Instructor's Key to Physics & Curved Space 101 Entrance Exam

Montréal, March 24, 2000

Intructor's key to PACS 101 Entrance Examination
Physics and Curved Space 101
March 24, 2000

Preamble to Interpretation of Examination Responses

Consistent with the reciprocity characteristic of the geometry of curved-space time, ... the 'instructor' must also be the 'student' (one must reciprocally assess one's own exam guided by the theory-based key) since the comprehension of relativistic curved space-time concepts involves the 'imaging' of complex conjugates, ... a process which must be achieved within a unified 'interpretation vessel'. In other words, curved space-time concepts involve complex interference effects which cannot be fully described in 'real' terms, thus it is a subject whose explanatory 'key' necessarily goes beyond an 'objective' explanation, placing the 'observer' in the hermeneutic circle between 'subject' (the causal agent) and the 'object' (the operands).

Western adults, because of acculturation, have difficulty assimilating the interference based view of phenomena wherein he can neither observe from the point of view of subject or object . That is, instead of the observer 'coming from' the vantage point of 'subject' or 'object', he must instead come from the vantage point of the 'medium' in which both 'subject' and 'object' are contained so that he is in a position for a relativistic viewing. That this shift of perceptual vantage point is both possible and necessary follows from the development of quantum physics, as articulated by Erwin Schroedinger [1] in terms of our having to perceive physical phenomena as objects interacting with objects and by our not 'fixing' the location of the observing mind to either one or the other of these; i.e.

"I cannot believe that [for example] the deep philosophical enquiry into the relation between subject and object and into the true meaning of the distinction between them depends on the quantitative results of physical and chemical measurements with weighting scales, spectroscopes, microscopes, telescopes, with Geiger- Mueller- counters, Wilson-chambers, photographic plates, arrangements for measuring the radioactive decay, and whatnot." . . . "... the mere contention that every observation depends on both the subject and the object, which are inextricably interwoven, is hardly new; it is almost as old as science itself. Though but scarce reports and quotations of th two great men from Abdera, Protagoras and Democritus, have come down to us across the twenty-four centuries that separate us from them, we can tell that they both, in their way, maintained that all our sensations, perceptions, and observations have a strong personal subjective tinge and do not convey the nature of the 'thing-in-itself' (The difference between them was that Protagorus dispensed with the 'thing-in-itself; to him, our sensations were the only truth, while Democritus thought differently.)"

Thus our scientific embrace of ancient philosophy persists in opting for the denial of our senses in favour of the abstract notion of an objective world 'in its own right'. Schroedinger goes on to speak of how this question has persisted in science and philosophy, ... the question of whether we construct our reality by means of our sensations (purely subjective), or whether the world 'is given' by its bottom-up 'atomic structure' (purely objective) or whether it is somehow a combination of the two. Schroedinger points out that while the subject of the "inextricable interweaving" of subject and object is old, ... that the physicists of the modern day could be making the point that new insights into the nature of the weave have emerged through quantum physics. He continues;

"I think it is most true that in previous centuries, when discussing this question, one mostly had in mind two things, viz. (a) a direct physicalimpression caused by the object in the subject, and (b) the state of the subject that receives the impression. As against this, in the present order of ideas the direct physical, causal, influence between the two is regarded asmutual . It is said that there is also an unavoidable and uncontrollable impression from the side of the subject onto the object . This aspect is new, and, I should say, more adequate anyhow. For physical action always is inter - action; it always is mutual. What remains doubtful to me is only just this; whether it is adequate to term one of the two physically interacting systems the "subject." For the observing mind is not a physical system, it cannot interact with any physical systems . And it might be better to reserve the term "subject" for the observing mind. (From Schroedinger's book 'My View of the World', ... 'Why Not Talk Physics?').

So, we can see a basic problem with the geometry of perception here, which is arising from both the theory of relativity and from quantum physics, ... dealing with the relationship between 'subject', 'object' and the space-time 'medium' of their containment, the 'aither' or 'field'. For example, 'relativity' says that we cannot determine, absolutely, the velocities of the 'subject' and 'object', ... only their 'relative velocity'. This is a fundamental problem because our notion of energy is that it comes in two forms, ... 'kinetic energy', which is DEPENDENT on the notion of a 'thing' and its velocity along a euclidian trajectory (one half the product of its mass times the square of its velocity), and 'potential energy' which is its energy of 'position' with respect to its containing 'field' (gravitational, electric, magnetic etc.).

So the kinetic energy of a baseball bat when it is held still (with respect to the euclidian reference frame used) is zero. This choice of reference frames clearly makes a practical difference; i.e. if the bat is being held out the window of a speeding car and I am a pedestrian who gets conked on the head by it as the car passes, ... clearly, from my perspective, the bat had a high kinetic energy.

So we have a 'weave', here, of the concepts of perception and energy which depends on our 'framing' of the phenomenon. As far as the terminology goes, it is not yet established; i.e. Schroedinger suggests that we reserve the term 'subject' for the observer, ... making the point that the observer does not have to be firmly fixed (in an exclusionary logic 'either/or' sense) to either the 'subject' or the 'object', but can be 'coming from' the medium of containment of the 'subject' and 'object', ... and thus seeing physical phenomena in terms of the interactions of 'objects' within a containing space-time medium (Note that Einstein and Schroedinger are also saying that 'space' is a 'participant' in physical phenomena).

The instructor should further take note here, that what is being struggled with is an 'incompleteness' in our notion of 'direction' which comes from the euclidian space assumption. In euclidian space, direction is bivalent and things can only be described in terms of their motion with respect the opposite ends of linear, rectangular axes (e.g. 'going and coming', 'past and future') of components thereof, .... and that 'direction' is seen very differently in non-euclidian space-time.

