Steven Louis Grillo 4/17/08
So you might be, like me, a little lost when it comes to conversational Special Relativity. It's certainly an interesting topic, but the details - well, they can be a bit daunting. So you, unlike me, have probably (and wisely) passed on making any serious efforts to digest the unique set of ideas it presents. You, like most, probably get by on the very basics of Special Relativity; some bits about shrinking space; a parcel of expanding time; a twin brother staying younger than you, just because he stows away on a rocket ship. All essential trivia for the 21st century. Granted. I am, however, more easily described in terms of my trivial pursuits. I have put nearly 30 years into unraveling the vagaries of Special Relativity. And yet I mean what I say in the opening sentence - I think I understand Special Relativity(SR) as well as any - but making easy conversation out of it is a whopper of a task. It's not so much the odd alteration of elusive elements like time, and space (and don't forget that "at rest" and "in motion" are really tough to parse) - it's more that the weave of reasoning seems to drop a stitch here and there.
Take, for instance, one of the first niceties of SR one might encounter. It runs along the idea that all non-accelerating frames must consider themselves "at relative rest". The next step in the larger argument then becomes the clever observation that relativistic calculations are/must be reversible between any two frames- meaning, they work exactly the same from either perspective. Einstein makes an elegant example of this when he discusses the two trains at the station. In his time, travel by train was common, and certain situations were also common at a busy hub. Often trains would decelerate very slowly as they neared the end of the trip, and by the time a given train reached the final few hundred yards, it was gliding very slowly, indeed, to its eventual stop. I suppose the actual sensation is hard to capture, but I'm guessing a well-made carriage car (and long string of same), gliding effortlessly on well-made wheels and track, would give almost no hint of motion. One would need a window and external cues to deduce that one would be still rolling along the track. If another such train were on the adjacent track, but not yet in motion, a certain trick of truth would be at hand. Since the trains are so close, the moving observer could only see into the adjacent carriage car, and not necessarily see its wheels. At this point the question arises, is the other car moving or standing still? In fact, if all the far windows in the other car are covered, blocking all the external cues, the moving observer might assume the car he's on is already at rest and the other car is in very early acceleration for its journey. Good proof of exactly which scenario is correct in this peculiar circumstance, would be difficult. And thus begins the famous dethroning of "at rest" (absolute rest) - when one takes into the consideration the orbit of Earth, and Sun's galactic orbit - one can't very easily grab a big bunch of rest. Things are, ah, well... more relative.
But here's where we seem to drop a stitch. By technical definition in SR, all non-accelerating frames are at relative rest. Thus, all relativistic calculations are always, so to speak, about the other guy (frame). You use the famous calculations to determine how a receding or approaching frame is altered by its apparent motion. When you determine by your standards how fast a frame is receding from or approaching you, you can use the relativistic calculations to determine how much the other frame has shrunk, and by how much its clock has slowed. This part is not such hard conversational fodder. It makes a certain coherent sense that the relative velocity alone determines the alteration, whether the frame is coming or going. But here's the rub: How does the Twin's paradox work? You see, anyone in the other non-accelerating frame must use the calculations just as you did, to determine how much your frame has shrunk, and your clock slowed- all due to the same apparent motion - just from their perspective(frame), because all non-accelerating frames are taken to be "at relative rest". I'm at rest, your at rest; each of us use Einstein's calculations to reveal how the other is in a different relativistic state. Do you see the rub? I don't want to tilt at windmills just yet, but something is wrong with SR, I assure you. If I am shrinking and slowing your frame with the same calculations with which you are shrinking and slowing my frame, how do we get out of balance? How does a twin get on a super fast rocket, travel for two years by his clock and return to find his twin brother many more than 2 years older? Discounting the rocket's acceleration, for most of the trip it was defined as a non-accelerating frame, with a specific apparent velocity relative to Earth. Two such frame's should each see the other as the "receding frame;" each should then slow the other's clock. Each should then shrink the other's frame. Each using the same velocity (apparent motion) in the calculations. What goes wrong to make the ages not match?
