# Special Relativity and Perception

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The Special Theory of Relativity (STR) holds sway as a theory of time due to its apparently successful predictive structure regarding time-related phenomena such as the increased life spans of mesons or retarded clocks on jets circling the globe, and due to the relativization of simultaneity intrinsic to this theoretical structure. Yet the very structure of the theory demands that such very real, physical effects be construed as non-ontological. The scope and depth of this contradiction is explored and, if these time-changes are indeed viewed as ontological effects within STR, an additional problem for the theory is introduced in the context of perception. The origins of this confused situation arise in an inadequate concept of time and in the failure to acknowledge the reality of simultaneous causal flows.

Special Relativity and Perception 3/21/2007 Special Relativity and Perception Abstract The Special Theory of Relativity (STR) holds sway as a theory of time due to its apparently successful predictive structure regarding time-related phenomena such as the increased life spans of mesons or retarded clocks on jets circling the globe, and due to the relativization of simultaneity intrinsic to this theoretical structure. Yet the very structure of the theory demands that such very real, physical effects be construed as non-ontological. The scope and depth of this contradiction is explored and, if these time-changes are indeed viewed as ontological effects within STR, an additional problem for the theory is introduced in the context of perception. The origins of this confused situation arise in an inadequate concept of time and in the failure to acknowledge the reality of simultaneous causal flows. Special Relativity and Perception Physicists mislead us when they say there is no simultaneity. When the camera pans to the heroine tied to the rails and then to the hero rushing to the rescue on his horse – these events are simultaneous. – James J. Gibson 1 1.0 Introduction In 1922, Henri Bergson engaged with Einstein in a spontaneous discussion under the auspices of the Societe de Philosophie (Gunter, 1969, pp. 123-135). Acquiescing to an invitation to make an impromptu comment, Bergson noted, in the course of about 15 minutes of remarks, that the concept of universal time arises from our own “proper” or experienced time in our immediate environment. He drew attention to the concept of the simultaneity of flows. Our experience of simultaneity, he observed, arises from our experience of multiple flows within a single flow, whether it be (using my own examples) multiple race cars racing side by side down the track, multiple melody lines within a single flow of a symphony, multiple musicians playing on the symphony stage, multiple women cooking in the kitchen, multiple family members eating at the table, a boat floating down a river with geese flying overhead, or Gibson’s hero coming to the rescue of a struggling heroine. This experience of multiple simultaneous flows within a single experienced flow is generalized to other perceivers, ultimately, he argued, to our concept of a universal flow of time. Further, this intuitive notion of simultaneity supports the very concept of relating an event to a specific time instant on a clock (as for example where an observer must relate a lightning bolt and a clock hand at 3PM as occurring simultaneously). [Gibson, the highly respected theorist of perception, made this statement in a talk at the University of Minnesota in 1975. He had read a paper by the author the previous day which at the time was satisfied with Capek’s (1966) view that relativity adequately preserves the “becoming” of the universe, and which attempted to fold in psychological time as part of the relativistic structure of time. Gibson, however, appeared to have none of this. He is in effect alluding to the concept of the simultaneity of flows of time, a subject discussed at length by Bergson in Duration and Simultaneity (1922/1965) in his analysis of relativity.] Now, he noted, a microbe observer could say to our observer that these two events (clock hand at 3PM, lightning bolt) are not “neighboring” events at all, but are vastly distant, and would not be simultaneous to a moving microbe observer. Nevertheless, to paraphrase his conclusion, he felt that this intuitive simultaneity must underlie the possibility of any time measurement at all in relativity, and was in fact the basis for reconciling the two notions. Einstein’s reply is worthy of complete quote: The question is therefore posed as follows: is the time of the philosopher the same as that of the physicist? The time of the philosopher is both physical and psychological at once; now, physical time can be derived from consciousness. Originally individuals have the notion of simultaneity of perception; they can hence understand each other and agree about certain things they perceive; this is a first step towards objective reality. But there are objective events independent of individuals, and from the simultaneity of perceptions one passes to that of events themselves. In fact, that simultaneity led for a long time to no contradiction [is] due to the high propagational velocity of light. The concept of simultaneity therefore passed from perceptions to objects. To deduce a temporal order in events from this is but a short step, and instinct accomplished it. But nothing in our minds permits us to conclude to the simultaneity of events, for the latter are only mental constructions, logical beings. Hence there is no philosophers time; there is only a psychological time different from that of the physicist. (Gunter, 1969, p. 133) This was the totality of the interchange. And so it rests. Bergson’s position is, to say the least, a minority opinion. Einstein’s “time of the physicist” has been the accepted criterion of reality. The simultaneity of perception is considered, at best, suspect, and in practice, invalid. Stein (1991) essentially reprised and expanded Einstein’s argument, attempting to explain ongoing misconceptions of relativity, as he saw them, in terms of our continued naïve belief in the perception of simultaneous events – an illusion based on the high velocity of light. Thus, he argued in essence, the naïve or intuitive simultaneity that perception provides is founded upon the “fleeting motions” of “masses of elements” in the brain, all subject to the limitation of communication via the velocity of light, and implying therefore that at a small enough scale of time, perceptive simultaneity would break down. 2 Special Relativity and Perception This is, in fact, a curious state of affairs. Let us allow that Stein expresses Einstein’s view in somewhat extended form. Then this exposition of relativity and its inherent, relativized simultaneity of events entails, or at least places a fundamental constraint upon, a theory of perception (cf. Hagan & Hirafuji, 2001). Stein is assuming a model, admittedly sketchy, of the processes in the brain underlying perception. Perception, however, is simply part and parcel of what Chalmers (1995) dubbed the “hard problem,” i.e., the explanation of conscious experience, the “world-out-there” in depth, in volume, in quality. As the problem fundamentally involves our consciousness, the problem surely cannot be divorced from our model of time. It is a problem become ever more acute, far more so than realized in Einstein’s time, and even just becoming so in Stein’s time. Neither Stein nor Einstein could claim to have a solution. We can ask an interesting question: what if the solution to the hard problem intrinsically relies on the simultaneity of events? Bergson had such a solution. As this has been discussed it extensively elsewhere (Xxxxxx, 2000, 2001, 2002, 2004a, 2004b, 2006a, 2006b, 2007), I will not repeat things here. Sufficient it is to say that this theory contains a prediction in the sphere of perception/action that contradicts the Special Theory, though it is a contradiction if and only if physics holds that the relativization of simultaneity is a real property of time, i.e., a real, ontological property of the matter-field and its temporal evolution. But this is the problem. 