That is, on the surface of sphere (we where indeed live!), direction is 'complex' and has an 'imaginary' component, in that a straight line on the surface of a sphere becomes a circle, so that something which is moving linearly along the surface of a sphere not only has a direction give in terms of the linear euclidian tangent to the sphere at that point, but also has a property of 'space-time phase' associated with it, which is informationally important, as the observer in question 1. of the examination (on circular causality) is going to discover if he dilly-dallies to much on the question of formulating the equation of motion of cascading collapse of dominos on a spherical surface. In our space-time sharing eco-reality, ... as indicated in all of the questions which follow the 'domino' question (areal distribution of nomads, subduction of the tent, sailboat navigation, forensic human relationships), we have many interfering space-time considerations and a view based on spherical surface area and 'reciprocal disposition' is more complete; i.e. as Einstein says;

"Therefore we say that the spherical surface of the globe is an unbounded continuum. Moreover, the spherical surface is a finite continuum. For if we stick the paper discs on the globe, so that no disc overlaps another, the surface of the globe will finally become so full that there is no room for another disc. This simply means that the spherical surface of the globe is finite in relation to the paper discs. Further, the spherical surface is a non-Euclidean continuum of two directions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane." (the more complete statement is in footnote [2]).

To tie this back to the simple physics that the student has learned and accepted, ... but where he has passed over the key philosophical question on 'directional relativity', we can 'take another look' at what we already accepted in our study of physics in the domain of wave motion. Since we need a terminology which removes confusion as to what is the 'subject', ... we can for temporary convenience refer to our three relative components (subject, object and container) as 'transmitting object', 'receiving object' and propagational medium. This puts things, at the same time, into the more general realm of 'information theory' and 'communications' wherein 'cause-and-effect' become 'transmit -and-receive'.

For example the baseball bat held still by the person in the speeding car can be said to have 'sent a message' to the pedestrian on the street who got 'whapped' on the head, rather than being 'the cause' of an 'effect' since this is simply an 'attitudinal view' which, by cultural convention, is associated with the observer with the 'transmitting object'. If the person in the car was 'innocently holding out the bat as he was moving his house and belongings, and a mischievious pedestrian pushed a heavy cart into position so that it would 'intercept' the bat as it went by, ... the notion of 'causality' shifts over to the object which was a moment ago seen as the 'receiver' of the message. In other words, ... 'transmission' and 'reception' are fully mutual or 'reciprocal' in a relativistic sense, and it is our attitude which positions our observation on 'one end' or the 'other', ... an attitude which derives, in the case of the western culture, from the primacy of euclidian space and linear time concepts, over the more complete geometry of our sensations.

For example, causality tends to be seen in a false logic 'post hoc ergo propter hoc' context; "after this, therefore because of this", though many actualities emergence from multiple interfering relationships, as in 'self-organized criticality', for example, where avalanche's are triggered by the addition of sandgrains on the crest of a sandpile whose flanks are already at the critical angle of repose and the tensions which keep the sandgrains in place relate to the 'reciprocal disposition' or shape of their interfering spatial configuration. The more complete geometry of a non-euclidian space-time nature, is needed to deal with this, as is very effectively described by Henri Poincaré in 'The Relativity of Space' (from Science & Method), a copy of the full essay which can be found at

So, it appear to be wise to to review the basic aspects of wave motion, as observed and measured in real-world experiments, in the context of this ambiguity of direction and vantage point amongst the trio of 'transmitter, receiver and medium'.

Such a review of the energy and direction characteristics of waves has been nicely done by Claerbout and Dellinger in regard to the apparent energy paradox of the 'reciprocity' of 'transmitter and receiver' in seismology ('Eisner's reciprocity paradox and its resolution' by Jon Claerbout and Joe Dellinger"

The authors, including several others who were involved in the dialogue on this paradox, point out that the fact we can reciprocate (a) the 'source' of the wave disturbance and (b) the 'receiving' point where the disturbance is observed, and observe the same signal, runs counter to what we would expect from a material distribution point of view. That is, the materials surrounding the point of transmission ('source' of the wave disturbance) may have very different physical characteristics than those materials surrounding the point of reception, and thus the 'sampling' of the materials in this materially asymmetrical situation, by the expanding spherical wavefronts originating from the source, will be very different depending on the situational context of 'source' and 'receiver' relative to their containing medium. In spite of this counter-intuition (emanating from our cultural pre-suppositions), it turns out that the signal observed is the same in both cases (i.e. where the source is at a point (a) with receiver at point (b), and where the source is at (b) with receiver at (a)) in spite of 'material asymmetry'. Thus if a signal (amplitude of disturbance as a function of time) is observed from a remote source at a particular point of reception, ... and the geometry of the experiment is reciprocally reversed with the disturbing source now located where the point of observation was, and the observation made from the point where the source had located in the previous experiment, ... the observed signals will be identical (within the practical range of the source generator and measuring instruments).

Clearly, our cultural presuppositions would have us start with the 'causal' view here in that the 'source' at point (a) causes the disturbance whose 'effect' emerges 'later on' in 'linear time' at the point (b), and that this 'emergent' effect should not be reciprocal because of the asymmetries in the medium and our notion that the effect has been 'transported' by a linear-in-time series of causal transactions. On this basis, a theoretical objection was introduced which postulated that 'transmitter-receiver reciprocity' would violate the law of conservation of energy, and why this was not so, is the topic of the Claerbout-Dellinger discussion. What was implicit in the discussion surrounding the hypothesis that reciprocity violated the conservation of energy was that wave energy has an associated space-time 'phase' character which cannot be handled in the euclidian space convention where all we can associate with a 'point' in space is matter and linear motion, ... whereas we need also the 'complex' notion of possibility (potential energy) versus actuality (kinetic energy).

