FORWARD THINKING IN ASTRONOMY [A series of eight lectures specially prepared for Compu-Serve Information Systems (CIS), for presentation in ASTROFORUM. Copyright 1990 by Tom Van Flandern of Washington, DC [CIS ID code 71107,2320]. Please seek the author's permission before reprinting more than two paragraphs. If redistributed in electronic form, must include this statement of source and copyright.] CHAPTER II. THE NATURE OF SPACE, TIME, AND MATTER ******** We will be using Scientific Notation for large or small numbers. For example, 1E5 = 10 to the power 5 = 10,000; 2E-7 = 2 times 10 to the power -7 = 0.000 000 2. Ask about this, or anything else you aren't familiar with. Use private messages, if you wish. But your questions are probably shared by others. A. Introduction Last week we saw the value of deductive reasoning for determining the nature of reality, provided that a suitable starting point can be found. This week we will reason deductively about the nature of the physical universe starting with a minimum of assumptions. In fact, I propose that we start out with nothing whatever: a universe completely empty of everything which exists. Is space an absolute thing, existing even without matter in it? Or does it depend upon the existence of matter to give it meaning? Let us define "substance" broadly as anything which exists, whether it takes the form of matter, energy, or "other". In order to answer the question of whether space itself exists in the sense of having substance of any kind, we need to introduce some additional useful properties of substance. B. The One Particle Universe Let our starting universe remain empty of everything except a single infinitesimal "stationary" particle. Now imagine the same particle in motion. How fast is it going, and in what direction? There is nothing for it to move relative to, and nothing to provide orientation. All directions are equivalent, and all distances are equivalent. The only way it can be otherwise is if space itself has a sort of "structure" to it, a framework to provide meaning to orientation and scale and motion. However we have postulated an empty universe. In it, there is no matter, no energy, no substance of any kind except the single particle. How can there be "structure" without substance? In the real universe there is a frame of reference to provide meaning to distance and direction. The reference frame is provided both by the presence of distant matter in the universe, as well as by seas of rapidly moving "agents", such as photons and neutrinos. The essential point is that the reference frame is provided by the presence of substance in the universe. I would not insist that MATTER is needed; but I take it as self-evident that some sort of substance is required, or there can be no reference frame in space. In the absence of other substance in the universe, our lone particle would be incapable of motion, for motion could have no meaning. Moreover (and this is something to note), the size of the universe would be indeterminate, even if our lone particle has "finite" dimensions. Indeed, it is impossible to say whether the particle has infinite dimensions, finite dimensions, or is infinitesimal (without size), since there is no scale to measure by. The number of such particles which can fit into the universe around it is infinite in any case. Our lone particle would even be incapable of spin. If it had parts, they might move relative to one another. But a uniform spherical lone particle cannot spin about any axis, because there is nothing outside the particle to spin relative to. By extension, the particle could not be made to exhibit the properties of spin, such as centrifugal force -- a tendency to hurl objects off itself due to spin; nor would it tend to flatten from very rapid spin. The origin of these "inertial forces" is surely rooted in the substance which defines the framework of space. Without a framework, without substance (except for the particle), without "agents" to produce forces, surely there could be no meaning to, nor consequences of, "spinning". (The idea that the presence of distant matter in the universe is the origin of inertial forces is known as "Mach's Principle".) Our example may start to seem a little less hypothetical if we postulate a finite limit to all of the substance in the real universe, with nothing beyond. (The "Big Bang" Theory in its simplest form is such a case, in which all substance remains inside a sphere whose surface consists of photons moving outward at the speed of light since the instant of the original explosion.) Under this assumption, the entire substance of the universe would be like our single particle; and all remarks about its size or motion in a larger infinity of space and time would be fully applicable; i.e. they would be indeterminate. C. The Two Particle Universe Consider again our simple lone particle in an empty universe. Now let us imagine a second particle just like the first at another location, not touching. Now, for the first time, we have "scale" in our universe, and can measure the dimensions of the particles themselves as a fraction of the distance between them. There is no such thing as "absolute length" in this universe -- we cannot tell if the two particles are "close together" or "far apart". Their separation is indeterminate relative to the universe beyond. It can only be measured in terms of the number of particle diameters. We have also introduced meaning to motion, since the separation measured in particle diameters can vary. But with only two particles, if the separation "varied", we could not tell whether the particles had moved, or perhaps only changed diameter (shrunk or expanded) -- either would give the same result. We can also now detect spinning. Note that the two particles cannot "see" or influence each other in any way except by collision, since our otherwise empty universe definitely contains no photons or agents to produce forces or actions at a distance, such as electromagnetism or gravitation. Consider a hypothetical pendulum suspended at the "north pole" of one of the two particles, taken as spinning. In the real universe, a suspended pendulum would continue swinging back-and-forth in the same direction in the universe, ignoring the spin of the body (e.g. the Earth) underneath it (as many museum exhibits of the Foucault Pendulum demonstrate). But our two-particle universe can have no such properties, because there can be no framework to provide a "remembered" preferred orientation for the pendulum. Indeed, the pendulum could not swing at all, because there is no gravity in this imagined universe. Now if the particle on which the pendulum is suspended is imagined to have local gravity only, so that the pendulum can swing; but gravity which does not reach out to influence the second particle, so that no framework is provided to the universe; then clearly the pendulum must keep its orientation with respect to the particle it resides on, since that is the only framework it has. But as soon as we imagine a sort of universal gravitation, this immediately provides a framework for the pendulum. The proximity of the pendulum to a spinning particle is then no longer relevant, since the pendulum "senses" only the universal gravitational framework, and must maintain its orientation in that frame. Nothing about the forces acting on the pendulum would tell it the particle, above which it is suspended, is spinning. By these constructions, we begin to see the origins of the what are called inertial forces, and the importance of a frame of reference to the properties of the universe we live in. We also begin to see why it must be that scale and motion are relative, not absolute, in nature. We have just seen that absolute motion has no meaning without a frame of reference; and that such a reference frame must logically be provided by some sort of substance. This gives us a basis for looking at a very famous dilemma called "Zeno's Paradox". D. Zeno's Paradox Zeno's Paradox deals with the ultra-small structure of space and time. In its essence, the paradox notes that, if a moving body is in a specific place at every instant, then there is no instant when it is in transition from one place to another; and therefore motion is impossible. Since this contradicts everyday experience, it is called a paradox. The same paradox can be expressed in a different form: to move from point A to point B one must first complete the trip to the mid-point. Having reached that far, one must next reach the new mid-point of the remaining distance. But however far one has travelled, one must first travel half the remaining distance before one can travel all of it. Hence one can never reach point B, because an infinite number of "half-the-distance" steps are required. It might be, of course, that space is not infinitely divisible -- that there is a smallest possible increment of distance. But this leads to all sorts of conceptual problems. Consider points X and Y, separated by the smallest possible increment of distance. Now consider another point Z, also separated from X by the minimum possible distance, but in a slightly different direction. Then the distance between points Y and Z is less than the minimum possible distance, contradicting the starting assumption. But if space were "grid-like", so that adjacent cells had no overlap, then motion in any desired direction would not be possible, unless one took a zigzag path from grid-point to grid-point! Clearly, the postulate of a "minimum possible distance" is problematical. If time is treated like just another dimension (a "fourth dimension" of space), the same remarks might be extended to include the concept of a "minimum possible time unit". Or we may make a separate argument about time. If there were a minimum possible time unit, then all existing substance would have one condition at one time moment, and some slightly different condition at the next time moment. By hypothesis, there is no possible interval in time, nor any moment in between when anything could have happened to provide a transition from the first condition to the second. It is therefore just exactly as if everything existing at the first time moment ceased to exist, and then was created from nothingness in its new condition at the next time instant. We conclude then that space and time must be infinitely divisible in order to avoid these dilemmas. But is this not also ruled out, by Zeno's argument? The problem is with our intuitions: while it is easy for us to imagine a whole as composed of an infinite number of parts, it is difficult for us to imagine an infinite number of components being assembled into a finite whole. As is well known in mathematics, an infinite series CAN have a finite sum. For example, there are an infinite number of possible fractions or decimal numbers between zero and one, yet obviously only a finite interval. In Gamow's book, "One, Two, Three ... Infinity", we learn how to count and compare things made up of an infinite number of parts, using one-to-one correspondences. Such a one-to-one correspondence can be set up between points in a space interval, and decimal numbers between zero and one. Since the interval from zero to one is finite by definition, the one-to-one correspondence shows us that the space interval is finite also. With another one-to-one correspondence we also conclude that it is possible to traverse the space interval in a finite time as well. This is an important point, even though our intuitions do not deal with it easily. The infinite mathematical series (1/2 + 1/4 + 1/8 + 1/16 + ...), where each new term is half the preceding one, has a finite sum of 1.0. It is clear that, in mathematics, there exist finite intervals with an infinite number of points between, and infinite series with finite sums. By placing these in one-to-one correspondence with the physical concepts of space, time, and mass, we can reason by extension that FINITE intervals and masses may actually be composed of an infinite number of divisions; and conversely, that an infinite number of divisions may have a finite sum. What about Zeno's objection, that if a moving body is someplace specific at EVERY instant, then at no instant is it moving, making motion impossible? One way to see the resolution of this paradox is by considering time to be another dimension, just like the three dimensions of space (although admittedly not exactly like a space dimension; e.g. we cannot travel both ways in time). Then a body travelling at a uniform velocity from point A at time 1 to point B at time 2 is travelling on a straight line in this space-time universe. To clarify this picture, suppose the body is at rest in space. It nonetheless takes a straight "line" in space-time to connect its position at one time with the same position at a later time -- the "line" representing an interval in time, instead of space. Viewed in this way, it may be seen that the body is at every instant at some specific point on a space-time line. And once again the points in the interval can be put into a one-to-one correspondence with numbers between zero and one. So even though the distance travelled by the body in zero time is zero, it is nonetheless possible to traverse a finite distance in a finite time, each interval consisting of an infinite number of time instants and space points. To put this conclusion more strongly, it is possible for substances to be unchanging at every instant, yet changed after a finite interval, ONLY if there are an infinite number of steps in the interval! ******** This "one-to-one correspondence" may be the toughest concept to understand in this entire course. Does anyone have any ideas on how to better explain it? E. Zeno-like Paradox for Matter There is another form of Zeno's Paradox which applies to masses: if bodies are infinitely divisible, then contact is impossible. For example, when macroscopic bodies seem to touch, they actually consist of mostly empty space at the atomic level; so it must be their atoms which actually touch. But atoms are themselves composed of smaller particles and mostly empty space, so it must be these smaller constituents which actually touch. But if matter is infinitely divisible, this argument can be prolonged indefinitely, and nothing can ever actually touch. One might use this argument to conclude that there is a smallest possible unit of matter or substance. Imagine such a "unit particle". It must be utterly uncomposed. It therefore cannot be broken or divided, nor even deformed by spin or collision -- since these are properties of bodies composed of yet smaller particles. What then are we to assume will happen when two such unit particles collide? What density will the unit particle have? Indeed, will there be anything inside it at all? What would the unit particle's "surface" be like? Could it be hollow? With what thickness of shell? Would two colliding unit particles have to stick, since they can't rebound elastically? If they rebounded, with what resultant velocity? Would the unit particles be spherical in shape? Why would they have finite space dimensions, yet infinite dimension in time? Or do they come into and go out of existence constantly? Where and when would they appear and disappear? It should be apparent from these considerations that postulating a "minimum possible unit of substance" is no more logically palatable than a "minimum possible unit of space or time". Substance must be infinitely divisible, as must space and time; or else the paradoxes quickly lead to unresolvable logical dilemmas. But how then can matter ever experience "contact", if everything which might experience contact is itself composed of smaller substances? The resolution of this paradox would seem to be analogous to that for space-time. If the substance of bodies always gets denser (more substance per unit volume) at smaller and smaller scales, then in the limit as dimensions approach zero, density approaches infinity and substances approaching each other must make "contact" (i.e., at infinite density, they cannot be "transparent" to other substance). In the real universe, the density of matter greatly increases as scale decreases. Hence the ratio of mass to volume in electrons is enormously greater (about 1E10 g/cc) than the same ratio for matter in ordinary human experience (of order 1 g/cc), which in turn is enormously greater than the ratio for the entire visible universe (1E-31 g/cc). "Contact" is therefore possible for infinitely divisible matter, as long as the smaller and smaller particles continue to increase in density with sufficient rapidity, without limit. I appreciate that it is very difficult for the intuition to grasp this concept. Consider the approach of one minute particle of substance to another. As the outer surfaces approach, the lesser particles (call them "second level" particles) of which each is composed begin to approach each other. After the original particles traverse only a very small distance, the third level particles of which the second level particles are composed begin to approach each other. After an even more minute traverse of distance, and after an ever smaller lapse of time, the fourth level particles begin to interact. Although this continues without limit, as we have already seen, the process takes place in a FINITE time and a FINITE distance. The penetration of each