1.1 The Problematic Status of Relativistic Effects The relativization of simultaneity is necessarily being considered a property of time when physics claims that STR is gainfully used in the explanation of “real asymmetric effects.” By “real effect” is meant a change that is considered a real property of matter, not simply something inherent in the measurement process. In the common parable of the twin paradox, were the stationary twin to have a gray beard and wrinkles relative to his pink cheeked, youthful brother just returned from his rocket travel, this differential aging is a real property of matter. If the clock carried in a jet around the globe shows definite time lag when later compared to its 3 Special Relativity and Perception stationary counterpart, this differential time reading (physically expressed in the clocks) is an inescapably real property of matter. By “asymmetric” is meant that the symmetry due to the reciprocity of reference systems is broken. The observers on the ground do not advance to the jet just landed on the runway and announce that the plane was in fact stationary and that the earth was moving at extreme speed relative to the jet, and thus it is their clocks – those of earth-bound observers – that have been retarded. The stationary twin does not announce that it was the rocket that was at rest or stationary and that the earth moved away and back at extreme speed, and thus it is he, the obviously pink-cheeked rocket-twin, that has the beard and wrinkles. The use of STR in explanation of this class of phenomena – the increased lifespan of mesons falling through the atmosphere, the retardation of the jet-carried clock – is the lynchpin upholding STR as a successful, useful, predictive theory of time. In this explanatory usage, it relies intrinsically on the relativity of simultaneity, or equivalently, on the concept that there is no absolute simultaneity of events, or equivalently again, that the relativity of simultaneity is a real property of the time-evolution of the universe. Employing STR in this fashion also implies that an intrinsic feature of relativity, namely the reciprocity of reference systems, is a feature that can be ignored. This was not so initially. It was clearly not so in regards to the Michelson-Morley experiment and the apparent contraction of the length of the apparatus arm which lay parallel to the hypothetical ether current. Let us remember that Lorentz, some years before Einstein’s 1905 publication, originally proposed that the contraction was indeed real. He advanced ether-based, electro-dynamical arguments in support of equations he developed for the fore-shortening of the apparatus-arm in the direction of motion as a function of velocity. His equations expressed the degree of contraction and accounted for the equality of the light-ray travel-times through both arms of the apparatus. The Lorentz equations had the exact form that Einstein’s eventually would take. But the contraction was unappealing to physics; it was rejected, or at least never accepted. Why was Einstein’s length “contraction,” using precisely the same equations, accepted? Because the length became a space-time invariant. 4 Special Relativity and Perception How does the length become such an invariant? By being subject to the reciprocal transformations of two observers in two different reference systems, either of which can consider himself at rest and the other in motion. Einstein’s perceived advance was to embed the Lorentz transformations within this symmetric, reciprocal framework, together with postulating the invariance of the velocity of light. Indeed, Einstein wished that his theory had been named “Invariantentheorie” rather than “relativity” (Horton, 2000). In Special Relativity, the Lorentz equations cannot be applied asymmetrically. They have no meaning with respect to just one observer considered the only such system at rest, and the other the only such system in motion. There is no invariance with just one observer. Some form of transformation is required for an invariant. This symmetric system is required, and within it, either observer can declare himself at rest, and then attribute the length contraction to the other (in motion), adjusting the other’s space and time units to preserve the invariance of the velocity of light. Therefore as A. P. French (1968) states in his textbook on relativity, neither the length contractions nor the time-dilations are a real property of matter, they are a measurement effect, “something inherent in the measurement process” (p. 114). Physics resolutely holds that the contraction of length in the Michelson-Morley experiment is not ontological; it is a measurement effect. On the surface, then, it would appear that the ascription of ontological changes is limited, inconsistently, despite statements such as those of French, to changes in time, for the physical difference in the jet-carried and earth-bound clocks is inescapable; reciprocity will not make it go away. And despite French, the meson effect can be seen as very real. Were the increased life of the meson just enough to let it travel an extra bit of distance to trip an electric switch which rings a bell, the ring is inescapable; reciprocity will not undo it. But for length this is different. It is stated quite clearly that length, for example, that of the arms of the Michelson-Morley apparatus, is a space-time invariant, therefore respecting reciprocity and the system of transformations in which the Lorentz equations must be embedded to be used at all. Another paradox, the “pole-barn,” is unhesitatingly applied to this teaching 5 Special Relativity and Perception point. In this paradox, we have a longish, say, telephone pole. In its resting state, it is too long to fit into a certain barn. However, when the pole is launched into motion at a velocity near the speed of light and flies through the barn, there is a period where the pole, due its length contraction, actually fits into the barn. This paradox is used as a parable for illustrating that we should not consider these real effects. It is unhesitatingly pointed out that the barn could be conceived to be in motion, and therefore the barn will contract. Now the pole does not fit. So the length contractions are not real, or in philosophical terms, they have no ontological status. This nicely holds the line with the interpretation of the Michelson-Morley experiment. One could ask something however. Just like the jet-carried clock experiment, why not perform a pole-barn experiment? We could rig a mini barn-like apparatus with front-end and back-end doors that open and shut at great speed, or some analogy. The device would capture a mini-pole moving at high velocity precisely when it fits inside due to its length contraction. If we can so unhesitatingly predict that the jet-carried clock will slow down, why would we not predict that the mini-pole would contract and be trapped in the mini-barn? Distance changes, in fact, are already intrinsic within the meson, twins and jet-clock cases, we simply choose to focus only on the time changes. But this would be admitting that the length contraction too is a very real effect. It would signal the end of any pretense of usage of the reciprocity of reference systems aspect of the Special Theory with the non-ontological status of effects that the theory inherently entails. Unspeakably worse, it might move us back to Lorentz. At present, physics deploys the reciprocity feature of STR for length contractions, and unhesitatingly dumps the feature for time-expansions. It therefore rejects the relativization of simultaneity as real and simultaneously (or not simultaneously?) accepts the relativization of simultaneity as real. At one level, the cause of this chaos is physics’ attempt to appropriate a theory, namely