While kinetic energy is energy that is already 'committed' and manifest in our perceptual view, ... the associated potential energy is available for use as other committments to actuality are precipitated. The choice of coordinate systems will determine the manner in which we allocate energy to 'potential' versus 'kinetic' as is apparent in the baseball bat example cited earlier. In order to avoid this arbitrariness which is left up to the observer (whether he wants to associate with the 'subject' or the 'object'), we need to bring the participative role of 'space' out of its euclidian geometry- forced- retirement, and model phenomena in terms of 'transmitter-receiver-medium' relationships (non-euclidian space-time which includes space-time phase relationships), a modeling which requires the notion of complex analytical signal (as pointed out by Dr. Denis Gabor in 'The Theory of Communications'.

Meanwhile, the Claerbout-Dellinger discussion sets out to make the point that reciprocity does not violate Newtonian physics, and achieves this purpose admirably, ... leaving as 'unharvested inference', the reciprocal insights on the topic of relativity and quantum physics. That is, the authors make the point that in seismic wave propagation (a particular case of general wave phenomena), the relative distribution of potential energy and kinetic energy is a function of direction of travel and hence the azimuth of interference. When 'two seismic waves' which are travelling in opposite directions meet at a point, ... the resultant amplitude of the wave doubles in terms of pressure but goes to zero in terms of velocity, ... and when 'two seismic waves' travelling in the same direction meet at a point, ... the resultant amplitude of the wave doubles in terms of velocity but goes to zero in terms of pressure.

In other words, there is a reciprocal relationship between kinetic energy and potential energy in interfering waves, .. the more the kinetic energy adds, the more the potential energy cancels out (i.e. the more a system is purely mechanical, the less flexibility there is for evolving the system). What the observer tends to focus on in reciprocal situations (e.g. in seismic observations as in the game of pool) is on the linear-time based causal transactions or 'ontological transformation' (in terms of 'states'), whereas the induced space-time transformation or 'ontogenetic transformation' in the 'potential energy space' or 'possibility space' tends to be ignored. Yet the domain of 'pressure' associates with more degrees of freedom since it is azimuthal or 'radial' rather than linear, and one can see from the geometry of relativity, that only the 'thrust and parry' motion of convergence and divergence is relativistic while the notion of a 'directional velocity' depends for its existence upon euclidian space. In non-euclidian spherical space, there can only be 'convergence and divergence' in a primary sense while the notion of a 'velocity vector' is a derived abstraction which is innately incomplete. This incompleteness shows up in Claerbout and Dellinger's summary statement"

In the end, the lesson of Eisner's paradox is that ``energy equals amplitude squared'' is not true when more than one wave is involved. We fall into the habit of thinking it is always true because we so often see single plane waves, where it is true. "

The authors appear to fall into the same trap as has been pointed out by Dr. Denis Gabor points out in his 'Theory of Communications' in that the notion of 'a single plane wave' is a physically unrealizable abstraction since quantum physics and Heisenberg's uncertainty principle (Pauli's reformulation in terms of time and frequency) show that time and frequency interfere and that space-time cannot be split apart. Thus it is our abstraction to think in terms of 'single plane waves' which is non physically realizable. So we might say, instead, that "In the end, the lesson of Eisner's paradox is that the MANIFEST notion of the 'velocity' of 'a single plane wave' seen as a moving 'ontological thing' is a non-relativistic euclidian abstraction which is a special case of the LATENT notion of 'pressure' seen as a relativistic ontogenically transforming 'space'.

Our self-deception comes from the fact that in dealing with interference phenomena, what are in the primacy in nature are the underlying latent flows (space-time phase information), while we tend to focus on the manifest aspects (e.g. 'amplitude'). This 'forgetfulness' has been much discussed over the centuries, as noted in the above citings from Schroedinger, and was rife in 520 B.C. in the time of Heraclitus, Lao Tsu and the Buddha (see 'Yin-Masters and the Doctors of Yin' and it's epilogue at and at yinmas.htm). An intuitive example can be seen in visual form in footnote [3] along with the relevant writing from Heraclitus and from Lao Tsu's 'Tao Te Ching'.

In common experience terms, ... the quantitative, measurable aspects of physical phenomena represent the precipitation of the possible (potential energy of space) into the actual (kinetic energy of material entities). Since the potential energy is 'not yet committed to moving matter about', it is available for use in the 'precipitation' of a new 'causal actuality'. In this context, the 'Tao' represents a 'tuning in' to the potential energy configuration so that one's material being can further one's ontogeny by using the potential energy flows as a sailor who is sailing in turbulent weather by the 'jib of purpose', having had to abandon a mechanically pre-planned trajectory, might use the winds and currents. In other words, the option to relativistically 'go with the flow' discards the euclidian frame of 'actuality space' with its notions of material-causal trajectories, and engages directly with the 'convergence and divergence' of non-euclidian spherical space.

The difference between 'navigating' in the Tao of non-euclidian space versus 'controlling one's trajectory' in clockworks of euclidian space is easy to 'feel' on a personal level and can be associated with 'not getting ahead of the beat' or with the adage of 'on ne va plus vite que le violon'. Similarly, the skilled pool player essentially opts to 'go with the flow' of the reciprocal disposition (possibility space latencies) of the space-time configuration so as to tap these relativistic, interferential energy latencies in his furthering his ontogeny (the ontogeny of the ensemble of balls he represents), rather than mechanically engineer a succession of 'ontological states' of actuality.

