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History and meaning of the most important idea

The wide spectrum of answers to the apparently banal question: “what is the difference between Cause and Effect ?” shows each one of us has a different idea in correspondence to a single concept.   However, whoever agrees the connection between Events is an extremely delicate and important subject.   Its direct applications in the scientific research, as well as the technological Problem Solving and Root Cause Analysis, being known examples.  

A point of view dated more than one century ago about causal connection or causal relation, hints to the validity of two conjectures thousands of years ago elevated to the rank of principles:

1. time-ordering of the Events;
2. someone or something here acts over someone or something there.

These ideas, however ancient they may be, are also the base for a plethora of non-scientific applications, easily recognized because of their capability to show the (apparent) veridicity of all and, simultaneously, of its opposite.   We’ll synthesise the point before from its original merely logic-phylosophical background, and later deeply in the most modern mathematical-physical.

Root Cause and Effect: Classic view

What are the “fundamental ingredients” of Cause and Effect ?  

The reasonable classic answer is:

• energy,
• space,
• time. 

It is impossible to find a general definition for Energy. 

It is however observed that Energy:

• is conserved,
• exists in different forms,
• can be stored,
• can be transferred through space
• can be transferred from one material body to another,
• can be transformed into other forms of energy.

A discussion of the problem of causality is dependent not only on an interpretation of experimental facts and theoretical conceptions of modern physics, but also on a comprehension of the category of causality itself.   Causality is interpreted in different ways, and this gives rise to misunderstanding and difficulties in any discussion of the  problem of causality, which at times disintegrates into arguments about words.   It is therefore deemed highly desirable, before going into a discussion of the matter, to clarify the concept of causality and elucidate the relationships between its different meanings.  

In the majority of the technological cases, causality is used in the sense of a relationship between cause and effect, in which the cause is the active agent which generates its effect.  

This is the definition of causality ordinarily encountered in the:

• technological literature (example: Root Cause Analysis);
• forensic literature (example: who or what caused what);
• philosophical literature;
• natural-science literature (example: the causes of phenomena).    

In synthesis, with reference to the representation below, Causes precede Effects.  

                                                                                            Agent

         Cause                   Effect

The explanations cite Causes, not Effects.  The Agents use Causes to manipulate (or, shape) their Effects; they do not use Effects to manipulate their Causes.  The Effects of a common Cause are correlated; Causes of a common Effect are not.  An ensemble of unidirectional rules underlying the existence of the named arrow of Time, since 1970 synonimous of the physical process named Decoherence.   

Classical determinism

In certain cases the concept of causality is  identified with the necessary relationship of  states of a system, when the initial state of the system of necessity determines its state  at any subsequent instant of time.  By the state of a system at any time is meant the totality of all the properties that the system is endowed with at that time.  

This kind of causality is ordinarily termed classical determinism.

Laplace determinism

In Theoretical Physics causality is frequently identified with the fundamental possibility of absolutely precise predictions of a future state of a system that uniquely defines its behaviour if the state of the system is known at some given instant of time.

This kind of causality is sometimes termed Laplace determinism. 

Mathematical determinism

The concept of causality is also used in the meaning of mathematical determinism. This occurs when the differential equation expressing the behaviour of a physical system in mathematical form has a unique solution under specified initial and boundary conditions. 

In the Quantum Field Theory (QFT), the concept of causality is identified with the  assertion that physical actions cannot be propagated in space-time with a velocity exceeding that of light.  In accordance with the Lorentz transformations, this ensures invariance of the temporal relationship of cause and effect.

Cause and effect

In elucidating the relationships between the various meanings of the term “causality” it is of fundamental methodological importance to recognise the objectivity of the relationships and interactions of phenomena in the surrounding world, which  relationships and interactions are reflected in the man-made concepts of causality,  regularity, necessity and chance, possibility and  reality, and so forth.

On the basis of our activities, we are convinced that there exist relationships of phenomena such that one phenomenon gives rise to (or, generates) another. Indeed, if we are capable of generating lightning by creating the phenomena under which it occurs in nature, if we are able to regulate the intensity and direction of a discharge of  atmospheric electricity by altering the appropriate conditions, and, finally, if we are able, in the laboratory, to produce “artificial” lightning, we have then demonstrated that there is a connection in nature itself between the given phenomena and lightning.  Certain phenomena generate, give rise to other phenomena. That phenomenon which generates another is called the cause, the phenomenon generated by the cause is termed the effect.

Thus, the concept of cause reflects an active agent, the concept of effect reflects a result generated by the active agent.