One logical place that might help resolve the problem is to consider the phase of extreme acceleration as responsible. Maybe, in such extreme acceleration, the rocket-riding twin goes back in time, gets younger for the acceleration phase, and then ages normally - as the perfectly interchangeable calculations really beg for us to do. But, no. There is no special application for extreme accelerations that can help here, but we can point to a trick that could be helpful here. What if the rocket were under a very gentle, and so constant, acceleration for the whole journey? At least then we could have some wiggle room with respect to our basic definitions. If the rocket were under constant acceleration, then it could never be said to be a non-accelerating frame; the whole assay would then be a comparison between a non-accelerating frame and an accelerating frame. At least there is enough fundamental difference to impute a difference. Not bad, but then you might want to use the basic logic you learned in SR to say, "Hey, how do I know which of us is accelerating?" Clearly, a frame in acceleration will see the other frame as receding at an ever increasing rate! I'll wait while you go take some aspirin... And by the way, here's the rub: Einstein attempts to marginalize absolute rest; he has some handy and solid reasoning for this, but to marginalize absolute rest requires the marginalization of absolute motion. The rub is that neither remain marginalized as the model tries to develop into its wider investigations.
Consider the the above conundrum of determining which frame is accelerating. It is an example of how slippery a slope it is to expand logic models. SR allows a frame the ability to "know" it is accelerating, and what Einstein means by this is an onboard and self-reliant method of determining such acceleration. So in the above example, you're not lost if you're the one accelerating... And so in the above section we left the meaningful analysis right after we asked, "What goes wrong to make the ages not match?" How does Einstein go from "interchangeable calculations" to "age discrepancies"? Remember Einstein is building an empirical model, and to build an empirical model one utilizes fitting components for the observed behaviors. The extendable elements of the model require relative rest and relative motion; the primary components of the model are the calculations which first apply to a known physical problem. When Einstein saw how niftily these calculations fit the problem at hand (See Michelson/Morley interferometer), he saw how they made impossible the term, "absolute rest." And, axiomatically, absolute motion loses meaning. The calculations implicated a phenomenon that made any non-accelerating frame appear to be at rest because there could be no experiment that could help ascertain any absolute motion; hence, all motion is relative. All non-accelerating frames are "at relative rest". It just fits. But now you have the problem of a given frame trying to figure out what's going on all around it. Here, we can see Einstein-types making the calculations work from their sole perspective. The high-velocity recession of a frame translates into an time-shifted frame - where the time passes more slowly for it than the the guy "at relative rest" making the calculation. It just fits. Empirically speaking, it really doesn't matter if the tag ends don't meet. When Newton figured out the computation for gravity, it really didn't matter how odd it would turn out that a cannon ball and a feather, in a vacuum, would fall at the same rate. Empirical formulas (and Newton's expression for the acceleration of gravity is one) do what they do, and the chips of expectation fall where they may.
But still it seems a glaring inconsistency that the rocket-riding twin doesn't calculate his Earthbound brother to be the one that ages more slowly. It seems certain from this that some relative rests are more special than other relative rests. I think it can be suggested without citation or reference that the reader examine any explanation, in any media, of the Twin's Paradox. After the reading, I ask the reader to say he is satisfied with the term, paradox. Not-very-internally-consistant is more my take. What seems clear to me is that there is some conservancy regarding the analytic roots built rather subtly into SR. Even if you turn time and space into types of scalars, motion and rest remain inherently real. Whereas, Einstein's model succeeds because of how confusing it is to have Earth, Sun, and Moon in constant, yet plausibly relative motion to one another ( it is clear there is no absolute rest as pertains to them), it seems to fail when sentience wants to wring out the details. What Einstein's model fails to consider properly (and this is the heart of the conservancy) is the fact that an emission point in space of a photon is a rest point - the photon will immediately leave that point, and the source will most probably immediately leave that point, but that point coordinates space for those two real entities (the source and the photon), and serves as a rest point for their future behaviors - no matter that human technicians have difficulty putting a pin in it. What we think of stars in the night sky follows this idea exactly. We know that light leaves a point, and the source (star) leaves that same point, such that we know we never see a star in its current location, we see the point in space it left when its light began to travel towards us. The star has long left the point in space we see twinkling. As we can see, these macro-structural logics collide with the empirical tenets of SR in very odd, and unsatisfactory, ways. I say the macro-view(general astronomic) should have a better fit with the micro-view(local frame). And we could more easily see how to do this if one were to examine SR through a wider lens, just to see if the analytic roots can help with the tag ends.