The notion of directional phase, as given by the relationship between potential and kinetic energy, is nicely illustrated in the Claerbout-Dellinger discussion, even though it is oriented towards a mathematical and computational explanation of the energy paradox of reciprocity of 'transmitter and receiver' and does not get into relativity or quantum effects. 'The (mathematical) 'way out', as given by Dahlen and Odom, and presented in this contextual narrative by Claerbout and Dellinger, is by assuming that the notion of a 'point' in space; i.e. the euclidian space notion, ... is too constrained to explain this phenomena and we must instead visualize each point in space as a sphere. Thus the solution is developed by referencing the phenomenon to non-euclidian spherical space-time, ... although this is never specifically stated in the article. This narrative discussion along with insightful illustrations and wavefield simulator playouts, has an elegant simplicity which makes it very easy reading.

This 'complex bi-directional' (circular) quality of space-time, essential to a relativistic perception of wave phenomena, is denied to us by our choice of the euclidian space convention, a convention which implies linear, non-recurring 'time'. This point is not that 'time does recur', ... it is that the notion of 'time' on its own is a false and deceptive notion, as both relativity and quantum physics assert, well as our own sensory experience. The conceptualization in the questions on the examination, of around-the-world dominos etc., ... while 'thought experiments', are 'real' do-able examples (difficult to do, perhaps) which our senses and our inductive reasoning informs us are 'correct'; i.e. the fellow in question (1) who kicked off the progression of 'domino'ing' interference effects is not merely the 'causal agent' as he would appear on the flat planes of the euclidian space convention, ... he is instead, and at the same time, the 'transmitter' of a message and a 'receiver' of the message. The 'linear time' 'delay' which comes to our mind in association with the dominos 'taking the time' to go around the world, epends upon the linear notion of time. If we were to think, instead, in terms of space-time, ... we would say that the 'transmitter of the complex message', SIMULTANEOUSLY WITH HIS MOTION, infuses a message into the latent energy potentials of the medium which contains him (and from which he was born and into which he will return). The medium, seen as a 'possibility space' is thus immediately altered (since any motion has a reciprocal effect in terms of space-time interferential latencies, there can be no 'lag', ... 'lag' being a distinctly euclidian notion, ... a derived artifact of the choice of space convention with its ontological material-trajectory context as opposed to the ontogenetic space-time transformation from whence it derives.

The space conventions we choose to impose on our sensory experience; i.e. the choice of a non-euclidian perception of spherical space-time constituting a 'SIMULTANEOUS unity and plurality' of energy-flow-features, as contrasted with an euclidian perception constituting a 'SEQUENTIAL unity and plurality' of rectangular space and linear time of 'material things' and their 'causal transactions', ... reflects what is most convenient to the purpose of interpreting our experience (Poincaré). This point was apparently 'missed' by Aristotle in his interpretation of Heraclitus, as noted by history of philosophy scholars in their study of conceptual baton-passing amongst the Pre-Socratics and the Socratic philosophers [4]. That this is an interpretational choice, in its related implication of 'space-over-matter' is also suggested by Einstein and Infeld in their 'Evolution of Physics', as follows;

"We cannot build physics on the basis of the matter-concept alone [a concept which is in the primacy in the euclidian space convention]. But the division into matter and field is, after the recognition of the equivalence of mass and energy, something artificial and not clearly defined. Could we not reject the concept of matter and build a pure field physics? What impresses our senses as matter is really a great concentration of energy into a comparatively small space. We could regard matter as the regions in space where the field is extremely strong. In this way a new philosophical background could be created. Its final aim would be the explanation of all events in nature by structure laws valid always and everywhere. A thrown stone is, from this point of view, a changing field, where the states of greatest field intensity travel through space with the velocity of the stone. There would be no place, in our new physics, for both field and matter, field being the only reality. This new view is suggested by the great achievements of field physics, by our success in expressing the laws of electricity, magnetism, gravitation in the form of structure laws, and finally by the equivalence of mass and energy."

Gerhard Grössing (Austrian Institute for Nonlinear Studies, ... a group of three independent researchers) gives an evolutionary view of the development of the general theory of relativity in this same context of material-independent percept as Einstein is speaking to, ... benefitting from our considerable experience since 1938 (when Einstein and Infeld published the above paragraph). He incorporates this evolutionary view in his paper 'Die Information der Physik: Subjektal und Objektal' at, and suggests, as Einstein also did, that there can be no path to general theory of relativity effects from models formulated in euclidian geometry. In fact, this same inference also follows from Schroedinger's earlier cited comments on the interpretation of 'transmitter-receiver reciprocity', in that there is no place in euclidian space for the 'observer to sit' if he does not sit on a material object; i.e. on a material 'subject' or 'object', since space is seen in the simple terms of material dynamics within a non-participating void in the euclidian space convention. Grössing puts this into perspective as follows;

"Mach and Einstein pointed out the assymmetry in Newton's physics: while space and time effected matter, ... the reverse was not true. The special theory of relativity succeeded in taking the first step towards integration of this reversal in the description of natural phenonemena: measurements for space and time are no longer absolute, but dependent upon the 'point of view' of the observer, and therefore on [dependent upon] a reference system bound to the presence of matter.

In the general theory of relativity, this [material] presence at last becomes the central theme. The logic of evolution (i.e.; symmetry - asymmetry - integration = meta-symmetry, and so on) also characterizes the development of our understanding of nature (i) subjective schema in Aristotelian physics and alchemy [the view of Protagorus]; (ii) asymmetry between space and time on the one hand, and matter on the other in Newtonian physics; (iii) Integration by means of RECIPROCAL influencing of space-time and matter in the general theory of relativity.

