the center of the cube and rotate the figure about that axis through 360º.  The cube shall return to the original configuration, but not the strings.  Strings shall be entangled and no way to untangle them.  A 360º rotation alters the orientation entanglement relation between the cube and its surroundings (abridged by   J. A. Wheeler et al./1973)

















   When two surfaces are joint by different wormholes, full justice is made for the use of the mathematical term multiply connected space. The example above shows what they can be:  1) two multiply connected leaves;  2) a single multiply connected leaf joint in an area not visible in the image (   S. Hossenfelder/2014)

Now, select any axis running through the center of the cube and rotate the figure about that axis through 360º.    You’ll see the cube returning to the original configuration, but not the strings.   Strings shall be entangled and no way to untangle them.  The 360º rotation alters the orientation entanglement relation between the cube and its surroundings.   Two successive rotations by 360º restore the original orientation of the cube and the strings can now be untangled.    The spinor is a particular mathematical operator accounting for the effects of these rotations with respect to the origin of a triad of coordinate axis.    Until now we spoke only of the rotation of a cube and its surroundings.    Now, we’ll try to figure out what shall happen when the cube is slowly rotated through 360º when passing along the entire space of other geometric surroundings.   We’ll discover that under the same identical rotation of 360º different 3-dimensional closed manifolds can be fully classified by total four properties:

  1. topology;
  2. differential structure;
  3. metric;
  4. spin structure.

The figure below, next section, represents different kinds of 3-dimensional points.  All of them are closed 3-dimensional geometries but six of them, labelled as:


        D+        D-        E++       E-+       E+-       E--    


present the handles which let them be identified as wormholes.   In a multiply connected space they have to be expected behaviours looking strange as seen from our many-3-geometries-at-once point of view. Imagine two identically coloured pink cubes.   Also, oriented the same way with respect to a single triad.   Let one of them enter the wormhole by one of its throats.    Let the other cube move along the flat surface out of the handle, from the same throat where the other get in, til the other throat.   You’ll discover they'll get rejoin themselves differently oriented with respect to the triad used as a reference when they get in!   

This kind of geometric structures are an unavoidable feature of the space.   The Readers may immediately consider the risk to introduce Closed Timelike Curves (CTCs) in the structure of the space.   These risks really exist applying the General Relativity theory without the conditioning of the modern version of the Principle of Superposition, of quantomechanical origin, object of deeper analysis in other sections.    As a matter of fact, CTCs really are constantly created, but on timescales extremely small (~10-43 s), giving effects completely uneligible in our macroscopic measurements.



Triggers and Time

Simplest instance of electronic inspection measurement devices: 

a way to differentiate leaves otherwise similar

“Triggers are a way to differentiate leaves otherwise undistinguishable”


The synthetic and technological meaning of what preceded is that General Relativity, a theory made of worldpoints named Events, constitutes a device to calculate and constructing a leaf of history that slices through the higher dimensional space S whose  dimensionality is ≥4.   What above has relation with the purpose of the Triggers adopted in the industrial automated Quality Control (electronic inspection) and in the Machinery.   Knowingly, the purpose is to acquire Information about the phase, state, dynamic or kinematic conditions of an object.   





“...Detectors occupy a multitude of 3-dimensional leaves along a single constructive interference crest”




 4D multiply connected manifold. The foliated character of space is here distinctly visible, jointly with those topologic peculiarities whose crest values in the multitude of constructive interfences, we name Events or Measurements.  Each one of the differently coloured volumes visible is 4D.  Each one black dot is a 3D volume: each one of the points A, B, C,… contains the infinite amount of points existing in a 3D space.  The smooth lines numbered 19, 20, 21, 22 are wave crests of probability amplitude function Ψ(x, t), where E is the Energy, x a spatial coordinate and t the time parameter.   Dashed lines, the same for an energy E + ΔE.    S(G) and S’(G) are the Hamilton-Jacobi dynamical phases due to different waves, their elliptic intersection points being regions of constructive interference, what we name particles.  In the year 2000, it has been discovered (Deutsch, 2000) that the information flow at the quantum level follows the basic rules of the flow of information between different 3-geometries in a multiply connected 4D manifold like the one here depicted (abridged by  Wheeler, J. A., et al./1973) 


That a triad:

  1. Clock,
  2. Encoder,
  3. Trigger,

is the necessary and sufficient precondition to let a microprocessor have information about the presence of objects and their kinematic, is a basic notion.   But what is the rationale for that necessity to “acquire Information” ?   The reason is that the Triggers, are a way to differentiate leaves otherwise undistinguishable.   There is always at least one Trigger introducing objects into the Shifting-Registers.  Shifting-Registers clocked (Time-related) and counting the Encoder pulses. 

 Portion of the multiply connected manifold here above, limited to the leaves A, B, C, D+ and D_ in a representation privileging the time evolution.  ΨE (x, t) is a probability amplitude function, E is the Energy and x a spatial coordinate



“...measurements (and, triggerings) of the Electronic Inspectors, are exclusively happening between different 3-geometries where we, our devices and Machinery, exist”

The figure above shows portion of the multiply connected manifold seen in the figure  immediately before.   It is a bi-dimensional representation limited to the leaves A, B, C, D+ and D_ in a representation privileging Time evolution.   Here, ΨE (x, t) is a probability amplitude function, E is the Energy and x a spatial coordinate.   What it is showing is a different way to visualise coexistence and separate evolution of several different 3-dimensional spaces.    Looks too complex ?   Not that much, when thinking that Nature adopts similar designs also at our macroscopic scales.    Below, an example of botanical branchings.    The cuts reveal the inner foliated structure:  a geometric dynamical reality referred to a foliated 3D space which, having existence extended along the Time-dimension, is a 4D space foliated with 3D spaces.



    Physics means “Nature”, and the foliations to whom General Relativity refers are natural.  Nature organizes also its botanical structures foliating the fiber bundles to occupy the available space


   The annual rings of the trees are one of the many ways a physical foliation may present itself



But, in 1967 it was first suggested that the space-time of 1915 does not exist at all.   On the opposite, it is just one of the ways we perceive the fundamental Quantum Field.   The idea received contributions by many, some of them nobelists.   Between them Paul Dirac, Higgs, Bergmann, Pirani, Arnowitt, Deser, Bryce DeWitt, Charles Misner and John Archibald Wheeler.    



Time Today




“We, and our Detectors, occupy a multitude of 3D leaves along a single constructive interference crest.   

And that’s why we feel the effect of Time named Dynamics


It is known that the nature of Time and its very existence are since thousands of years object of innumerable philosophical discussions.   Yet 1-2 centuries ago mathematicians and physicists as Leibniz and Ernst Mach debated against the then dominating idea of absolute Time.   Until recently, the evidences against the idea of Time as a fundamental of Physics, were only philosophical or logic.   In 2013 the concept of Time had been experimentally proved derived, rather than fundamental.   

Time reduced to just an apparent effect of that particular kind of relation named Entanglement of a multitude of individual measurements happenings between different 3D spaces, foliated to form a 4D.    We, and our Detectors, occupy a multitude of these 3D leaves along a single constructive interference crest.   And that’s why we feel the effect of Time named Dynamics.   In this way, spatial geometry itself becomes a physical clock, and the program of Leibniz and Mach may finally be fully taken into account by completely eliminating any relic of absolute time. 


Links to the pages:


                                                                                                            Copyright Graphene Limited 2013-2019