STR, for the explanation of very real phenomena for which the theory simply is not suited as an explanatory vehicle. At another level, the origins of this usage already reside in Einstein’s 1905 paper. At a deeper level yet, there is a fundamentally inadequate concept of time. 6 Special Relativity and Perception 2.0 Bergson and Time Let us begin with the heart of the difference between Bergson and Einstein. The “microbes” in Bergson’s comments are an index, in essence an index to the process of thought leading to the “objective” that Einstein must take to its logical conclusion. Bergson, in introducing them, had asked just what is the concept of “proximity” or “neighboring events” used in relativity to relate clocks to events? A microbe consciousness questions whether the clock and lightning bolt of the system of some observer are “neighboring.” A micro-microbe questions the microbe’s judgment of what is “neighboring;” a micro-micro-microbe does the same to the micro-microbe, and so on. Logically, we are forced to take this to its conclusion. There can be no accepted judgment of neighboring (and therefore of simultaneity) as we descend scales until we end at the mathematical point. The mathematical point is the essence of complete abstraction. The question is, is time found at all at this abstract point-event? At the foundation of Bergson’s theory (1896/1912) was a critique of the abstract space and time implied in Einstein’s vision. Abstract space, Bergson argued, is derived from the world of separate “objects” gradually identified by our perception. It is an elementary process, for perception must partition the continuous field which surrounds the body into objects upon which the body can act – to throw a “rock,” to hoist a “bottle of beer.” This fundamental perceptual partition into “objects” and “motions” is reified and extended in thought. The separate “objects” in the field are refined to the notion of the continuum of points or positions. As an object moves across this continuum, as for example, my hand moving across the desk from point A to point B, it is conceived to describe a trajectory – a line – consisting of the points or positions it traverses. Each point momentarily occupied is conceived to correspond to an “instant” of time. Thus arises the notion of abstract time – the series of instants – itself simply another dimension of the abstract space. This space, argued Bergson, is in essence a “principle of infinite divisibility.” Having convinced ourselves that this motion is adequately described by the line/trajectory the object traversed, we can break up the line (space) into as many points as we please. But the concept of 7 Special Relativity and Perception motion this implies is inherently an infinite regress. To account for the motion, we must, between each pair of points supposedly successively occupied by the object, re-introduce the motion, hence a new (smaller) trajectory of static points – ad infinitum. It is the core of Zeno and his paradoxes. Zeno, Bergson held, was forcing recognition of the logical implications of this infinitely divisible, abstract space and time. With each step, Achilles halves the distance between himself and the hare, but he never catches the hare; there is always a distance, no matter how minute, between pursuer and pursued. In the paradox of the arrow, the flying arrow occupies, at each instant, a static point in space, therefore, “it never moves.” In all four of the paradoxes, it is the infinitely divisible space traversed which is the focus. Motion, Bergson argued, must be treated as indivisible. We view the indivisible steps of Achilles through the lens of the abstract space traversed, and then propose that each such distance can be successively halved – infinitely divided. Achilles, never reaches the hare. But Achilles moves in an indivisible motion; he indeed catches the hare. 2 But the abstraction is further rarified. The motions are now treated as relative, for we can move the object across the continuum, or the continuum beneath the object. Motion now becomes immobility dependent purely on perspective. All real, concrete motion of the universal field is now lost. But there must be real motion. Trees grow. People age. Stars grow cold. Galaxies collapse. Bergson would insist: Though we are free to attribute rest or motion to any material point taken by itself, it is nonetheless true that the aspect of the material universe changes, that the internal configuration of every real system varies, and that here we have no longer the choice between mobility and rest. Movement, whatever its inner nature, becomes an indisputable reality. We may not be able to say what parts of 2 There is a mythology that these paradoxes have been resolved by Russell (1903) and/or modern mathematics. While Bergson showed that all four paradoxes have exactly the same root cause in an abstract space, Russell, having missed the point, actually accepted the fourth paradox as a physical reality. The mathematical “resolutions” are inherently limited to a spatial treatment and, in “taking a limit,” simultaneously invoke hand waving over infinity in the operation (cf. Bergson, 1944, pp. 335-340). 8 Special Relativity and Perception the whole are in motion, motion there is in the whole nonetheless. (1896/1912, p. 255) He would go on to note: Of what object, externally perceived, can it be said that it moves, of what other that it remains motionless? To put such a question is to admit the discontinuity established by common sense between objects independent of each other, having each its individuality, comparable to kinds of persons, is a valid distinction. For on the contrary hypothesis, the question would no longer be how are produced in given parts of matter changes of position, but how is effected in the whole a change of aspect…” (1896/1912, p. 259) Within the global motion of this whole, the “motions” of “objects” now become changes or transferences of state. The motion of this whole, this “kaleidoscope” as Bergson called it, cannot be treated as a series of discrete states. Rather, Bergson would argue, this motion is better treated in terms of a melody, the “notes” of which permeate and interpenetrate each other, the current “note” being a reflection of the previous notes of the series, all forming an organic continuity, a “succession without distinction,” a motion which is indivisible. In such a global motion, there is clearly simultaneity. The process of “objectification” which Einstein, in his response to Bergson, describes and accepts as leading us to the “real,” to objective events, and which leads Stein to his “fleeting motions” of masses of “elements,” is exactly the process warned of by Bergson. The “objects” of perception – purely practical partitions carved by the body’s perception in the flowing universal field at a particular scale of time – are reified into the concept of abstract, independent “objects” and their “motions,” and this is further rarified to “objective” space and time, with its objective, separable “events”. And following this path, Einstein is consistent. These “objective,” separate events are only mental constructs. They and their simultaneity are fully subject to the relativity logically inherent in their birth. Hence, to Bergson, Einstein’s “time of the physicist” is an artificial time. It can be argued, however, that this (artificial) path is exactly the opposite of what physics has found itself to be following. The concept of abstract space and time – this “projection frame” for thought 9 Special Relativity and Perception originating in perception’s need for practical action – has been the obscuring layer which is slowly being peeled away. As Bergson argued, “…a theory of matter is an attempt to find the reality hidden beneath…customary images which are entirely relative to our needs…” (1896/1912, p. 254). The customary images are dissolving. The trajectory of a particle no longer exists in quantum mechanics. If attempting to determine through a series of measurements a series of