The activity of the cause is ordinarily expressed by the verbs “generate”, “give rise to”,  “produce”, “alter”, “operate”, and so forth.  In the scientific literature and in colloquial  speech, the dynamic character of the cause is frequently denoted by means of such  words as “motive  force”,  “impulse”, “source of motion”, etc.

The notion of “effect” is often expressed by such words as:

• “caused”,  
• “generated”,  
• “having as its cause”, 
• etc. 

This definition ignores the most essential feature of a causal relation, the activity of the cause.  The effect not only necessarily accompanies the cause but is also generated by the cause. 

Conditioned effect of a cause

This definition is  exceedingly broad, it satisfies more than a causal relation. The effect  generated by a cause depends on the conditions.

The conditions are the totality of all the  dynamical relations of  the given body with other  bodies, with the exception of its causal relation.  A bottle in free fall to the surface of the earth experiences the gravitational action of the earth and other celestial bodies; it is also acted upon by the air, etc.   The gravitational interaction with the earth which causes a bottle to fall is the cause of this phenomenon.  The remaining dynamical factors operating on the bottle make up the conditions of the action of the cause.

Under certain conditions, a given Cause gives rise to an Effect, under other conditions, the same Cause gives rise to phenomenon Effect'.  If the conditions are varied, the effects generated by the cause will also vary.  

As an example, we may imagine that:

1. we are blowing a serie of pre-heated PET preforms by one and the same mould;
2. all preforms entering taht unique mould are identical;
3. the blowing sequence is automatic and enacted by mean of identical timings and air pressures;
4. following each blowing, the mould is returned to the original state.  

Though the cause, the pressured air blowed into pre-heated preforms, is the same, we are convinced after a number of blowed bottles that the outcoming bottles are not identical.  The eyes of whoever or an Electronic Inspection device confirm differences in several of their geometric, mechanical and optical properties.  This is due to the changed conditions under which the each originally supposed identical preform is blowed.  As an example, changes on the ambient temperature when blowing the sequence. 

The same cause operating on the same object generates, under different conditions, different effects.

Causes and conditions' relative character 

It is readily seen that the distinction between condition and cause is not absolute but  relative in character.   In a definite relationship, every condition is a cause, and in another relationship, every cause is a condition.  For instance, the action of the wind causes a flying dart to be deflected in its parabolic flight and is, in this respect, a cause for its divergence when targeting a precise point.  Then all the other relations, including the action of the hand on the dart at the time of the launch, will be conditions of this cause.

Difference between Cause and Condition

The cause is a dynamic  factor generating a given phenomenon.  But the conditions, though they affect the behaviour of a thing,  do not  generate the  given change brought about by  the cause.  A condition alters the behaviour of a thing in a different respect  than the cause, but  this alteration  influences the effect brought about by the given  cause.  For example, the elastic stresses produced in ther stainless steel tank of a Filler Machine during CIP-phases with hot water, are essentially dependent on temperature conditions, the variation of which alters the physical properties of the steel, and this in the final analysis affects the distribution of internal stresses in the tank.

One might say that any cause is, in a specific respect, a condition, and any condition, in  another respect, is a cause.  However, in a fixed relationship, the distinction between cause and condition is definite.  The cause is a relatively active factor generating a given effect.  The condition is a relatively passive factor which affects the result brought about by the cause but which is not the cause of the effect.

Cause as:  Sum over Conditions 

But it is a fact that some interpretations identify the cause with the  totality of conditions  necessary and sufficient  for a given phenomenon to take  place.  Cause reduced to sum of the conditions positive and negative taken together.   A definition ignoring the  difference between the relatively active and the relatively passive relationships.  Every  phenomenon is regarded as the consequence of an enormous number of other phenomena of diversified significance.  Thus, we would then be compelled to include in  the concept of the cause of a falling bottle not only the interaction of the bottle and the earth, but also its interaction with all other celestial bodies, including even the absence of any kind of relationships impeding the fall.   We then have to include in the composition of the cause other conditions as well: the specific weight of the air must be less than the specific weight of the bottle, the gravitational force between the body and the earth greater than the gravitational force acting between this bottle and the sun, and so on.  A rigorously scientific method of exposition knows no “causes” but only law-governed relationships….   

This point of view originated after the breakthrough of the physics nobelist Richard P. Feynman, dated 1946.   It logically introduces the actual multiversal perspective, where “causes” and “effects” do not have a factual existence, rather mere sensations arising by the arrow of Time.  All possible constructive and destructive superpositions constantly happening without any causative relation, other than that of a certain outcome of the sum of a superposition of many terms.    Only by us named “causes” and “effects”.  Thus relegating to the role of “noise” all those many more “effects” that we are not capable to relate. 