The way is opened, with the development of the causality concept in describing natural phenomena, which leads to the notion of integrative, circular co-dependency, not only in the general theory of relativity but also in the quantum theory; (i) subjective 'explanation' through the personification of nature (mother, spirits, god etc.); (ii) 'linear' objective description of nature (for example, observer-independent planetary orbits in Newtonian physics); (iii) circular, 'wholenesslike' causality (for example the connection between space-time and matter in the general theory of relativity)

Only with circular causality does an essential characteristic of quantum theory become understandable: now we can no longer (in the metaphorically extended sense of 'linear') describe an 'object', unless we put the hermeneutical circle between 'object' and 'observational instrument' in the center of our awareness (whereby the detached independence of these two components shall be understood merely as a thought-prop)."
* * *

What is 'bottlenecking' us of the western philosophical outlook from pulling out of the dive into discord and getting back 'in the Tao', ... the way from which a harmonious future emerges, is our insistence on putting the manifest in a primacy over the mystery (i.e. putting our quantitave observation of kinetic energy into a primacy over our implicit observation of potential energy patterns.). While the eastern and indigenous philosophy opts for the 'simultaneous' rather than 'sequential' unity and plurality in its relativistic declaration; "there is no path to harmony, ... harmony is the path.", we in the west persist in our discounting of the 'ordering power' of potential energies. In other words, one has to 'live' this possibility space of co-resonance and coevolution, rather than developing strategic plans to 'get to it'. The linear euclidian evocation of 'history, plans and progress', which are fine and dandy if relegated to a supportive role, ... when put into the primacy, ...doom all chance of coming into a coresonant, coevolutionary (with the potential energies) 'transmitter-receiver-medium' mode (i.e. container-constituent-coevolution) mode.

This needed 'shift' in perception is, more precisely, a 'reciprocation', ... or 're-reciprocation' since the perceptual mode of the child is relativistic and it is by the imposition of cultural teachings that our culture inverts it to the euclidian and non-relativistic perceptual mode of 'material things', 'causal transactions' and 'linear time histories', ... a perception and inquiry mode which is consistent in itself but which is radically 'incomplete' with respect to the interferential way in which the world works.

It is for this reason that this pre-amble to this 'PACS 101- examination- results- assessment- guide' has provided pointers back to basic physics principles (the energy principles in wave motion) as a reminder of our having passed by some important questions which we must revisit in order to see why this shift in perception is needed and to achieve the necessary reconciliation between natural phenomena and our natural powers of perception and understanding.

As a child, when we hear a nursery rhyme such as;

Twinkle, twinkle, little star,
How I wonder what you are.
Up above the world so high,
Like a diamond in the sky,
Twinkle, twinkle, little star,
How I wonder what you are.

... we are thinking in terms of a 'pet star', ... and wondering about the nature of the how something relates to us, ... rather than a sterile 'what you are' in terms which address such things as chemical composition and 'a brief (linear) history of time'. It is the awesome discovering of life, rather than analytics per se, which captivates us as children, ... we bathe in the wonder of how things 'are' And though we may use the verb 'to be', the grammar of existence, ontology, ... our minds are scarcely constrained to what is known to be (actuality), ... and our fancies run free and high in the domain of possibility. When we start school, however, it is likely that the Thomas Gradgrinds of Charles Dickens 'Hard Times' will be there to help 'set us straight'.

"No little Gradgrind had ever seen a face in the moon; it was up in the moon before it could speak distinctly. No little Gradgrind had ever learnt the silly jingle, Twinkle, twinkle, little star; how I wonder what you are! No little Gradgrind had ever known wonder on the subject, each little Gradgrind having at five years old dissected the Great Bear like a Professor Owen, and driven Charles's Wain like a locomotive engine-driver. No little Gradgrind had ever associated a cow in a field with that famous cow with the crumpled horn who tossed the dog who worried the cat who killed the rat who ate the malt, or with that yet more famous cow who swallowed Tom Thumb: it had never heard of those celebrities, and had only been introduced to a cow as a graminivorous ruminating quadruped with several stomachs."

The stars sometimes seem to have less twinkle in the 'year of our Lord' 2000, ... and it does not appear to be Jesus or Mohammed or Moses who has taken the twinkle out of it, since all of the aforementioned clearly had the ability to 'tune-in' to the Tao, as the theory of relativity and quantum physics would say that we must. But as we drive our SRV up into the mountains, above the pollution, to observatories financed by the logging of the forests which once lay below them, ... in our focus on ontology, on the state of things, ... on the historic state of the cosmos and the various states which mark the progress of the universe from the supposed 'big bang' till today, ... something seems to be missing. Even as our factual knowledge builds, there is an emptiness in this ontological pursuit which grows starker and colder with each linear passage of day and night.

The 'circular causality' of curved space-time as contrasted with the linear causality of euclidian space convention, ... a reciprocal energy relationship which is at the very base of nature according to modern physics, ... is caricatured in the first question of the exam and is emerges in various different forms in all questions.

With this pre-amble review of the basic physics of motion in hand, the 'instructor' may be in a better position to assimilate the body of the 'key' which follows, and to reciprocate with himself as 'student' to discover whether he is the receiving same signal at both locations (in both roles), and not coming into contradiction with himself as a result of his own choice of interpretational simplifications [5]


Key to Exam: (in dipolar non-euclidian space, since teaching and learning are simultaneous reciprocals, ... this goes to all participants in the exam process).

Physics and Curved Space 101

March 23, 2000

Instructor's Key to PACS 101 Entrance Exam

The key inference in the questions on this entrance exam is that in all of the questions, the impression is given, at the beginning of the problem formulation, that the problem is 'out there' and that it can be fully described as being 'out there'.