instantaneous positions, simultaneously we renounce all grasp of the object’s state of motion. In essence, as de Broglie (1947/1969) would note, the measurement is attempting to project the motion to a point in our abstract continuum, but in doing so, we have lost the motion. Motion cannot be treated as a series of “points,” i.e., immobilities. Thus Bergson noted, over forty years before Heisenberg, “In space, there are only parts of space and at whatever point one considers the moving object, one will obtain only a position” (Bergson 1889, p. 111). Lynds (2003), echoing Bergson, now argues that there is no precise static instant in time underlying a dynamical physical process. If there were such, motion and variation in all physical magnitudes would not be possible, as they would be frozen static at that precise instant, and remain that way. In effect, such an instant would imply a momentarily static universe. Such a universe is incapable of change, for the universe itself could not change to assume another static instant. It is only the human observer, Lynds notes, who imposes a precise instant in time upon a physical process. Indeed, Nottale (1996), noting Feynman and Hibb’s (1965) demonstration that the typical paths of quantum particles are continuous, but non-differentiable, now questions the fundamental assumption that space-time is differentiable, laying out a fractal approach to space-time, i.e., indivisible extents. A matter-field in a global motion, wherein the motions of objects are changes or transferences of state, implies a simultaneity of causal flows. We shall initially examine the consequences of this for Special Relativity in the context of perception. 10 Special Relativity and Perception 3.0 Special Relativity and Perception For Bergson, the perceived world is the reflection of the possibilities of bodily action. As he put it succinctly, perception is virtual action. The fly buzzing by, his wings ablur, is an index of the possibility of the body’s action. Were the fly flapping his wings slowly, like a heron, this would be an index of a yet different possibility, in this case, reaching out slowly and grasping the fly by the wing tip. Note that in each case, this index is simultaneously reflective of a scale of time, also a feature of our perception. That perception is indeed virtual action is indicated by our modern understanding of the processing areas of the brain with their reentrant connections. For example (simplifying greatly), visual area V1, which initially receives the retinal signals, projects to V4 (simple form processing) and V5 (motion processing). Simultaneously V4 and V5 project diffusely back to V1, modulating V1’s processing. While the visual areas project to the motor areas, simultaneously the motor areas feedback to the visual areas, modulating visual processing. In fact, counterintuitively, if we simply sever the connective tracts between the visual areas and the motor areas, the subject goes blind (Nakamura and Mishkin, 1980; 1982) But supporting this resonating feedback in the neural architecture, there are underlying chemical velocities. It is the base rate of these chemical velocities that determines our normal scale of time, e.g., the world of normally “buzzing” flies. Chemical velocities are subject to modification by catalysts. Were a catalyst (or catalysts) of sufficient strength introduced into the systems underlying the computation and preparation of action, increasing the velocity of chemical processes, then we could expect that the time scale of perception would change. In principle, catalysts of sufficient strength would now allow the system to specify a heron-like fly, barely flapping his wings. By the principle of virtual action, this view of the fly is precisely a specification of how the body can act. The change of scale and form for the fly is not merely “subjective,” or a “subjective modification” of experience. This is an objective effect. Virtual action, straightforwardly, 11 Special Relativity and Perception makes a prediction on action relative to the increase or decrease of the velocity of underlying processes. Albeit difficult today, in principle, this is a testable consequence. The question is, does Special Relativity also make a prediction, and if so, what? Figure 1. The Minkowski diagram. Let us consider the case of two observers, X and Y. We take the X system to be stationary, and Y moving relative to X at high uniform velocity. Assume there is a fly in X’s system. X, at his normal velocity of processes, i.e., at his time-scale, perceives the fly as a blur. The fly, which X is observing, travels one of X’s distance units using sixty wing-beats. It does this in one of X’s time units, say a second. Y, moving at great velocity, has much expanded time units (and contracted space units), the time units increasing as he moves nearer to the speed of light. However, this is as X computes these units relative to his stationary system. The complimentary case is Y’s (in motion) view of the space-time of X. The Minkowski diagram (Figure 1) shows this situation. The rhombus OFGH is gradually collapsing like a scissors as the velocity of Y increases. The tangent to the hyperbola, GF, drops lower and lower below X’s time unit, displaying that the time units of X, as Y sees them, are contracting steadily. Eddington (1966) had us imagine that at O, X lights up a cigar which lies along x1 and has a very longish length of one space unit. The cigar burns one of X’s units of time, being represented by the line t1 and extending to its first unit. Y would now see the cigar as burning longer for X, in fact, as the tangent drops as v increases, it would last many units of X as assigned by Y. This could equally be X himself, aging (a form of “burning”) many more time units than Y. Simultaneously, the 12 Special Relativity and Perception space units of X, as Y sees them, are increasing. Thus note that GH would fall outside the space unit of X – the cigar is longer. .Now it might be said that the fly, flying the length of the cigar lying along x1, is flying a longer distance as far as Y is concerned since he determines X’s space units have expanded. But the distance that the fly traverses in sixty wing-beats, however great or small the distance is measured to be – this distance holds a fundamental “causal flow” or invariant that relativity and its measurement procedures cannot alter. If we mark this distance by two markers, A and B, the fly will buzz from A to B in sixty wing-beats, no matter what the reference system from which he happens to be viewed. It is the “sixty wing-beat distance invariant.” We start from this. The fly flies this distance every day, from the cereal bowl to the sticky spoon on X’s table, in sixty wing beats. Relativity, simply because Y goes into motion, contains no inherent justification for altering this. Assume that the rocket is moving at 80% the speed of light. Given Y’s view of X as having contracted time units, the same sixty wing beats require 1.66 seconds as assigned to X by Y. So, now we partition this sixty wing beats (an invariant causal flow) across the 1.66 seconds. In X’s normal system, at sixty wing beats/second, there are six wing beats in each 1/10th second, and X can normally perceive or discriminate one wing-beat per 1/10th second. Thus at six beats per each 1/10th second, he sees a blur. In the new partition assigned by Y, with sixty beats partitioned over the 1.66 seconds, X sees only 3.6 wing beats in each 1/10th second. It is less a blur. The fly appears to be buzzing more slowly. X’s time (his perception of the rate of events) is slower, despite the fact that his velocity of processes has not changed. This is clearly absurd, yet this is exactly what is required of the world of X if we ignore reciprocity, and if these transformations are ontological enough to support Y’s eventual return as more youthful than X. On the other hand, there is the effect on Y, whose time units are expanded and space units contracted. In Y’s moving system, a fly is buzzing across the table in the rocket cabin, again using sixty wing beats from A to B. It requires only .6 of the expanded Y-second for the distance 13 Special Relativity and Perception to be covered. The invariant sixty wing beats are partitioned across this amount, therefore becoming ten beats per each 1/10th second, and thus the fly is now more of a blur, despite the unchanged velocity of processes. It can be argued, just as Eddington notes, that due to the rocket’s velocity, Y’s processes are retarded. But in fact everything in Y’s reference system is retarded, to include the fly and its buzzing from A to B. In effect, we have simply subtracted a constant across all motion values of the system, and the problematic modification of perception just noted still holds. In essence, psychology contradicts physics. In this analysis, I have stayed consistently within the implications displayed in the Minkowski diagram, that is to say, within the case where Y is consistently the system in motion, and X stationary. If we want to set X in motion, we need another diagram, and the situation simply reverses. 