A law-governed process or state is never determined with complete precision by “one exceptional cause” but always by a sum of conditions, all of which are equivalent, for  they are equally necessary.   In this perspective, it is a field, named quantum field, what plays the centric role.  This today mainstream idea, argues that a global wave function is sufficient to describe the entire reality.   Thus, not meaning that the traditional way to connect the quantum formalism with measurement outcomes is wrong, but rather that its pragmatic use of “complementary” or “uncertain” classical concepts can be avoided and instead justified in a consistent way in purely wave mechanical terms.  Here different outcomes (“Effects”) arise by a precedent, time-arrow oriented, state of the system.  State lying in our memory and having been recorded by our instruments and detectors, which we name “Cause” simply because part of the entire set of conditions which let the present state (“Effects”) exist.

Classic view: same causes carry out same effects 

Practical experience points to the fact that certain conditions generate phenomena,  while other conditions do not possess this property.  Likewise, it is incorrect to assert  that  the principle of causality invariably requires acceptance of the fact that every phenomenon has a unique cause.  For instance, a change in the volume of a bottle may be due  simultaneously to a variety of  thermal and mechanical actions. These various factors can act on a  body in one direction, reinforcing one another, or in different directions, diminishing the resultant effect, or, finally,  they can can eel out entirely  producing no resultant effect whatsoever.

This classic point of view was openly advocated by the French physicist Jean Bernoulli, when writing: “Nor would the same fruits be constant to the same trees, but would be  changed; and all trees might bear all kinds of  fruit.”  It is important to understand that Bernouilli is writing what appeared true, what get out of repeated experiments, using instrumentation available over two hundred sixty years ago.

Indeed, if identical pieces of metal, when heated, behaved differently, expanding, contracting, melting and so forth without  rhyme or reason, it would be impossible to  predict the behaviour of metal under altered temperatures or make use of it in any way. If  the same set of conditions operated in different ways on identical organisms, stimulating  vital processes or inhibiting them or even killing them outright, no living thing could exist, for it would be encountering unforeseeable and mortally dangerous events at every hand.

Practical human activities and the purposeful actions of human beings using the  instruments of production are possible only insofar as identical conditions give rise to identical effects. All of modern natural science, at any rate that engaged  with macroentities, essentially rests  on the view that under the same  circumstances, identical causes give rise to identical effects.  

In the realm of classical mechanics, identical forces acting on bodies of the  same mass generate identical  accelerations; in the theory of elasticity, the same external actions  affecting the same objects  give  rise to identical deformations; in the field of classical electrodynamics, identical current sources and charges placed in identical media generate electromagnetic fields of the same intensity, etc.  

What above may be resumed in the observation of another great French physicist and mathematician, Henri Poincare’ (see figure above), over one century ago considered “The Living Brain of the Rational Sciences” of the entire World.  He famously claimed that: “if two organisms are identical or simply similar, this similarity could not have occurred by accident, and we can assert that they lived under the same conditions”.

Classic view: motion and change create different conditions 

Of course, the idea of an absolute  identity of conditions is an abstraction.  In nature we do not find two identical leaves from a single tree, and if they were identical,  then they  would be one leaf.  Too, there are no two objects in nature which would be in absolutely identical conditions.  What is more, one and the same object cannot be twice in identical conditions, for the conditions of every object are the actions of other objects, which, like the given one, are in a state of motion and change.

If we take into account the absence in the actual world of even two absolutely identical  phenomena, then the necessary character of causal relations should be understood as an expression of the fact that the fewer the differences between the causes and conditions,  the fewer will be the differences between the effects produced by them.  In the limiting  cases where the  causes and  conditions are  identical, the effects will also be identical.  

From  the  necessary  nature  of  the  relationship  of  the cause and its effect there  follows the conclusion that  if definitely identical causes  give rise to different effects, then  they are operating under different  circumstances.  

If causes operating under the same  circumstances generate different effects, then the acting causes are different.

Classic view:  Phylogenetics

It was on the base of considerations like all those presented above, that since centuries Phylogenetic studies were Time-ordering Causes and Effects in tree-like structures.  However, it is just since only twenty-three years, that it has been finally understood what really changes what.  

Phylogenetic trees like the one below (click-to-enlarge this extremely high definition map) in the reality are showing the way followed by the genetic data (i.e., in the DNA) to structure a complex topology.

Classic view

Homogeneity of space and time and isotropy of space are associated to the causal relations 

The necessary character of causal relations is closely associated with the homogeneity of time and space and the  isotropy of space.  If for instance one and the same action  of an industrial steam hammer on an ingot is the same, irrespective of whether the time is today or tomorrow, it  then follows that  time is homogeneous relative to causal relations.   True, during the time lapse both the hammer and the ingot may have changed, but this change is not the  result of the action of time on things, it is inherent in the nature of the interacting entities.

