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The definitions of “Events”, whatever kind of Event, triggerings, measurements and “observations” included, 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 majority of the evidences hint to a common origin. One characterised by extremely high values for energy density (then, high temperature) and geometrical curvature for the greater Environment where all, measurement instruments and Machinery included, operates.  With reference to the figure below, four wordlines a, b, c, d intersecting today a 3D hypersurface, for example the worldline a at the point Q, are a System.

The worldlines of all elementary constituents of all Triggers, measurement instruments, electronic inspectors and Observers, were joint in an extremely small volume when the initial condition acted.  In the figure are showed four worldlines a, b, c, d originated by the same Event O in the Past, as seen in the classic perspective: 4 distinct histories.  All technological and analytical applications happen into the slices existing between two Events, e.g. P and Q.  Each slice or leaf is 3D.  Time indicates the location of a chosen 3D space.  An hypersurface in the infinitely wider 4D space (   abridged by Wheeler, Thorne, Misner/1973)

It is frequently over looked the fact that all the world-lines 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 more homogeneous, not locally curved as it is today.  This classic point of view, mainly derived by the ideas of Einstein, Minkowski, Lemaitre and Gamow, one time backed by Hubble and Eddington’s discoveries, started to hold as a paradigm.  And it still holds today, at least in part.  To reduce all this to mere theory, in opposition to facts, is no more possible.  So many they are today its direct applications in our life and Machinery.   As an example, all smartphones’ GPS location devices integrate routines with formulas of General Relativity to be so precise as they are, evidencing the correctness of the Theory. 

Higher Order Infinities

How many 3D surfaces (or leaves, or sheets) contains the 4D solid above?  The answer is the sum of:                

• ∞³  points;  

in a coordinate system making diagonal the metric:

•  3 diagonal components of the metric specifiable per space point;

then, ∞³ choices of the metric per space point. Therefore, a total amount of possible 3D leaves equal to: 

Amount later heavily reduced by the added dynamical condition of constructive interference, but however an impressively 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.  

Classic 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, electromagnetic or nuclear, is possible with particles lying in the space out of the bicones: all what lies in that space (the “Elsewhere”) is causally disconnected by 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.  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. 

 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 Gasperini/2007)

The figure presented in the precedent section was published in 1973, before the discovery of the Inflationary mechanism by Starobinski, Linde and Guth.  The figure here above, 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.  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.  That initial causal connection is 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.   

Focus on the Degrees of Freedom

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, whose essence is visible in the figures below.  The curvature parameter Ω0 encodes the system density, a relation between mass + energy and volume.   But, the volume strictly depends on the number of degrees of freedom.  And that’s why the amount N of degrees of freedom is since then considered an input parameter at the most fundamental level of physics.    

 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 

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.  A perspective built over the new concept of Alexandrov interval.  Alexandrov's intervals are commonly named Causal Diamonds.   

   The curvature parameter Ω0 encodes the system density, say the relation between (mass + energy) and the volume they occupy.   Today the experimental evidence favours the idea that the curvature Ω0 is that of an hyperbolic hypersurface.  Hyperbolic because is the one guaranteeing strict causal relations between Events in the Past and Events in the Future.  All signals, electromagnetic and gravitational, propagate through a 3+1D spacetime, immersed in the de Sitter spacetime.  De Sitter is a manifold reproducing, after a renormalization, a 5D Minkowski spacetime in the limit of zero curvature.  Green-coloured 4D-surface in the single hyperbolic space is a projection mapped by the corresponding points in the 5D de Sitter space. With reference to the point P (t = t0) at the vertex, all what happened in its Past is causally related with it since t = 0, the origin of Time.  The graphic is evidencing that today’s idea of causal relation between Events, considers signals travelling in a 5D space composed by an infinity of 4D spaces 

Examples of these in the four figures below, 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 objects in the relativistic frame in which, as an example, also all smartphones’ GPS 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.   

With reference to the diagram above, imagine how many superposed green-coloured patches may coexist in the hyperbolic 3+1-dimensional space. The diagram here at right side, evidentiates how a Present human observation or choice, or a measurement by a Detector in a Machinery, includes observations, choices and measurements happened in the Past (  abridged by J. A. Wheeler/1983) 

Alexandrov Intervals and Causal Diamonds

With reference to the three figures below, 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.  