3.1 The Role of Reciprocity What is wrong here? There is the strange picture of Y’s view of X’s altered perception of events in X’s own system. But let us ignore this. One aspect of the problem is more elementary. As noted, when we represent the situation of X and Y in the Minkowski diagram, we have fixed on one observer, X, and set all other systems in motion relative to him. The Minkowski schema represents the adjustments in time and space units necessary to preserve light-velocity invariance for all other systems. But it cannot represent reciprocity. We could equally have fixed on Y and set all other systems in motion with respect to him. This, again, requires another diagram, and so on for each observer upon whom we fix. Given this, we must ask the fundamental question: is the effect on either X or Y a real effect? Y, we know, could equally declare his system to be at rest, and X in motion relative to him. Clearly, the effects cannot be real from this perspective. The different “times” and “distances” represent only the observer’s method of keeping his measurements consistent with light-velocity invariance. STR, from this perspective, fails to justify, either for X or for Y, a different perception of the fly based on the observer’s motion. 14 If we respect the inherent reciprocity of Special Relativity and Perception reference systems in STR, there is no contradiction with the relativity of perception. STR is at worst neutral with respect to a causal flow in time (the fly) invariant to both X and Y. Only if we insist that STR implies a real effect is there a contradiction. It must be clearly understood here that I am not denying the empirical facts, e.g., increase of life spans in mesons, or the retarded clock carried by the jet, or increases in mass. The empirical evidence is not in dispute. These are real effects. What is in dispute is the use of STR to explain the empirical evidence; it is used inappropriately in attempting to do so. The structure of reciprocity intrinsic to STR is being ignored, and if we ignore it, we have a contradiction in the context of perception. 3.2 Half-Relativity “Half-relativity” is what Bergson (1922/1965) termed the asymmetric use of STR. The Lorentz equations are applied to the meson; the life span increase falls out via t’. End of explanation. As noted already, A. P. French (1968), in a textbook which attempts to maintain clarity, in a section entitled “Relativity is Truly Relative,” flatly states that the time dilation (just as the length contraction of the Michelson-Morley apparatus) as observed for a meson is not a property of matter but something inherent in the measurement process. He goes to the rare extent of actually showing two Minkowski diagrams, one for each observer (as though there were a small observer on the meson), to show the symmetry of the changes in each system. Just as Bergson (1926/1965) once argued, French notes that were an observer to compute t’ as the meson falls to the earth, the tiny observer on the meson is equally allowed to say that he is stationary and the earth moving towards the meson. This is to say we have here, in French’s terms, at best a “measurement effect.” Thus, when French treats the twin paradox, he invokes the asymmetry introduced when the twin on the rocket turns around to return, therefore introducing a new inertial frame (pp. 155-156). STR is used to compute the different (shorter) “time” of the traveling twin for each leg of the trip, thus ascribing the magnitude of the difference to v. But he assumes, in conjunction with this, that it is the asymmetry introduced by the turn-around that is required to 15 Special Relativity and Perception support the real (aging) effect, i.e., as a real property of matter. Clearly, if one twin is now gray and has a long beard, we have a change that is a real property of matter. Thus he argues that STR, factoring in this asymmetry associated with the turn-around and its acceleration, and due to the fact that a time difference value can be derived due to v, can indeed handle the twin paradox. Yet he has earlier painstakingly built the case, to the point of doubled Minkowski diagrams, that the structure of STR demands symmetry (reciprocity), and given this symmetry, it does not explain any changes as real properties of matter. 3 In essence, the entire explanatory burden for aging as a real effect now falls on the asymmetry introduced by the change in inertial frame. But where is this theory, i.e., where is the theoretical structure supporting how and to what magnitude introducing an asymmetry affects the physiological processes underlying aging? Or why the asymmetry can be introduced into STR? More precisely, where is the theory which explains how introducing an asymmetry now allows the use of the Lorentz equations independent of, or outside of, the symmetric, reciprocal structure provided in STR? In the comparison between X and Y above, we only asked Y to be in uniform relative motion at velocity v, just as in the meson case, just as in the Michelson-Morley case. This comparison could care less about Y’s return or differential accelerations. We don’t need a rocket. While X sits by the kitchen table watching the fly, Y could travel by on his tricycle, and the same relativistic laws hold.5 Nevertheless, there are those that would simply classify this case as the twin-paradox, invoke the existence of accelerations, and move the problem and the effects involved into the General Theory. All of the effect can then be assigned to acceleration(s). This reaction is extremely problematic. If we seize upon any accelerating component of a motion 3 I have been posed one objection or “solution” to this problem stated as follows: “The twin leaving and returning on the rocket ages less because his worldline between departure and return is shorter. And the length of the worldlines is observer invariant.” This is a strange misconception and misstatement. The “observer invariance” is only defined within the structure of symmetric (reciprocal) transformations created by both observers. There is no “invariance” with but one observer. But then it is this very symmetry that makes it impossible to use relativity to explain changes as real properties of matter. 16 Special Relativity and Perception (which one can always find, even for the startup of the tricycle) to allow us to get to the safety of the GTR, then what if anything is the province of STR? The physics would be in danger of becoming a shell game, shuffling an explanatory pea between STR and GTR. If we are doing this to avoid reciprocity, then the argument that STR, with its inherent reciprocity, fails to explain any of these effects is effectively conceded, and this lynchpin in its being a theory of time – its ability to explain these effects – is removed. 4 Note again, it is not the aging effect, it is all asymmetric effects – jet carried clocks or long living mesons – that would have to be so moved into GTR for consistency. One dismisses the above comparison of X and Y into the GTR then only with difficult consequences. 