  The character of the causal relations is associated with the homogeneity of time and space and the isotropy of space.  Something evident when trying to imagine the outcomes of measurements and observations defining causes and effects in the space on side, where the colours define boundaries of volumes with different energy.  No homogeneity nor isotropy (image credit A. Arad, et al., The Australian National University, 2014)

A  given set of conditions gives rise to one and the same effect, irrespective of the time at which the set of conditions operates.   The  important thing is that the set of conditions and the time intervals during which they are realised be the same.   The same thing goes for space as well.  One and the same set of conditions generates the same effects, irrespective of the region of space in which they are realised.   

To take an example, one and the same quantity of gasoline in a calorimetric bomb will release, upon being burnt, the same quantity of heat wherever the burning takes place (whether on the equator, at the north pole, or elsewhere on the earth), so long as the other  combustion conditions are the same.   Carrying the conditions from one region of space to another does not alter the corresponding effect.   The behaviour of a body in specific conditions does not change either if we rotate it through some angle and thus alter all the conditions upon which the behaviour of the body depends.  

That causal relations are independent of translation in space and time and of rotation through a fixed angle might be expressed on the basis of the concept of symmetry.

Classic view: symmetry cause homogeneity and isotropy

The German mathematician and physicist Herman Weyl is known for important insights in the meaning of a concept today nodal: symmetry.  Following him, we will say that an  object is symmetrical if it remains the same as before after being subjected to some kind of operation.

Then we could say that the causal relations of natural phenomena, or at least the causal relations of physical phenomena, are symmetrical with respect to a transfer in space and time and relative to a rotation through a fixed angle.

Transition from Classic to Modern: 

phenomena are independent of transfer in time

The German mathematician Emmy Noether (see figure below on right side) demonstrated in 1915 that it is possible to obtain the law of conservation of energy in its ordinary mathematical form, recognising that phenomena are independent of transfer in time.

She also demonstrated that the law of conservation of momentum is due to the fact that a phenomenon is not dependent on the region of space in which it occurs. It was also  found that the law of conservation of angular momentum follows from the isotropy of space.  From these theorems it follows that recognition of the necessity of a relationship between cause and effect under fixed conditions is essentially related to the laws of conservation of energy, momentum and angular momentum.

If one assumes that even one of these conservation laws is violated in some kind of  process, this unavoidably will lead to a change in the form of manifestation of the  necessity of causal relations.   For instance, to presume the law of conservation of momentum is violated would force one to acknowledge that absolutely identical conditions realised at different times would be accompanied by different effects.  Time  would then be capable of operating physically on processes, and then time would have to be included in the conditions (causes) of these processes.   The supposition that processes exist for which the laws of conservation of momentum do not hold would compel acceptance of the fact that the same conditions realised in different regions of space would generate different effects.  This would mean that space exerts a physical effect on these processes, and then space would have to be included in the conditions of these processes.

In the same way, violation of the law of conservation of angular momentum would mean that a rotation of certain phenomena through some angle in space could physically effect the behaviour of the phenomena.  If in the actual world there were entities that did not obey the laws of conservation of energy, momentum and angular momentum, then the idea of the necessity of causal  relations determining the behaviour of these entities would have to be stated as follows: the behaviour of an entity is of necessity determined by its interactions with the environment and with space-time.  This relation to the environment as part of the experimental configuration, is exactly what started to be hinted by theory and experiments around ninety years ago.  

And now we’ll leave the descriptive approach to examine closer, the Logic-Mathematical background of what is a Cause, what an Effect and what really are correlated Events. 

In the following, we’ll examine two different points of view:

1. classic Physics, due to the mathematician Hermann Minkowski (1908) and extended by Albert Einstein. It entered in crisis one decade later, after the discoveries of Werner Heisenberg, Erwin Schroedinger and others.  Classic point of view, however falsified, still today the laymen point of view.
2. modern Physics, dated 2000, mainly due to physicist Rafael Bousso.

The definitions of “Events” (whatever kind of Event, included the triggerings, measurements, actioning of motors or solenoid valves) and of the derived concept of causal relation between Events, are part of the investigational fields of Theoretical Physics, Quantum Physics and Relativity.   

Why ?   Because the cosmological and high-energy Physics’ experimental and theoretical evidences converge to a common origin of all objects, energy, space and time.  One characterised by extremely high values for energy density, temperatures and geometrical curvature for the Environment.  The Environment where today all, measurement instruments and Machinery included, operates.  