level of particle into its counterparts in the approaching particle continues until the density of matter in the approaching particle is too great for it to penetrate deeper. Then the smaller particles at the next level penetrate until the density becomes too great for them to make further progress, and so on. By one-to-one correspondence with terms in our infinite series with a finite sum, we see that the depth of penetration has a finite limit, and requires a finite time, after which the original particles react with resistance to the intrusion of new substance into their ranks JUST EXACTLY AS IF THERE HAD BEEN A COLLISION! By analogy with the proposed resolution of Zeno's paradoxes for space and time, the paradox for mass is resolved, apparently necessarily, by the conclusion that substance must be infinitely divisible, and that it must approach infinite density as size decreases toward zero dimensions. This conclusion is reached by reasoning alone; it is reinforced by the observation that matter does in fact increase rapidly in density as scale becomes smaller over a range of 40 orders of magnitude in the observable universe. From the preceding considerations it seems altogether reasonable, and in a way compelling, to deduce that space, time, and substance are all infinitely divisible; because the consequences of the alternative are logically absurd. But if they are infinitely divisible on the smaller scale, what about the larger scale? Recall our earlier argument that the entire visible universe would have undefined scale in space, time, and mass, unless such scale is provided by the presence of other substance in the greater universe beyond. That argument must remain true without limit. The upper limits to the structure of substance, the dimensions of the universe, and the extent of time, must all be as unbounded on the high side as they need to be on the small side. This will become even clearer as we further examine the nature of substances. F. Meaning of Space and Time Let us return again to our empty universe which contains no substance, and therefore no frame of reference, except for a single uniform particle of substance. But as we have just seen, the particle must itself be composed of an infinitely divisible variety of sub-particles. We could have chosen a single particle at any of an infinite number of sub-levels to be our single particle. To avoid the issue of the arbitrary size of the particle we select, let us conceive of it as having zero radius. Although it does not, this conception will allow us to introduce one scale of distance at a time. As remarked earlier, motion and orientation have no meaning for a single particle in an empty universe. Now introduce a second infinitesimal particle. This gives meaning to orientation, since angles can be measured from the line joining the two particles. It also provides a single measurement of length, the distance between the particles. It does not, as before, provide a scale for the empty universe, since the distance cannot be measured in units of particle diameters, which are still being assumed to have no dimensions. Therefore there is no way yet to determine whether our particles are separated by a microscopic or a macroscopic distance. There is as yet still no meaning to motion in this two particle universe. The two particles cannot change direction, since all directions have meaning only relative to the particle-to- particle direction. And the two particles cannot change distance, since all distances have meaning only relative to the particle-to-particle distance. In a very real sense, this universe without the possibility of motion or change has no time. Time can have no meaning if there cannot be events or change to mark its progress. Put differently, if there were such a thing as an absolute time which existed somehow in addition to our two particles, the lapse of a microsecond or a million years would be just the same and utterly indistinguishable. But the existence of something with substance, such as an absolute time scale, violates the assumptions of our construction, that nothing exists except our two infinitesimal particles in an empty universe. Remember, we refer to "substance" rather than "matter" to cover ANYTHING which exists. An absolute scale of time, just as for a structure or framework in space, would have substance in this broad definition. Perhaps you have thought about one possible event or change which might occur in our two-particle universe up to this point. We might imagine that the two particles coincide, which is a distinguishable condition from non- coincidence. It might be fair to say that the first coincidence of the two particles marks the beginning of time; and that the interval between any two coincidences marks an interval of time. This interval still has arbitrary and indeterminate length. We cannot tell if the interval to the next coincidence is longer or shorter than the last (that implies an absolute scale of time to measure against). We can merely mark the progression of time by counting coincidences. This brings us to an important point of our mental construction. In an empty universe consisting of two elementary units of substance, the ordinary properties of the universe (time, space, matter) do not exist outside of the particles and between events of coincidence. It can therefore be said in a logically meaningful way that space and time which are empty of particles and events DO NOT EXIST! This eliminates a logical fallacy we have been skirting around up to now about whether the empty space and time surrounding our particles exist. In our construction they do not. Therefore our use of "substance" to mean "anything which exists" is logically correct, since a true void would not exist (in either space or time), in the