5 Thus others (as well as French) have argued, as Eddington (1966) appeared to believe, that the twin-effect is perfectly consonant with STR. But to stay fully within the context of the Special Theory without bringing in gravitational field changes, Salmon (1976) envisaged a rocket ship (A) departing earth and passing another (B) coming in the opposite direction at the same velocity. At the point of meeting, the two exchanged signals to coordinate their clocks. B continued on to earth where clocks were compared, and of course, in a triumph for the theory, an earthbound observer’s clock showed a greater passage of time than B’s. This appears to be ironclad, yet there is a problem. Reciprocity has not been avoided. The observer in A takes with 4 Brillouin (1970) would argue that a reference system must be very massive to reduce all actionreaction effects. The tricycle, let alone an abstract “coordinate system,” would not qualify in his opinion. The same point however can be made with a more massive system going by the table. But I do not believe that Einstein was concerned at all with this distinction, the geometry being the overriding consideration. 5 The comfort of assigning this to the GTR arises from the tenet that acceleration breaks the symmetry or reciprocity of systems. I am aware that this is a fundamental tenet of GTR, but it is yet possible that the original analysis by which this tenet was derived is subject to question. Bergson argued simply that acceleration cannot be distinguished from velocity in the sense relativity claims – velocity is a rate of change in position over time, acceleration simply the rate of change of the rate of change of position. Wang (2003) refines this argument, deriving the generalized Lorentz equation for t’ in the context of acceleration. If we cannot integrate over infinitesimal velocities, he argues, as did Bergson also, we have undercut all of physics. Wang’s equation completely undercuts any appeal to the GTR due to acceleration in the twin paradox; in fact it implies a question to the foundation of GTR. 17 Special Relativity and Perception him his own reference system. Since no reference system is privileged, he has equal right to declare himself at rest and everything else in motion relative to him, including the earth, the earthbound observer, and the earthbound observer’s clock. When B passes A and signals are exchanged, will they then reflect a decrease in the rate of A’s time? Hardly, given A is at rest. Only the author of the argument happens to believe A is in motion, but he forgot to ask A. 6 The twin-paradox is disturbing precisely because it epitomizes, very concretely, the inconsistency relative to standard use of STR. It highlights a very real effect, e.g., a youthful man versus a hoary old one, that cannot simply be assigned to a measurement process. Interestingly, Einstein himself, in a (little known) 1918 article, attempted to preserve reciprocity and the asymmetrical effects together by arguing that indeed the rocket ship could be considered stationary, its motors only neutralizing the pull of the earth as the earth recedes. 7 But he then argued that it would require such tremendous field changes to move the earth and bring it back that the earth twin would undergo rapid aging. The reciprocity and the paradox denying the reciprocity appear resolved (just as French argued). But now, ignoring the ad hoc, physically 6 Davies (1977) resolves the twin paradox by flatly assigning the aging differential to the turn around at the target star and the homeward acceleration of the rocket (pp. 43-44). Yet, like French, he applies the Lorentz equations, claiming that he has also preserved the symmetry, a fact his table of durations (p. 44) obviously belies, for only the rocket clock shows a consistent, timeexpanded 4.8 light years for each leg – the rocket is clearly the only object moving to Davies. Davies (1996) drops the clear emphasis on acceleration as the root cause of the aging. He does declare there is no paradox because the symmetry is broken due to accelerations in the necessary stop and return of the rocket, but never mentions this again. Ignoring the consequent inapplicability of STR, he again proceeds to apply the Lorentz transformations (with what justification?). In essence, he notes that that at 80% of the speed of light, earthbound twin Ann would see the clock of the rocket-twin (Betty) as running .6 of earth-Ann’s. Symmetrically, rocket-Betty, viewing herself as stationary, sees earth-Ann’s clock as running .6 of Betty’s. This symmetry holds for each leg – the outward and the homeward bound. In Davies’ scenario, it is rocket-Betty who returns having aged less, not earth-Ann, and he claims that he has resolved Dingle’s (1972) critique that in this case, “each clock runs slower relative to the other,” in other words, a critique which says precisely that there can be no ontological status here. Given the symmetry he took great pains to describe, Davies conveniently never tells us why earth-Ann does not also have the distinction of aging less. 7 A translation of this paper is discussed in Dingle (1972, pp. 191-200). 18 Special Relativity and Perception unrealizable fields, it is not clear of what use relativity is here at all. Its mathematics, with its intrinsic reciprocity, now does not accurately describe the phenomenon – we can clearly distinguish the two systems via gravitational effects – and it would seem logically prior to have a theory relating gravitational changes to a model of the physiological processes driving aging – this in itself being sufficient to account for the phenomenon without appealing to changes of “time” itself. The one-way application of the Lorentz transformations would then appear in retrospect to be but a convenient empirical description of these events, but a deeper theory would provide a model of the processes involved (as Lorentz himself attempted). 3.3 The Undertone of Half-Relativity in 1905 Einstein, for all practical purposes, began assigning real effects due simply to v in 1905. In the paper, he quickly invokes the reciprocity implied in the first postulate, having us envisage a rigid sphere of radius R, at rest in the moving system (1905/1923, section 4, p. 48). At rest relative to the moving system, he notes, it is a sphere. Viewed from the “stationary” observer, the equation of the sphere’s surface gives it the form of an ellipsoid, with the X dimension shortened by the ratio 1:(1 – v2/c2)1/2. He notes (the reciprocity) then immediately: “It is clear that the same results hold good of bodies at rest in the ‘stationary’ system, viewed from a system in uniform motion” (1905/1923, p. 49). Two paragraphs from this point he notes the “peculiar consequence” that were there two synchronous, separated clocks A and B in the stationary system, and if A is moved to B with velocity v in time t, it will lag behind B by ½ tv2/c2 (section 4, p. 49). The structure of reciprocity is already being voided here – we are dealing only with an effect in the stationary system, not relating the two systems. The observer in the stationary system can simply move the clock from A to B to fulfill Einstein’s condition, and the effect is simply ascribed to v. This conclusion is quickly reinforced. Within another paragraph Einstein, extending this to “curvilinear motion,” states flatly that this result implies that a clock at the equator must go more slowly, by a small amount, than one situated at the poles (p. 50), i.e., again two clocks in the same system. Physicists accept this equatorial clock retardation naturally as a real effect. The effect 19 Special Relativity and Perception had to be factored in to Hafele and Keating’s jet-carried clock experiment. Yet reciprocity demands that the clock on the equator be stationary, the observer at the pole spinning around. Now it is not a real effect. This is likely not very tasteful. Yet this conclusion regarding v