  The worldlines of all elementary constituents of all systems and Observers, were joint in an extremely small volume when the initial condition acted.  Showed 4 worldlines a, b, c, d originated by the same Event O in the Past, as seen in the classic perspective: 4 different histories.  All technological applications happen into the slices existing between 2 Events, e.g. P and Q.  Each slice or leaf is 3-dimensional. Time indicates the location of a chosen 3-dimensional space, an hypersurface in the infinitely wider 4-dimensional space (image abridged by J. A. Wheeler, K. Thorne, C. W. Misner, 1973)

With reference to the figure above, several wordlines a, b, c, d intersecting today a 3-dimensional hypersurface (i.e., the worldline a at the point Q) are a System.  It is frequently over looked the fact that all the worldlines were joint in a single dot, back in the Event O, origin of Time at t = 0.  The nearly spherical surface in the past, crossed by the worldline a at P, represents a phase of the historical evolution when space-time was not locally curved as it is today. 

How many 3-dimensional surfaces (or leaves, or sheets) contains the 4-dimensional solid above ?

The answer is the sum of:                

                    ∞³  points;  

and, in a coordinate system making diagonal the metric:

       3 diagonal components of the metric specifiable per space point;

say, ∞³ choices of the metric per space point. 

Therefore, a total amount of possible 3-dimensional leaves equal to: 

Amount later heavily reduced by the added dynamical condition of constructive interference, but however a huge amount.  These numbers are not shown as a sterile exercise of infinitesimal calculus, rather to enforce that since at least one century Nature answers to the continued questions by the humanity inviting us to replace infinities with Numbers so big to be easily confused with infinities.  

Relevance of the history of the Events

Events strictly or slightly related

In the following we'll introduce the relevance of the history of the Events. The long and, at a first sight, time-ordered chaining of Events which are considered causes for the Events in the present time.  A relatively complex sequence of bicones illustrating why in General Relativity the spatial dots all around a single Event, are related to the Event.  The figure below shows a situation constantly present in the reality lived by us and our measurement devices.  Here, on the base of Minkowski geometry of 1907, an entire spatial volume lying out of the light cone should be causally disconnected by the Event a in its Past.  In the reality, it shall result still related, however slightly.  

We start observing the time-ordered sequences where:

1. the Event a lies in the Past of the Events b and c;
2. the Event a lies in the Past of the Events y0,  x, s and y;
3. S(a, b) is the hypersurface separating the Future of a by the Past of b;
4. S(a, c) is the hypersurface separating the Future of a by the Past of c.   

The Event s in the Future of the Event x, visibly lies on the external boundary of the Future of the Event a, part of the same hypersurface of constant Time of the Event b.  Due to this reason the pink coloured hypersurface centered on the Event s results partitioned. The inner portion hosting a 3-dimensional space-like volume causally related to the Event a much more strictly than the portion lying out of the hypersurface S(a, c) external boundary.

 A relatively complex sequence of bicones illustrating why in General Relativity the spatial dots all around a single Event, are related to the Event.   The image shows a special case: here on the base of Minkowski geometry of 1908 an entire spatial volume lying out of the light cone should be causally disconnected by the Event a in its Past. In the reality, it shall result still partially related.  Event a lies in the Past of the Events b and c.  Also, a lies in the Past of the Events y0,  x, s and y.  S(a, b) is the hypersurface separating the Future of a by the Past of b.   S(a, c) is the hyper surface separating the Future of a by the Past of c.  Visibly, the Event s in the Future of the Event x, lies on the external boundary of the Future of the Event a, part of the same hypersurface of constant Time of the of the Event b.  Because of this reason, the pink coloured hyper surface centered on the Event s results partitioned. The inner portion hosting a 3-dimensional space-like volume causally related to the Event a much more strictly than the portion out of the hyper surface S(a, c) external boundary (diagram abridged by H.-J. Borchers, R. N. Sen, 2006)

Classical version pre-2000

We’ll start to list, as seen by the point of view until 2000 considered standard, what is and what is not possible in terms of Events' causal connection.  

No interaction, gravitational nor electromagnetic, is possible with particles lying in the space out of the bicones: all what lies in that space is causally disconnected with respect to the object placed at the Triggered Event.  In this classic approximation, only what lies in the Past light cone is causally connected with what lies in the Event position. The Future light cone traces the paths of light rays emitted in every possible direction from the Event at the origin, and the Past light cone indicates the paths of light rays arriving at the Event's location at the Present moment.  In this classic view, the Event has a purely geometric meaning: a dot of space-time.  It is not necessarily the place where also happens an interaction.  