operationally-defined meaning of the word "exist" as used here. Of course, for actual particles with finite dimensions, events of coincidence do not occur. Instead we have what may be operationally described as "collisions", in the sense already discussed. Two particles interact "collisionally" when their sub-particles at all levels approach the infinite density limitation and are forced to retreat. Notice, however, that if we were to imagine an infinitesimal volume of space IN OUR REAL UNIVERSE within which there were only two uncomposed infinitesimal particles and nothing else (including forces), then all that we have concluded about distance and time not existing between events of coincidence would still be true. No time or time interval would exist until an event occurred, with the only possible events being collisions with other elementary particles of substance. Therefore, on the most microscopic levels, time must proceed "instantly" from one collision event to the next. Reflection on this construction, which implies the non-existence of space and time between events in a region, begins to provide some insight into why the universe seems to behave as if space and time were relative, not absolute. We have reasoned to the conclusion that they must be. To emphasize the point that true vacuum implies non-existence, we are asserting that every point in the perceptible universe is at every moment of time filled with contiguous substance at some infinitesimal level. If substance could be imagined to become absent anywhere at any time, time there would cease and the perceptible universe would collapse until the "vacuum" was filled. Put another way, a particle reaching one edge of a "vacuum" would skip instantaneously to the opposite edge, just as if the "vacuum" had zero dimensions, because there is no substance to mark the passage of time inside of the "vacuum", and no absolute time without substance. G. Implications and Discussion Pausing for a moment to digest some implications of our reasoning, it would be fair to conclude that the only logically imaginable way in which substance can come into, or pass out of, existence (in this model) is for it to "enter" or "leave" the region of collisional interactions with other substance. But if there were such regions where matter density is so low that no collisional interactions between units of substance occurred, then all substance on the edge of such regions would instantly dissipate itself into the non-interacting regions, followed by substance slightly further in, and so on. All substance in this universe would dissipate instantly into the void. We suppose that even solid bodies are held together by the action of agents which would disperse if not continually held together by the presence of other substances, so that even solids would dissolve. Since this does not happen, we conclude that this universe has no such regions where collisional interactions between units of substance do not occur. The same reasoning applies to time. A cessation of collisional events would bring a cessation of time; but with matter existing everywhere with sufficient density for collisions, it follows that time continues forever, in both the future and the past. But couldn't substance redistribute itself so that densities no longer approach infinity anywhere, thereby ending collisional events? By analogy with the dissipation of substance in space, if it could so dissipate (for example, if the amount of substance in the universe were finite), it would have already happened, virtually instantaneously. Conversely, if substance does not start out with density which approaches infinity as dimension approaches zero, it could not assemble itself into such an infinite-density configuration in a finite time. We may therefore be reasonably certain that the "universe" (in our model) is infinite in space, time, and mass or scale. It must be the case that every bit of space is occupied at all times by a continuum of substance; and that wherever substance is not, existence of time, space, and matter is not. The substances whose presence "define" space-time must be infinitesimal compared to the substances in our experience, such as baryons or photons, or even neutrinos. Distance scales must be purely relative, with no absolute meaning to "large" or "small". Likewise we should not be surprised by very large velocities. If distance and time scales are unlimited on the large side, then velocities must be also. We will discuss in future weeks how this can be reconciled with Special Relativity, which postulates that the speed of light is a maximum speed in the universe. Our conclusions are deductive, not inductive. So they can be invalidated only by faulty reasoning or an incorrect starting point or assumptions. They do not at first glance appear to lead to descriptions of the REAL universe; for example, they do not easily reconcile with the sort of universe inferred using the Big Bang as a starting point. But we will see in coming weeks that, although they do imply changes in some of our theories, the descriptions from this new starting point seem entirely reconcilable with REALITY. If they add understanding and make successful predictions, I argue that is sufficient for them to be worthy of consideration AS HYPOTHESES in the field of astronomy. ******** If this model is unclear, ask questions to help clarify it. If the model is clear, it must lead to a description of the real universe and make successful predictions to be of value. It is still too early to compare this model with observed reality. But let's see some discussion of the implications of the model to this point, and perhaps anticipating the next step by guessing the nature of FORCE using this model. 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