as already producing real effects in 1905 is doubly reinforced when it is considered that the equatorclock is an exact analogue to Einstein’s future thought experiment of the rotating disk. Now the observer leaves the center of the disk, moving along a radius to the rim and back, while carrying a clock. Upon his return the clock is retarded. The thought experiment used this result as a very real effect. Yet why? The observer takes with him, at every point he occupies, his own proper time. He should return with the clock unchanged. Why is the problem of “real effects” significant? There are three reasons. Firstly, if STR is being used inappropriately as an explanatory device where the one-way use of the mathematics just happens to work, then physics should be searching for the true explanation. It could be extremely instructive, if only for the apparent return of the ether, which formerly housed some of these effects (again, in Lorentz’s mind for example), in more sophisticated form as the quantum vacuum. Secondly, there is now the contradiction with the psychology of perception just discussed and which I hope would merit at least some review. Thirdly, if we cling to the idea that STR can explain real, asymmetric effects, then we are equally clinging to the reality of the relativization of simultaneity, i.e., to the real breakup of simultaneity into successive moments in time, and vice versa. It is this implication that I wish to further question. 4.0 The Relativity of Simultaneity In Figure 2 we picture three points, A’, B’, and C’ in Y’s moving system placed along the direction of this motion. Each will be a distance L from each other. We will assume Y is at point B’, and the system is moving with velocity v. From the viewpoint of the stationary X, these three events are not simultaneous. The clock at A’ registers a time slightly behind that of B’, while the clock at C’ is somewhat ahead. The greater the value of v, the greater this lag and lead 20 Special Relativity and Perception time respectively. Both times are given by Lv/c2 seconds. As v approaches the speed of light c, the maximum difference becomes L/c seconds. Figure 2. Planes of simultaneity (cf. Bergson, 1922/1965). If we drop a perpendicular from A’ to K’, this line will symbolize all the past events at A’. Since we see that the clock is slow at A’, and Y then supposedly looking at past events, this line displays the maximum reach into this past. Likewise the line upwards from C’ to H’ shows the maximum of the future. Now we can draw yet another line of simultaneity, this one running to (hypothetical) points D’ (between C’ and H’) and E’ (between A’ and K’). Its divergence from the original line A’B’C’ is a function of the speed v. Further, were the difference in v between the X and Y systems infinitesimally small, there would be a line barely divergent from A’B’C’ representing the fact that at even the most infinitesimal velocities, we see the breakup of simultaneity begin, radiating from the most minute point or distance from B’, increasing in degree towards A’ and C’. There are any number of such lines. What is the reality here? Imagine that Y is moving at an infinitesimally small velocity relative to X. For practical purposes, X’s line ABC and Y’s line A’B’C’ are virtually coincident. But yet, even at the most minute velocity, simultaneity has begun to break up at the most infinitesimal point or distance from B, increasing in degree as we approach A’ or C’. Now Y moves at a much higher velocity. X now notes the difference in Y’s clocks. He is forced to assign events at A’ deeper and deeper into Y’s past as v increases, and to assign events at C’ farther into the future. He does this by the very fact that he needs to keep the velocity of light invariant as per the 21 Special Relativity and Perception Lorentz transformations. But Y can equally say he is at rest. He continues to note the simultaneity of events at A’, B’, and C’. He now notes the same breakup of simultaneity for X. Again the question becomes, is the conversion of simultaneity to succession real? Is it more than a notational convention required for the consistency of measurements between the two systems? Can this possibly be true of the flow of time? Elementary considerations say that it cannot be true. 4.1 The Simultaneity of Flows Figure 3. Two football players (e1, e2) converge on the ball (e3). The intuition of a universal flow is partially preserved in relativity in the conservation of a “causal order.” On analysis, we will find multiple causal orders or flows within this flow as Bergson noted or even, as Gibson insisted in the opening quote, where hero rushes to save the endangered heroine. The simultaneity of flows is integrally bound to causal order. Consider two football players running down each sideline of the field at precisely equal velocity. A physicist (O1) at the fifty yard line notes the time against two synchronized clocks on each sideline as the players run by and ascertains that they have passed the same point simultaneously (Figure 3, e1 and e2). Of course a second physicist (O2), thinking the first in motion and noting this observation says the first is in error, the events were not simultaneous. Yet the two football players continue on, converging on a football equidistant from both which they both kick simultaneously (e3), kicking the ball twice as far as just one would have achieved. From the 22 Special Relativity and Perception perspective of an instantaneous measurement, i.e., abstract time, their simultaneity is relativized. From the perspective of the two causal flows, the simultaneity of the flows is absolutely real. The second physicist cannot deny the effect of the simultaneous kick. One cannot simply relativize multiple causal flows. It can be argued that e1 and e2 are not truly simultaneous just as O2 states, that simultaneity is achieved only at the point-instant of the kick. But we could replace the football players equally well with a huge cue stick sweeping down the field towards a billiard ball. Positioned at each yard line are O1’s measurement clocks. If the cue’s outside edges truly fail to pass the measurement clocks/ points simultaneously, it will hit the ball at a slant sending it off at an angle. In sliding the x1 , x2 and t2 axes upwards towards e3, it can be seen that there will come a point as our very wide cue nears the ball at e3, that e3 will fall in the causal elsewhere of the light cones of each of the edges (e1, e2). This implies that the two outer edges could not possibly be squared in time for a flush contact of the entire cue surface with the ball if they are as non-simultaneous as claimed by O2. The global causal flow led by the cue’s frontal surface is fragmenting under STRs treatment. Yet the cue strikes the ball precisely perpendicularly. Only one strand in this flow, one local flow, the causal order in STR invariant to both observers, is ultimately preserved. This is the chain of causal relations, <, the relation determining time-like and space-like events, defined upon a sequence of infinitely minute point-instants extending through the time line t1 to e3. Were we considering the fly, no matter how infinite the “points” we place on this line, or the in fact multiple lines comprising the fly, this will remain sixty wing beats – an indivisible movement or flow. A global flow, whether fly or cue stick or hero and heroine, cannot be an invariant to all observers in STR. 8 8 A comment on concepts expressed in Myrvold (2003) is appropriate here. Myrvold considers the relation eRe’ (where R = “realized with respect to”) in the context of extended objects. This requires taking a spacelike slice – in effect an instantaneous stage along some foliation of the object’s history. Failure to do this results, he notes, in paradoxes like the “pole and barn,” where, with the barn at rest and the pole in high velocity motion through the barn, there is a period where the pole just fits inside the barn, and conversely, with the pole at rest, and the barn 23 Special Relativity