The rules about Events considered valid by Special Relativity, between 1905 and 1915, were:

• light beams emitted from the Event location travel exclusively along the Future light cone.  (On the opposite, as we’ll see later with more details, in the modern Quantum Physics view, trajectories do not exist and particles are emitted in both directions, Future and Past);
• arriving at the Event's location at the Present, all fall within the Past light cone;
• launched from the Event's location, all travel on trajectories bounded by the Future light cone;
  
• they cannot exist two Events superposed in the same worldpoint: all Events are unique identifiers;
• Past and Future light cones of other Events, at different spatial locations, have light cones which are offset from those of the Event we have arbitrarily designated as being at the origin;
• the areas where the Past and Future light cones of two Events overlap, are in their common Past and Future respectively, but each Event will have regions of spacetime from which they can receive and send Information that are not shared with the others.

After 1915, with the advent of the General Relativity, the point of view changed sensibly.  Every worldpoint is still the origin of the bicone of the active Future and the passive Past but:

• the two zones are no more separated by an intervening region ef Events causally disconnected;
• it is possible for the cone of the active Future to overlap with that of the passive Past;
• it is possible to experience Events now that will in part be an effect of our future Events or decisions (Trigger and Measurement Events included).

Adding to General Relativity the Principle of General Covariance, Einstein made a precise statement about the fact that global evolution does not exist.  From his point of view, the time t is just a label we assign to one of the coordinate axes.  

The figure presented in the precedent section titled: "Events’ causal connection.  Introduction” was published in 1973, before the discovery of the Inflationary mechanism by Starobinski, Linde and Guth.  

The figure below, on the opposite, includes also this phase and is the model considered standard from 1982 til ~1993.   Here, the qualitative evolution of the Hubble horizon is showed into the couple of dashed lines and of the scale factor with the external solid curve.  The time coordinate is on the vertical axis, while the horizontal axes are space coordinates spanning a two-dimensional spatial section of the cosmological manifold. The inflationary phase extends from  ti  to  tf,  the standard cosmological phase from tf to the present time tc.   The shaded areas represent causally connected regions at different epochs.  At the beginning of the standard evolution the size of the currently observed Universe was larger than the corresponding Hubble radius.   

 Qualitative evolution of the Hubble horizon (dashed line) and of the scale factor (solid curve).  The time coordinate is on the vertical axis, while the horizontal axes are space coordinates spanning a two-dimensional spatial section of the cosmological manifold.   The inflationary phase extends from ti to tf , the standard cosmological phase from tf to the present time tc.   The shaded areas represent causally connected regions at different epochs.   At the beginning of the standard evolution the size of the currently observed Universe is larger than the corresponding Hubble radius.   All of its parts, however, emerge from a spatial region that was causally connected at the beginning of inflation (abridged by image credit Gasperini, 2007)

All of its parts emerged from a spatial region which was causally connected at the beginning of inflation, implying that the causal connection is maintained also after the inflation.  

It is important to understand it is the initial causal connection what still today let a motor in the Blowformer Machine run following the same Physical Laws as the motor in the Palletiser, 150 m afar.   The figure above also shows that there is a wide all-around sector which was causally connected but that receded so fast to have since long time moved out of our horizon.   Then, disconnected also with respect to actual events.   

Modern view

In the year 2000 Tom Banks and Willy Fischler discovered that the mass density parameter Ω is determined as the inverse of the number N of degrees of freedom.  Refer to the figure below: the curvature parameter Ω0 encodes the system density, say the relation between (mass + energy) and volume.   

But, the volume strictly depends on the number of degrees of freedom.  

This is now considered an input parameter at the most fundamental level of physics.  Shortly later, Raphael Bousso, Oliver DeWolfe and Robert C. Myers have been capable to derive a new definition, today standard, of causal connection between Events. 

  Curvature determines the sum of the inner angles of a triangle, which in the three cases here depicted shall be:

   Ω0  >  1,     Inner total angle  >  360˚

   Ω0  <  1,     Inner total angle  <  360˚

   Ω0  =  1,     Inner total angle  =  360˚

Matter plus energy density parameter Ω0 is translated in a spatial curvature.  Curvature depending by the number of degrees of freedom of the system. From this discovery of the year 2000 it has been possible to derive a new definition of Causal Connection between Events, close to everyday measurements, referred to our macro scopic scale. Causal connection exists only between points in a small subspace of the bicone.  This, in turn, implies that the Environment, much smaller than expected, cannot be ignored 

The perspective establishes the new concept of Causal Diamond, an example of these in the figure down, carrying on the scenario of everyday practical measurements (or Triggerings) evaluations of causal relation until a few years ago extremely theoretical. It is an approach looking to macroscopic (and not mesoscopic or microscopic) objects in the relativistic frame in which, as an example, also smartphones’ GPS commonly operate.  All what we’ll refer  for a Trigger is true also for measurement instruments like all the electronic inspections, like triggering nearly exclusively electromagnetic.  Imagine that p and q are two points on an Trigger’s world line, with q later than p.  One can think of p as the beginning and q as the end of the Triggering (or, measurement).  