Our unquestioned use, in the West, of the euclidian space assumption, by excluding the observer from participation, splits apart space and time and gives rise to the bivalent notion that things either exist or do not, ... thereby removing the innate dipolar aspect of the immersing space-time of our experiential reality, which brings things into connection in an unbounded continuum. That is, the nature of our observational experience is an 'immersed one' in which 'we thrust' while our containing world simultaneously and reciprocally 'parries' and in which 'we inquire' and our containing world 'informs', ... as is very insightfully described by Henri Poincaré in 'The Relativity of Space' (from 'Science & Method', a copy of the full essay can be found at

The 'immersed' dipolar view of space-time, ... and to characterize the immersed world of our real-life experience including ourselves in our experience as well as the containing 'other', we can no longer split space and time apart but must see it is a space-time continuum we are immersed in, ... is a 'complex' (real + imaginary) informational view since we are comprehending, simultaneously, both what is 'out there', and what is 'in here'. In order to perceive in the full dimensional 'immersed mode' of our experience, then, we must utilize a dipolar space convention such as spherical space [1]. As Einstein explains, what we see happening on the dipolar surface of non-euclidian space will not be consistent with what we see happening on the monopolar surfaces of euclidian space; "Further, the spherical surface is a non-Euclidean continuum of two directions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane".

In other words, as the pool player knows from his mental exercising of curved space thinking in the game of pool, ... in our real-life sensory experience, SIMULTANEOUS with any 'thrusting motion' is a parrying by the containing environment; i.e. the parry and thrust are RECIPROCALLY related and space-time is, in this manner, 'dipolar' or 'bi-directional', ... the 'shape' of space 'transforming' simultaneously with our 'thrusting'. In other words, the non-euclidian perception of space is of an ontogenetic transformational space whose contained and dependent reciprocal view of space is the euclidian one of ontological transactions in linear (euclidian) time (i.e. a progression of ontological 'states').

Ontogeny is 'felt' as a 'living presence' as in the poetic myth of native americans [2], the 'sacred Here and Now' as inferred in the following lines from the story-telling of the indians of the pacific northwest, captured poetically by David Wagoner;

. . . . . . . . . . . . . . . .Wherever you are is called Here.

. . . . . . . . . . . And you must treat it as a powerful stranger.

. . . . . . . . . . . Must ask permission to know it and be known.

. . . . . . . . . . . . . .The forest breathes. Listen. It answers.

. . . . . . . . . . . . . . . .I have made this place around you.

. . . . . . . . . . . . . .If you leave it you may come back again.

. . . . . . . . . . . . . . . . . . . . . . .saying Here.

Thus, IT IS MERELY A PSYCHOLOGICAL PREFERENCE OF THE WESTERN CULTURE, rather than a characteristic of our world, to abstractly PURGE the dipolar quality of our world in our perception and inquiry by our choice of the simplistic euclidian space convention in our educational system and in our science, ... a point which has been made via very effectively through caricature by Charles Dickens in 'Hard Times'.

To paraphrase Poincaré, however, 'we can impose this psychological preference on our culture, ... culture being constituted by a suite of psychological and methodological preferences of our own making, ... but we cannot impose it on nature.' Thus the domino's of question (1), while perceived and mathematically formulated in Newton's laws as being 'out there', ... will persist in being 'in here' (immersed along with us within a unified space-time), ...the nomads, while seen and mathematically formulated as heading off in euclidian directions, ... will persist in self-referentially 'bumping into themselves' while 'coming and going' in a dipolar context, the moreso as the population on the surface of the earth builds. While the scholar sees his containing world as informing him remotely and sterilely (action at a distance) from 'out there', ... the world will persist in tapping him on the shoulder and saying, 'you are swimming in me, brother'. Similarly, while the sailing boat captain is trying to make sense of things on his radar screen in a euclidian 'out there' mathematical context, ... he finally comes to the realization that it is this very nature of 'immersion', ... of a dipolar 'thrusting and parrying' within a containing world, which is the essence of 'relativity' upon which he relies for an understanding of his own behavior, ... that his own behavior can never be strictly 'his own' because it is simultaneously and reciprocally included in the relational geometry of his containing environment.

In the case of the homicide investigation, while the DA insists on imposing the euclidian notion of there being an 'objective reality in its own right' 'out there', which contains the 'truth' in the form of material-causal fact, ... the defense attorney sees those same sources of information (the citizen's testimony) as scatterings from the 'larger reality' of the space-time continuum in which they are all immersed including the victim before and after and during the murder. The defense attorney thus 'images' the information by a backward (reverse-ontogenetic) continuation (or better, an 'inward continuation' from 'out there' as seen in voyeur mode to 'in here' in the immersed space-time continuum) of the data seen as wavefield data.

This 'inward continuation', ... into the layers of experience of the collective deposited like layers of sediment on the earth, ... 're-connects' the past with the present in a 'seamless' or 'unbounded space-time' manner. In other words, as in question (1) which caricatures 'circular causality', we can reciprocate the space-time positioning of the 'observer-as-transmitter' (subject) with the 'observer-as-receiver' (object) at either 'end' of an ontogenetic cycle. In this mode of perception, the observer associates with the 'containing space' or 'field' or the 'hermeneutic circle' which spirals forth ontogeny, and thus avoids a dependency on a material vantage point, a dependency which, as Gerhard Grössing points out, remained in physics up through the development of the special theory of relativity and was not fully removed until the formulaton of the general theory of relativity.