and Perception The fly, as a coherent biological system doing his sixty wing-beat trip, is precisely a global, indivisible flow. Were he taking his sixty wing beat trip to e3, the tips of his wings will stop precisely simultaneously, O2’s measurements to the contrary notwithstanding. When it was insisted earlier that this sixty wing beat flow be treated as invariant to both X and Y, this weakness inherent in STR’s treatment emerged. In the above, I have not attempted a formal definition of a causal flow. I am leaving this at the intuitive level where, for example, a fly, as a complex system in motion, is comprised of multiple processes acting in concert, be this multiple muscle systems, neurons firing, or chemical flows. Such a system could be as large, and larger, as a weather system such as a hurricane, or an evolving galaxy, or a collection of individuals all working together to play a symphony. The two football players with which we began were two seemingly isolable local flows. They could however have been two sailboats moving in unison before a vast pressure front. Or this could have been a vast magnetic flux sweeping the earth. The point is that we must ask if any such local flows, any more so than “objects” and their “motions,” are truly isolable from the global in motion, there is no such “pole-inside” state. This conflict is resolved, he argues, “by remembering that the states of the extended system of which one account speaks are states along spacelike slices of the system different from those of which the other account speaks” (p. 478). This is a not a justifiable modification of STR. The reference system of Figure 2 would be treated as a set of points, α. Another set, β, would be definite or realized with respect to α if in α’s causal past. Though seemingly applying to the cue stick example, we could not extend the system indefinitely, or it would extend across the entire universe, providing a plane of simultaneity. But, given Myrvold, what prevents this move? Would the two football players simply be one Myrvold-slice? Why not? But where does STR contain the a priori principles to make this causality saving determination? In fact, my earlier analysis relative to Figure 2 shows that the simultaneity of α begins to break up at the most minute interval relative to an observer in motion. But, there is a simpler reason why Myrvold is not a resolution. If the length contraction of the pole is being taken as a real effect in this paradox, the (very testable) implication is that we could actually trap the pole inside the barn, different spacelike slices or not. Such a real result (captured pole) is as much a contradiction as the twin paradox. If it is not considered a real (possible) effect, this is due to giving the reciprocity of reference systems its appropriate status, which is to say there is no ontological status to the relativisticcontraction, and no “paradox” in the first place. Myrvold dismisses the paradox, considering it an example of misunderstanding, yet it is no more a misunderstanding than the twin-paradox where the “time-change” should have equally as little ontological status. 24 Special Relativity and Perception flow of the universal field? Are they more than transferences of state within the global motion? This global transformation is the classical “flow of time” invariant to all observers. 5.0 Conclusion There have been other examinations of STR, of both its explanatory status in physics and as a theory of time. Bergson was perhaps the earliest. His argument in Revue Philosophique with physicist Andre Metz circa 1924 centered on the use of STR in explaining asymmetric effects (cf. Gunter, 1969, pp. 135-190). Metz could not accept that STR is an inappropriate explanatory vehicle, nor could he conceive of the possibility that the increased life spans of mesons could be explained without resorting to STR. Deleuze (1966/1991) would reprise Bergson’s (1922/1965) general argument on time with respect to relativity. Dingle (1967, 1972) would make interesting critiques, particularly on the invariance of light. Brillouin (1970, pp. 77-85) would give a nonrelativistic explanation of the retardation of atomic clocks (and of the red shift). Earman (1989) would note that there has yet to be a relational, let alone a relativistic explanation of Newton’s humble bucket. Rakić (1997), in proving certain logical inadequacies of the Minkowski metric, is reduced to declaring Special Relativity to be not an ontological theory, but concedes it a status as a “temporal” theory. Whatever meaning this concession might have, a theory with no ontological status is of little use; it is certainly not relevant to a science of conscious perception. 25 Special Relativity and Perception References Bergson, Henri (1896/1912). Matter and Memory. New York: Macmillan. Bergson, H. (1907/1944). Creative evolution. New York: Random House. Bergson, Henri (1922/1965). Duration and Simultaneity With Respect To Einstein’s Theory. Indianapolis: Bobbs-Merrill. Brillouin, Leon (1970). Relativity Reexamined. New York: Academic Press. Capek, Milic (1966). Time in relativity theory: arguments for a philosophy of becoming. In J. T. Fraser (Ed.), The Voices of Time. New York: Brasiller. Chalmers, D. (1995). Facing up to the problem of consciousness. Journal of Consciousness Studies, 2(3), 200-219. De Broglie, Louis (1947/1969). The concepts of contemporary physics and Bergson’s ideas on time and motion. In P.A.Y. Gunter (Ed.), Bergson and the Evolution of Physics. University of Tennessee Press. Deleuze, Gilles (1966/1991). Bergsonism. New York: Zone Books. Dingle, Herbert (1972). Science at the Crossroads. London: Martin Brian & O’Keeffe. Earman, J. (1989). World Enough and Space-time. Cambridge: MIT Press. Eddington, A. (1966). Space ,Time, and Gravitation. Cambridge: MIT Press. Einstein, Alfred (1905/1923). On the electrodynamics of moving bodies. In H. A. Lorentz, A. Einstein, H. Minkowski, H. Weyl. The Principle of Relativity. New York: Dodd Mead. Einstein, Alfred. (1918), ‘Dialog über Einwände gegen die Relatvitivätstheorie’, Naturwissenshaften, VI, 697. Feynman, R. P. & Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals. New York: MacGraw-Hill. French, A. P. (1968). Special Relativity. New York: Norton. Gunter, P. A. Y. (1969). Bergson and the Evolution of Physics. University of Tennessee Press. 26 Special Relativity and Perception Hagan, S., and Hirafuji, M. (2001). Constraints on an emergent formulation of conscious mental states. Journal of Consciousness Studies, 8, 109-121. Horton, G. (2000). Einstein, History and Other Passions. Cambridge, Massachusetts: Harvard University Press. Lynds, P. (2003). Time and Classical and Quantum mechanics: Indeterminacy versus discontinuity. Foundations of Physics Letters 16(4): 343-355. Myrvold, W. (2003). Relativistic quantum becoming. British Journal of the Philosophy of Science, 54, 475-500. Nakamura, R.K. and Mishkin, M. (1980). Blindness in monkeys following non-visual cortical lesions. Brain Research, 188, 572-577. Nakamura, R. K. & Mishkin, M. (1982). Chronic blindness following non-visual lesions in monkeys: Partial lesions and disconnection effects. Society of Neuroscience Absracts. 8, 812. Nottale, L. (1996). Scale relativity and fractal space-time: applications to quantum physics, cosmology and chaotic systems. Chaos, Solitons and Fractals, 7, 877-938. Rakić, N. (1997). Past, present, future, and special relativity. British Journal for the Philosophy of Science, 48, 257-280. Russell, B. (1903). The principles of mathematics, London: Allen and Unwin. Salmon, W. (1976). Clocks and simultaneity in special relativity, or, which twin has the timex? In P. K. Machamer & R. G. Turnbull (Eds.), Motion and Time, Space and Matter. Ohio State University Press. Stein, H. (1991). On relativity theory and openness of the future. Philosophy of Science, 58, 147167. Wang, L. (2003). Space and time of non-inertial systems. Proceedings of SSGRR 2003, L’Aquila, Italy. 27

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