Then, it is observed that the total Entropy is bounded by the inverse of the mass density and that it cannot be observed at all out of a small subset of the much wider bicone, say:

• increasing Entropy (second Law of Thermodynamics valid);
• to consider only the Trigger’s (or, measurement instrument) causal Past and causal Future and ignore everything else.

The last introduce sensible restrictions: 

1. at the point q, the endpoint of the Triggering (or,  measurement instrument), the Trigger can only have received signals from the Past of q. The rest of space-time has not yet been seen.  For the purposes of the measurement in question, its Entropy is operationally meaningless and can be ignored;
2. the Trigger’s Past is bounded by the Past light-cone from the point q, and that Signals within the Trigger’s Past must pass through this cone.  

Thus, if one wishes to encounter the boundary of the observable Entropy, it will be sufficient to bound the Entropy on the Past light-cone of the endpoint, q.  It is not enough for Entropy (or, Information) to lie in the Trigger’s Past.  To be observed, it actually has to get to the Trigger, or at least to a region that can be probed by the Trigger. 

But, a measurement that starts at p can only detect what lies in the causal Future of p.     

 Bidimensional lateral cut of the Causal Diamond. After 1990, Multiverse replaced the classic concept of Universe: a multitude of coexisting non-communicating bicones and new Events, derives by each one single Event.  Each one of the infinite yellow colour dots in the image, is part of the paths followed by the object to move itself from the point p to q in the Future of p (abridged by image R. Bousso, 2000) 

This modern perspective implicitly assumes p and q joint by at least one world-line, which corresponds to both points of view, classic Hamiltonian and modern Quantum Mechanical. 

To go from a point in the Past to a point in the Future, different paths are:

• possible         (Hamiltonian perspective);
• actual             (Quantum Mechanical perspective).

Causal Diamonds and histories

The descriptions visible in the figures above are named Causal Diamonds:

• above, what happens in the 2-D yellow colour region (a lateral cut of the 3-D Causal Diamond) of the form C(p,q) for some pair of points (p,q);  
• right side, their derivation by the wider Minkowski causal bicone, in a 3-D geometry;
• below, the only Causal Diamond in a 3-D geometry, and there is not to forget they are objects whose dimension is ≥ 4.

  3 dimensional view of a Causal Diamond, with two space dimensions plus time.  In the example, Past and Future light cones (light violet colour) are super imposed around a 3-dimensional sphere depicted above as a dark violet circle because we cannot show the third spatial dimension. The light violet volume named “causal diamond”, after 2000 became a portion smaller than the entire bicone in 1907 imagined causally related.  The oblique lines represent three of the infinite paths joining p and q.  Each one path a different history in the sense attributed by Feynman (abridged by image R. Bousso, 2000)

In the figure above, we are showing three of the infinite and actual worldlines conceived by Quantum Mechanics. There are innumerable curves joining two events, and none is privileged. 

If q is in the Future of p, there will be several world lines connecting them, otherwise the entire region C(p,q) will be empty or degenerate. To evaluate the phase difference and the coupling between two events, we have to account for the contribution from all paths. Colloquially, we can say everything between two Events contributes to their coupling. The pervasiveness of the agents of interaction further affirms the demise of empty space; what now may appear obscure shall become more clear in the following sections.   

Causal Diamonds are bounded by a:

• top cone, a portion of the Past light-cone of q;
• bottom cone, a portion of the Future light-cone of p;

occupying this way the entire subset of points into the (pink coloured) bicones in the couple of figures above. 

The cones usually intersect at a bidimensional spatial surface A (see the upper figure above), the edge of the Causal Diamond. In any case, the Entropy in the Causal Diamond must pass through the top cone and, we repeat, all Signals must have entered through the bottom cone. 

The net positive result of this line of reasoning is that the:

• Entropy within a Causal Diamond is under strict theoretical control; 
• 4-dimensional volume is much smaller than what it was conceived in 1907. Being much smaller, the causal relation inferences becomes closer to the scales of dimensions and durations of our everyday measurements.