 2D lateral cut of the Alexandrov Interval, also named Causal Diamond.  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.  After 1990, Multiverse replaced the classic concept of Universe: a multitude of coexisting, slightly interfering or non-communicating bicones, referred to Events along different branches of the foliation (   abridged by R. Bousso/2000) 

Thus, if one wishes to encounter the boundary of the observable Entropy, it shall 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, with some exceptions where this is not possible.  Refer to the figure at right side.  It represents the world-lines of two objects in the space-time U.  The point q lies in the causal Future of the space-time point p, say p ≺ q  (logic symbol “≺“ meaning “precedes”).  The curve joining p and q is a causal curve.   The Event p is the main cause for the Event q.  

The world-point q can to be mapped as a geometric transformation of the world-point p in its Past.  To be observed, light and gravity have to get to the Trigger !  Or, at least to a region that can be probed by the Trigger.  It is not enough for Entropy or Information to lie in the Detector’s Past.   Removing just a single 4D world-point between a cause and an effect, is enough to prevent the effect from seeing its own cause.  The effect continues to interact with the cause only if this is a material particle or a collection of them named physical body.  Interaction however no more electromagnetic nor gravitational, because it does not exist any more a geodetic curve joining cause and effect.  Relation mainly reduced to the statistical dependance (  abridged by F. De Felice, C.J.S. Clarke/1990)

The two world-lines, left- and right-ward, had an intersection at the space-time point p.  Implying the strictest interaction and statistical dependance of their physical properties of the objects in their respective Futures, along the time-like world-lines depicted.  Later they diverged but, for example, the right-ward curve (the history of an object) at the world-point q still has its physical properties partially defined by the informations acquired by the left-ward, during the strict interaction at p.  The world-point q can to be imagined as a geometric transformation of the world-point p in its Past.  Imagine to remove a single 4D world-point.  You’ll discover that now the object represented by the right-ward history, at the point q: 

• continues to be part of the causal Future of the world-point p,
• is still statistically dependent on p, 
• can be reached by massive particles, moving in the space J⁺(p,U) into the Future light-cone,
• cannot be reached any more by electromagnetic or gravitational signals emitted by the point p, moving in the space E⁺(p,U), external null-surface of the Future light-cone.  On practice, in laymen language: “from q is no more possible to see p”.

What precedes may be synthesized in that it is not enough for Entropy or Information to lie in the Detector's (i.e., a Trigger) Past.  To be observed, light and gravity have to get to the Trigger!   Or, at least, to a region that can be probed by the Detector.  Removing just a single 4D world-point between a cause and an effect, is enough to prevent the effect from seeing its own cause.  The effect can still interact with the cause only if the cause is a material particle or a collection of them (a materal body).  The relation is reduced to the statistical dependance.  In this perspective we assumed implicitly that p and q are joined 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 p to a point in the Future q, infinite different paths are:

• possible, in the Hamiltonian perspective;
• actual, in the quantomechanical perspective (Quantum Democracy Principle).

Causal Diamonds and Histories

The descriptions first introduced in the figure before, and detailed below with violet colour, are named Alexandrov intervals, or Causal Diamonds:

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

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

3D view of an Alexandrov interval (Causal Diamond), with two spatial 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 R. Bousso/2000)

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. The Alexandrov intervals 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; 
• 4D volume is much smaller than what it was conceived in 1907.  Being so small, the causal relation inferences becomes closer to the scales of dimensions and durations of our everyday measurements.  How close?  The causal diamonds can be also much smaller than, for example, the distance covered by an electromagnetic wave along 1 picosecond of time, say 0.2997 millimetres.

Given a couple of Events, p, q, their Alexandrov interval (causal diamond) is the set of all Events chronologically ordered between p and q.  When p and q may be chronologically connected, then the causal diamond is composed by those Events into the volume of the bicone depicted before.

 3D external view of an Alexandrov interval or, Causal Diamond. Given a couple of Events, p, q, their Causal Diamond is just the set of all Events chronologically ordered between p and q.  These spaces have dimension ≥ 4. In evidence the worldline γ passing through the point p and proceeding toward the future at q (  abridged by Wolfram Research/2013)

Examples of Events’ Causal Connections 

Exchanging Light Signals

All Triggers, in the most general and real case, operate in the Factory's 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 at 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.  

  Exchange of light beams between Trigger and container, to determine container's kinematics (  abridged by F. De Felice, C.J.S. Clarke/1990)

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′; 
• 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.  Now, 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 3   

Doppler Shift

A third example of causal connection pertains to the measure of a container velocity 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.  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.   

Also the measurement of a frequency shift of the exchanged photons, through the application of the Doppler formula, fails to provide an alternative way to measure containers’ velocity (  Lookang/CC BY-SA 3.0/2015)

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