This non-euclidian mode of perception does not have to be seen in terms of 'mysticism', ... as it would be seen when one is standing on the ladder of euclidian space. Since euclidian space is bivalent, any presence which is 'not a thing' could only be a 'ghost' or 'spirit'. Whereas, in the non-euclidian view, space-time is a continuum and one can conceive of an unboundedness between earlier and later ontogenies; i.e. one need not think in terms of a 'hard boundary' between the present and the past. In other words, our experiencing of reality can be seen in terms of 'ontogenetic time' or 'evolutionary time', as in the native americal story of 'Here', ... rather than thinking of our ongoing experience in terms of 'linear', 'historic' time. The ontogenetic conception of time assumes the primacy of our containing environment rather than our transient 'subjective' perspective, ... a notion of time wherein our containing environment, being unbounded, includes our ancestors etc. and in its continuing ontogeny (metamorphosis or transformation), continues to enfold the ontogenetic 'old' within the ontogenetic 'new' but in terms of infolded potentials rather than in the static iconic imagery of independent bodies and independent personalities as in the material-kinetic-actuality phase of their ontogeny. It is in these manner they they can be 'seamlessly included' in the space-time continuum, ... the implicit 'possibility space' of the defense attorney, ... without having to be seen as 'ghosts', as is the artifact of utilizing the constrained, bivalent, euclidian space convention.

Finally, in the Charles Dickens story 'Hard Times', ... the cultural attempt to constrain the view of reality to euclidian space and linear time is caricatured via the personage of Thomas Gradgrind, whose students are in various stages of capitulating to the pressures of this cultural mind-draw.

Today, as McLuhan has pointed out, we have become more sophisticated in our methods, and perhaps less aware, ... so that it has become that the 'medium is the message', and when youth sees Nobel prizes being awarded to the quark-seeking physicists, ... this induces them to believe that the secret to understanding the way the world works is to understand the 'euclidian world 'out there''. In other words, there is an imbalance in the culture in focusing on the manifest rather than on the latent which is the sourcing precipitate of the manifest, ... the many Thomas Gradgrinds of the past, in the western culture have 'done their job', putting us into our current 'maintenance and intensification' of the 'manifest-in-the-primacy' mode.

In the dipolar world of our real-life experience (as contrasted with the reduced world of our euclidian abstraction), teaching is at the same time, learning, ... and instead of Gradgrind's one-way spiels to Sissy, ... he might well have 'thrusted and parried' on a peer or 'reciprocal' basis with Sissy, ... flipping the transmitter-receiver geometry so as to keep the reciprocity in tune, ... learning more about 'what it is like' to 'belong to the horse-riding' in the process, and transforming the tension which obviously comes with his ethic of mind-control into energizing harmony.

Thus, the last question, which brings in a discussion of personal experience and reconciles it with the dual modes of perception (euclidian/voyeur and 'non-euclidian / immersed), can be the most fruitful part of the continuing, shared learing experience catalyzed by the PACS 101 'exam', if it is done in relativistic, reciprocal mode.

While it is implicit in all of the above geometrical comments, the shift in perception from non-relativistic rectangular space and linear time to relativistic spherical space-time does not 'take anything away' from the benefits of the former realm of Newtonian physics with its convenient bivalent precision, ... but instead, extends our post-conceptual perception in a way which includes and extends the euclidian view, and makes use of holographic (immersed) and whole-and-part harmonic aspects of our sensory perceptions.

* * *

[1] "For philosophy, the real difficulty lies in the spatial and temporal multiplicity of observing and thinking individuals. If all events took place in one consciousness, the whole situation would be extremely simple There would be something given, a simple datum, and this, however otherwise constituted, could scarcely present us with a difficulty of such magnitude as the one we do, in fact, have on our hands. I do not think that this difficulty can be logically resolved, by consistent thought within our intellects. But it is quite easy to express the solution in words, thus: the plurality that we perceive is only an appearance; it is not real. ... Yet each of us has the indisputable impression that the sum total of his own experience and memory forms a unit, quite distinct from any other person. He refers to it as 'I'. 'What is this 'I''? If you will analyze it closely, you will, I think, find that it is just a little bit more than a collection of single data (experiences and memories), namely, the canvas upon which they are collected."

[2] Einstein, Albert, 'Geometry and Experience', ... An expanded form of an address to the Prussian Academy of Sciences in Berlin on January 27th, 1921.

Can we picture to ourselves a three-dimensional universe which is finite, yet unbounded?

The usual answer to this question is ``No,'' but that is not the right answer. The purpose of the following remarks is to show that the answer should be ``Yes.'' I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental image to which, with some practice, we shall soon grow accustomed.

First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To 'visualise' a theory, or bring it home to one's mind, therefore means to give a representation to that abundance of experiences for which the theory supplies the schematic arrangement. In the present case we have to ask ourselves how we can represent that relation of solid bodies with respect to their reciprocal disposition (contact) which corresponds to the theory of a finite universe. There is really nothing new in what I have to say about this; but innumerable questions addressed to me prove that the requirements of those who thirst for knowledge of these matters have not yet been completely satisfied. So, will the initiated please pardon me, if part of what I shall bring forward has long been known?

What do we wish to express when we say that our space is infinite? Nothing more than that we might lay any number whatever of bodies of equal sizes side by side without ever filling space. Suppose that we are provided with a great many wooden cubes all of the same size. In accordance with Euclidean geometry we can place them above, beside, and behind one another so as to fill a part of space of any dimensions; but this construction would never be finished; we could go on adding more and more cubes without ever finding that there was no more room. That is what we wish to express when we say that space is infinite. It would be better to say that space is infinite in relation to practically-rigid bodies, assuming that the laws of disposition for these bodies are given by Euclidean geometry.

Another example of an infinite continuum is the plane. On a plane surface we may lay squares of cardboard so that each side of any square has the side of another square adjacent to it. The construction is never finished; we can always go on laying squares if their laws of disposition correspond to those of plane figures of Euclidean geometry. The plane is therefore infinite in relation to the cardboard squares. Accordingly we say that the plane is an infinite continuum of two dimensions, and space an infinite continuum of three dimensions. What is here meant by the number of dimensions, I think I may assume to be known.