Examples of Events’ causal connection 

Triggering of containers by mean of Laser light

Events’ causal connection, in its classic version pre-2000, encounters an important field of application with respect to the exchange of light signals.  This, in the Electronic Inspectors is commonly associated to Triggering of typically accelerated and fast moving containers.  In the following, we’ll outline the true rigorous scenario around a Triggering.  This is useful to understand why Triggering, an action strictly related to the kinematic of a container, results on the opposite always referred to an external speed reference, since decades an Encoder.  This, when it is well known that only direct measurements provide absolute values, those whose relative errors are  minimised. The choice toward external speed references is forced by the complexity of internal (direct) measurements.  

To use an Encoder, in the perspective of modern Physics, means to coarse-grain a container position into a Shifting-Register cell.  

It is a coarse-graining operation because the Shifting-Register cell has typically an extension, measured along the container's direction of movement, bigger than a container diameter.  Coarse-graining, for definition, is clearly not a synonimous of increase on precision, just the opposite, and in our case kinematic uncertainty means False Rejects.  Curvature effects described below are in the Shifting-Register emulated and widely reinforced by accelerations and decelerations of Conveyors and by the contacts of the container with the lateral guides causing container sliding.

What should happen if we’d be measuring containers' speed in a direct rigouros way ?

Example 1.   Exchanging light signals

Triggers, in the most general and real case, operate in the Factory accelerated reference frame, meaning they operate along worldines which are not geodetic nor orbits.   Because of this reason the Trigger has to be considered in motion along a curve γ (see figure on right side) with tangent vector u and proper time s as parameter.  What follows is referred to direct reflection Trigger photosensors, however valid also for the widespread thru-beam Trigger photosensors.  This last case carries similar results, because also the interruption of the light beam is not instantaneously received by the Trigger u on the worldline γ.  By mean of the Trigger, the CPU processing the informations encoded in the electric signals infeeding by the Trigger, on the base of a dedicated software routine, can only deduce the spatial velocity of a passing container relative to its own local rest frame.  

Meaning, by the exchange of light signals:   

• at the event A on γ the Trigger sends a light signal to the container, which receives it at the event P on γ′; 
• at P the light signal is reflected back to the Trigger which receives it at the Event B on γ.   

The Line from P to A0 is curved to indicate the fact that the spacetime itself is curved.  Denote as Υ and Υ′ the null geodesics connecting A to P and P to B respectively.  Let A0 be the event on γ, subsequent to A and antecedent to B, which is simultaneous with P with respect to the Trigger u and such that the space-like geodesic ζP → A joining P to A0 is extremal with respect to γ.  

Repeated reading of the time of: 

• emission of light signals at A;
• recording of the reflected echo at B;

allows one to determine the length of the space-like geodesic segment connecting P to A0, which represents, by definition, the instantaneous spatial distance of the container at P from the Trigger on γ like in the see the figure on right side.  The relative velocity of the container with respect to the Trigger u is then deduced, differentiating the above spatial distance with respect to the Trigger’s proper time.  The relative velocity so determined is along the Trigger’s local line of sight, so it is a radial velocity, as an example, a velocity either of recession or of approach.  The measurement process involving the events A, P, and B is non-local insofar as the measurement domain is finite.  

Example 2.   Doppler shift

An alternative way to measure the container velocity is based on the measurement of a frequency shift of the exchanged photons through the application of the Doppler formula. The image on right side hints to the process.  But, the velocity so determined, should be an equivalent velocity because the frequency shift can also be caused by geometry perturbations.  Say, something unrelated to the container’s motion. 

As already stated, curvature effects are in general entangled with inertial terms resulting from the choice of the reference frame, so we shall just term as curvature any possible combination of them.  The measurement of a relative velocity is the result of a local measurement, which does not contain curvature terms, and a nonlocal one, which depends explicitly on the curvature.   

Links to the pages:

Total Cost of Ownership of a Full Containers Electronic Inspector, based on its Fill Level Inspection technologyCounting the number of Technologies existing for the measurement of the fill level in Bottling Lines, we encounter at least seven different. …

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Introduction to a fundamental questionThe purpose of any photodetector is the conversion of light (photons) into electric currents (electrons). Photodetectors are among the most common optoelectronic devices; they automatically record pictures in the Electronic Inspectors’ cameras, the presence of labels in the Label Inspectors or the fallen bottles lying in a Conveyor belt. …

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First In First Out Application to Food & Beverage packaging an ideadeveloped to handle the highest production speed FIFO (First-In-First-Out) concept started to be applied some decades ago to industrial productions, specifically to manage the highest speed production lines. …

IntroductionThe language of Optoelectronics adopts its own special terminology, born along the developement of this branch of technology. In the following we are synthesizing in few lines the definition of the fundamental terms. …

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