If we accept the emission point of a photon as a rest point, then it becomes axiomatically true that the finite speed of light (the fastest thing we can measure) becomes an index against which all other motion can be compared. [Yes, I know that some readers are aware that the M/M interferometer was, at heart, just such an attempt to reveal Earth's velocity as indexed against light - and it failed to do so; therefore, SR comes into being as an explanation... hold, and read on, I implore you.] In that all motion can be indexed against the finite velocity of light, we can defend the term," absolute motion," even if we might want to use a small "a." We can at least say a frame is "absolutely indexed against the speed of light." A more preternatural sense of Absolute, we leave for philosophers. And so, if you leave here at .4c (c= the speed of light), then there are two roots to your journey. One is the true indexed velocity you have against light speed, (let's go ahead and say, .4c); the other is the relative velocity you have against any real velocity I have. If I am drifting in the same direction as you, but I am doing so only at a light indexed velocity of .02c, the we are apparently separating at only .38c. And this observation is interchangeable - I see you receding at .38c, and you see me receding at .38c. The improvement we are building-in here concerns the extrapolation [watch carefully] that your compact with thermodynamics is related directly to your true indexed velocity - you age as anyone would traveling, in any direction, at .4c. I age as anyone would traveling in any direction at the indexed velocity of .02c. (I can tell you that Einstein's model does this for the Twin's Paradox, but the base tenets of his arguments really can't give any satisfactory linkage as to how. I often wonder if the paradox lies in the ambiguous cross-threads of his model - and not the simple oddment that comes with twins aging differently - surely as it is though, a great puzzle for the average guy on the street.) This forces the coordinating structures of the emission-point model to do this: if I turn around and drift exactly opposite your direction, still at a true indexed velocity of .02c, we will now be apparently separating at .42c (and, of course, this observation is interchangeable). This new velocity would have great meaning in SR - the relative aging should change as the relative velocity changes from .38c to .42c. But under emission-point logic, this change is dismissable. Our indexed compacts with thermodynamics keeps our comparative aging the same - I am still aging at the .02c rate, and you are still aging at the .4c rate. Yes, I am still getting older than you, but it makes no difference that our relative recessional velocities have changed from .38c to .42c - the increment by which I get older than you is the same. Our relative velocity to one another is fairly meaningless. In fact, when our sentience sees the change in direction as responsible for the new relative velocity, we are using the underlying analytics which Einstein's model tries to dismiss. Einstein would simply claim from SR that we are looking at two different events and we would have little power under the tenets of SR to determine the fine point of a reversing of direction. That's the whole point with SR; it scrams basic ideas like absolute rest and absolute motion - and any fixed reference point they imply, so that we can't reckon with the basic analytics we so inherently rely upon. Under emission-point logic, direction matters; under classic analytics, direction matters; under SR, direction doesn't.
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Two things are in play with this extrapolation. The first is the clinically clean methodology of fixing a quantitative behavior (time-alteration) to a thermodynamically pure element (velocity - as indexed to light speed) The second is that we must presume to return to Cartesian (Classical) analytics for general cosmology. This second part demands that we succeed where Michelson and Morley failed. We must be able to ascertain our own frame's "true indexed velocity against light speed." Einstein says there is no experiment that can do this. I say there is!!! I've written two papers detailing the design and general analytic underpinnings. [These papers are following a different path]
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Continuing to explore comparative frames, we take triplets to the North pole. Send one east; send the other west; and keep one at the pole. The eastbound and westbound of the triplets each travel at .4c relative to the earth. After several years pass on their clocks they find themselves returned to the north pole. Under SR, they should be the same age, and the pole brother should be much older. But look at the problem from the point of view of either of the travelers. The pole brother is receding at .4c, his space counterpart is receding at .8c. How can the slower relative recession age a frame more than the faster receding one? This is exactly opposite the relativistic prescriptions. His counterpart receding at .8c should be older by far than the pole brother. At this point, I can offer no real hope for simple relativistic calculations; they too easily fall into such messes. Our more resilient comparative analysis can go on to say this: Above, we can say the earth had .0c true indexed velocity. But what if earth were traveling at .06c in the westerly direction. If the space bound brothers again achieve only a .4c relative separation from earth, their respective true indexed velocities will not be the same. The westbound brother will be traveling at .46c in the true indexed sense, the eastbound brother will travel at only .34c in the true indexed sense. When all meet again, they are at three different ages.