Now we take an example of a two-dimensional continuum which is finite, but unbounded. We imagine the surface of a large globe and a quantity of small paper discs, all of the same size. We place one of the discs anywhere on the surface of the globe. If we move the disc about, anywhere we like, on the surface of the globe, we do not come upon a limit or boundary anywhere on the journey. Therefore we say that the spherical surface of the globe is an unbounded continuum. Moreover, the spherical surface is a finite continuum. For if we stick the paper discs on the globe, so that no disc overlaps another, the surface of the globe will finally become so full that there is no room for another disc. This simply means that the spherical surface of the globe is finite in relation to the paper discs. Further, the spherical surface is a non-Euclidean continuum of two directions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane.

[3] Visualizing curved space cannot be achieved voluntarily, ... the mind must be relaxed. As Lao Tsu says;

"Ever desireless, one can see the mystery. ... Ever desiring, one can see the manifestations. These two spring from the same source but differ in name; this appears as darkness. Darkness within darkness The gate to all mystery".

Heraclitus makes the same point using the reciprocal geometries of sleeping and wakefulness;

"Of the Logos which is as I describe it men always prove to be uncomprehending, both before they have heard it and when once they have heard it. For although all things happen according to this Logos men are like people of no experience, even when they experience such words and deeds as I explain, when I distinguish each thing according to its constitution and declare how it is; but the rest of men fail to notice what they do after they wake up just as they forget what they do when asleep." ... "Therefore it is necessary to follow the [yin] common; but although the Logos is common the many live as though they had a private understanding." ... "And as the same thing there exists in us living and dead and the waking and the sleeping and young and old; for these things having changed round are those, and those having changed round are these."

The subtle situation being pointed to by both of these contemporary philosophers is that the manifest represents only that fraction of the invisible potential energies which have been precipitated into actuality, ... and precisely how the manifest and material can 'move' with respect to the potential energy distribution will modulate the emergent form of what is being precipitated into actuality. Thus, if one 'tunes in' the the potentials which underly manifest actuality, one can move so as to be in harmony with them, rather than 'clashing with them'. This ability to move in harmony with the energy potentials is the 'lost ability' of the western culture which is leading to the engendering of massive discord in our modern society, and is being amplified by the mechanical, unconscious manner we use technology. The problem is not 'technology per se', since technology can also be used to help us move in harmony with the potential energies in which we are immersed constituent-participants.

This following image which includes interference effects demonstrates the problem of visualizing patterns which go beyond the bivalent 'it exists or it does not', ... into the inclusionary 'yin/yang' domain of 'it exists and it does not'.

The Tao that can be told is not the eternal Tao.

The name that can be named is not the eternal name.

The nameless is the beginning of heaven and earth.

The named is the mother of ten thousand things.

Ever desireless, one can see the mystery.

Ever desiring, one can see the manifestations.

These two spring from the same source but differ in name;

this appears as darkness.

Darkness within darkness

The gate to all mystery

--- Lao Tsu, 'Tao Te Ching'

* * *

[4] The following quote from 'The Presocratic Philosophers', Second Edition, G.S. Kirk, J. E. Raven and M. Schofield, Cambridge University Press, ... makes clear that the subtlety of the curved space-time view of the cosmos, which cannot be conveyed directly in our euclidian 'thing-oriented' language, ... may well have been lost 'in translation' between Heraclitus and Aristotle, ... just as Darwin's similar intent may be lost in Sipper's 'recasting' of passages from Darwin 'within the modern evolutionary computation framework'. The omission shifts us from the domain of SIMULTANEOUS HARMONY also noted by Johannes Kepler in connection with the system of sun and planets and dropped out by Newton, to the domain of SEQUENTIAL TIME PERIODS, .. that is, the 'recasting' takes us from a curved, relativistic space-time continuum, to a rectangular (non-self-referential) non-relativistic disjoint view of independent things populating an inert and non-participating void, ... i.e. material existence out of the context of a unified whole-and-part harmony oriented space-time container. The referenced quote is as follows;

"Plato ('Sophist' 242D, DK 22 A.10) clearly distinguished between Heraclitus' SIMULTANEOUS unity and plurality of the cosmos and Empedocles' separate PERIODS of Love and Strife. At the same time, they are mentioned together as both alike in believing in the unity and plurality of the cosmos; and Aristotle's coupling of the two might conceivably have been motivated by the Platonic comparison, the important distinction between them being overlooked. See also Guthrie, 'History of Greek Philosophy',HGP1, 455f, and 458, with further references, and D. WIggins, 'Heraclitus' conceptions of flux, etc.' in Language and Logos, ed. Scholfield and Nussbaum (Cambridge, 1982), 1ff."

[5] Johannes Kepler, 'Epitome of Copernican Astronomy':

"As regards the academies, they are established in order to regulate the studies of the pupils and are concerned not to have the program of teaching change very often: in such places, because it is a question of the progress of the students, it frequently happens that the things which have to be chosen are not those which are most true but those which are most easy. And by that division in things which makes different people form different judgements, it so happens that certain people are in error contrary to their own opinion."

While many scientists view Kepler's work as being fully deducible from Newtonian physics, ... that is scarcely the case. Kepler's principle of space being in the primacy over matter (i.e. that the harmonic behavior of the system is beyond deduction from the behaviors of the components; the basic definition of a 'complex system') and the role of the ordering influence of the potential energies in space went well beyond Newtonian physics and presaged both relativity and quantum mechanics.

[6] Einstein and Infeld, 'The Evolution of Physics: From early concepts to relativity and quanta';

"To use a comparison, we could say that creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting point and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles of our adventurous way up."

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