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It was stated earlier that it seems Einstein's model really can't marginalize absolute motion as readily as he might wish. We are speculating that the age paradox is only meaningful if absolute motion has a thermodynamic component - and such truth hides in the empirically unimportant gaps in his model. With the help of a few animated graphics we can paint an even broader picture of how important it is that we try to align the "macro" to the "micro" on our way to a better view of thermodynamics and its analytic elements... I have reflected on what would make a relativist take a close look at the gaps in the SR model. Perhaps a scenario that illustrates that all motion is inherently index-able against light. And to ignore the quest for such indexing, leaves us with the Einstein of the Gaps...
We begin with three elements: a spacecraft on Earth, Alpha Centauri, and a distant pulsar. For convenience we are going to allow that Alpha Centauri is two light years away. The pulsar is thousands of light years away - its distance away isn't important to us, but we like it because it sends out one pulsar burst (blip) every second as measured on Earth. That's a total of 31,558,464 blips per year. And what's important to this scenario is that there are 63,116,928 blips inscribed into the gap between Earth and Alpha Centauri at all times, all traveling at light speed. see Scenario 1 and 2
Our mission is to send out the spaceship with an observer to make a lap around Alpha Centauri, and return to base. We will issue the observer a brand new craft - the Mark .5c Starlapper. It can accelerate to .5c relative to Earth. It can achieve its top speed in a negligible 2 Earth days. A back-of-the-envelope calculation reveals that it will take 4 years for the craft to reach Alpha Centauri and 4 years to return. Overtime starts for the observer after 3 years, but I digress... see Scenario 3
A more telling calculation can be performed even before embarkation... Using SR's calculation for time elongation, t' = t/SQRT 1-u~2/c~2, we can see that 1.155 seconds will pass on Earth for every one second that passes on the Starlapper's clock while on its journey. Not seemingly a large difference, but after 8 Earth years, the Starlapper's clock will have counted off slightly less than 7 years worth of seconds. The observer onboard will be a bit more than a year younger than the Capcom he left at launch. The observer, by the SR calculations, will be 1 year and 26.86 days younger than all he left on Earth. These are easily collected details. Now as it turns out, we have chosen a very special observer for this trip. He is a young grad student that has undergone a new sort of program where he has been given the essential elements of several cosmologic models; two that have little in common with good fundamentals, and one that is the bare bones but "complete enough" model of SR. It will be his final exam to conclude which of the three cosmologic models is the most widely accepted by the experts at large in his field. He is given the hint that maybe his onboard clock will "act funny" - so he must stay sharp.
He is given the distance to his target, 2 lyrs. He is given his craft's terminal velocity relative to Earth, .5c. Back-of-the-envelope ciphering tells him that this is an 8 year exam. There is a simple blip counter on the ship. His onboard clock is state of the art. A team calibrates it on the the launch pad to perfection. They are able to note that the background pulsar is blipping away at 1 blip per second. And it will happen as this trip goes forward that the tenets of SR will be in play; his clock will slow, but he of will have no way to prove this - just as SR would have it. Whoosh... all is away.
After the requisite 48 hours of acceleration, the engines cut and the observer enters free float - the most natural physical state according to SR. He sets to work on his exam. The cottage cheese model seems way off - there isn't one curd to be found anywhere. The "scream at the top of your lungs and you'll hear a duck's quack in the echo" model seems not very germane. And try as he may, he can't seem to find anything wrong with his clock. "Things will behave normally.." he remembers from somewhere in his studies; so far, so good. He sets up a simple test. He will count 3600 blips on the blip counter (an hour's worth by Earth standards) and see what his clock says. Whoops, when he consults his clock it reveals that 3117.6 seconds have elapsed for the trial. His clock is running slow compared to what he expected. In fact, he notes that 3600/3117.6 = 1.155 - which was the preflight relativistic calculation by Capcom. One might think this approaches proof of motion because the student observer got the right answer. Not really; Einstein's challenge was meant for anyone in a wholly enclosed frame (no external cues - which the blips are). The student could conclude that clocks on Earth are now clicking and clocking along faster than his is. [Points on his exam] Or he could assume the pulsar is speeding up, and simply sending more radiation to Earth per unit time. But then again every thing outside his window is picking up the pace compared to his sense of normalcy. Frequencies are everywhere showing up out of their norm. All other pulsars are quickening. Is it possible that his frame is NOT aging as fast as everything else? Seems so. But, then again...
His observation that 3117.6 tics have gone by while 3600 blips were counted, leaves him with a bit of insight into SR, but the limited scope of that calculation does not help him see through a wide enough lens to say he's head level with all the cosmos has to present. Assume there are way-points (markers) every .5 light years along the way. There would be three, and the fourth would be Alpha Centauri, itself. Counting up groups of 3600 blips will not help much when he gets to the first way-point. The first way-point represents the first year's journey for the Starlapper. On Earth, the blip counter at base headquarters has just tallied 31,588,464 blips, and the base clock reports 31,588,464 seconds have passed, and all the base personnel expect the Starlapper to be at the first way-point as it traveled at .5c for a year.
The student at the first way point reels a bit. His blip counter reads a grand total since launch of 47,337,696 blips. His clock reads 27,329,629 seconds have elapsed. His first simple calculation shows that he experiences 1.732 blips per second per his clock. And that checks out with his bare bone knowledge of SR - things age faster than you. And he really is aging more slowly if his clock is in sync with his thermodynamics. But does anyone reading along really think that a simple relative relationship with Earth has anything to do what this student is experiencing. And again, how is it that the interchangeability of the model does not have the Earth slowing relative to the non- accelerating frame that is the Starlapper? You see, all of this analysis is absolutely dependent on the integration of the frame that is the Starlapper with the pulse train that represents a form of absolute motion. You can't do this very simple test without integration being critically implicated. And yet Special Relativity was designed to completely ignore the integration of a frame with light. That's where the famous Michelson/Morley interferometer experiment comes in. It was trying to do just that - capture the integration of Earth's velocity against the velocity of a spatially inscribed light ray. The Earth has a finite velocity; the light ray has a finite velocity - they must integrate. And yet this test failed in 1887 - to the shock of physicists everywhere. 18 years later, Einstein, in 1905, supplies the best fitting answer since. In this example, the pulsar blips integrate with the Starlapper; there is no way around it. The integration supplies all the meaningful roots, even if there might be a way to negate certain artifacts of calculation that apply to it - like SR does...
*** The Starlapper 2 sets out without a clock. It does contain a blip counter. The pilot notes in the control room a board that has the target at 2 lyrs away, and that the Starlapper 2 will cruise at .5c on its way. He notes that there are way-points at .5c intervals. He notes that the pulsar emits a pulse every 1 second. The pilot of the Starlapper 2 is the French mathematician Des Carte. As he blasts off, the first, and pivotal, concern he has regards the integration of the two elements: the Starlapper 2 at .5c, and the pulses inscribed at c in the frame of space. Within minutes of his launch, he deduces that if the blip counter reads 47,337,696 at the first way-point, the Starlapper 2 is well and excellently on its way. He knows that a year shall have past on any clock that respects the finite value of c. He knows that by such a clock he will be experiencing 1.5 blips per second due the absolute integration of the two primary elements: the Starlapper 2 and the pulses from the pulsar. All Des Carte needs is someone to design a test that will properly resolve the integration of a frame with the inscribed spatial behavior of light. He doesn't really need such a test to remain a sentient creature, but it would be nice to know that sentience can be in agreement with good tests.
To complete the journey, the student must circle Alpha Centauri and return for his grade. It bodes poorly for him, as he returns to .5c heading home, that the original pulsar blip-rate falls off sharply. He notices that things in front of him behave considerably differently than things behind him. He consults the cottage cheese theory- nothing there. As luck would have it, a genie appears and tells him that enough is enough - it's time to get home. With that, the genie charms the engines to work the miracle of pushing the Starlapper to very nearly light speed. Tallying anymore of the pulsar's blips becomes useless, and the rule against factoring to integrate is not allowed. What he will put on his report, he does not know.
But luck was never kinder to grad student in need. Just up ahead, he sees a strange configuration of elements. He sees what is certain to be a cube, and just behind it a large mirror. As he passes just beyond the cube, as if by magic, the panel that was farthest away from him as he approached the cube appears, for a moment, in front of him. He has passed the cube and yet right in front of him is a panel that was fairly obscured when he passed the cube. This report will get him an A, if he simply doesn't mention any sort of integration in the summary.
A freeze-frame shows the moment the panel appears before the student, a purported phenomenon for Special Relativity.
Clinging to the bare-bones of SR, the student may have seen on this journey bits that will keep him in good stead with the grading panel. How he will keep integration out of his summary is anyone's guess...? The primary issue that was key to his experience was the fact that every blip he encountered approached and then left him at 1.5c. The blip's velocity integrated spatially with his velocity. At each way-point the blip count was correct for just such integrated activity. He may have been under a thermodynamic condition which slowed all in his frame, including his clock, but the wider lens of his experience leaves an impression...
After presenting his summary, he is granted his certificate by the panel. He continues for a time to extend his knowledge of SR. After scholarly reflection, he produces an independent analysis. He is taken aback by the fact that Earth saw his spacecraft (frame) as shrunken while it traveled away at great relative velocity. We find the student is duly emboldened by his experience; he will make an effort to add an interpretation to the model of SR. It will concern the possible thermodynamic roots of shrinkage...
Scenario 9 is an animated graphic that illustrates the general idea of frame shrinkage, if reduced to an atom. Never mind that it needs to be on a frame moving relative to you to be seen, as such; but keep in mind that interferometers here in the Earthly frame are imbued by the shrinkage of SR, it will just never be perceived. [and yes, the solution of SR was generated to fit the M/M anomaly, but properly understood, if the model works you'll never see evidence that such an interferometer is in motion. So doesn't the SR fix imply a more fundamental motion. More simply put: SR masks motion for a local frame (as Einstein would have it for the M/M experiment) or it does not. It becomes the "weaker conjecture" to suggest that a frame is always to be taken "at relative rest" instead of saying its motion is masked by a beautiful phenomenon.]
But the student has problems with this simple shrinkage. If the integration of frames and energy waves inscribed in space has a wider value, then it would lead to a more strict interpretation of how atoms would behave under absolute motion. Under SR's broad mechanics, one can average the charge balances between nuclear components, such that the round trip distance to any point requires the same amount of time due relative motion, but how does an electron that has moved closer to a proton know to shed some of its charge? If strict integration were allowed, then the orbit of the electron would require ever-changing axial limits as absolute motion were increased, but its orbital diameter would remain intact - and so, the charge relations.... ( see Scenario 10)
Of course, with increasing axial limits, comes the essential shrinking of the atom... (see Scenario11). A broader speculation suggests that, for heavier atomic weights, this axial shifting would load the weak force, and perhaps any collision of a hi-V atom against a target would, of course, unload some fractional nuclear energy along with any kinetic energy inherent to the mass.
Now the reader would have to attend the fact that this axial shifting is not completely in line with the strictest integration mechanics. What we are looking at here is an invocation of Occam's Razor as it would apply in the face of relativistic mechanics versus integration mechanics. It simply exists that on the arc described by the orbit, the extension of the cycles moves from 1-u~2/c~2 down to SQRT 1 - u~2/c~2 as one moves from the leading edge around to the side edge. Within all probable parameters, the point on the arc that would satisfy the relativistic shrinkage would be very near the elevation one would calculate for the relativistic shrinkage. (see Scenario's 10 & 11) The shifted axis keeps the final round trip in harmony with the simple shrinkage produced in the SR model, and of course the entity (atom; frame; etc.) would appear similarly shrunk... For the larger case, we can show clearly in these final frames just how the strict integration model insinuates an even deeper requirement for a complete analysis, and reveals an important oversight in the SR model.
We begin with the essential design for the famous Michelson/ Morley interferometer experiment, which became the root of the relativity era. It relies on the idea that a circle at rest will reflect uniformly moving subjects (pulses, particles, widgets) at the same time and return them to the source at the center in equal lapsed times. (see Scenario 12)
The larger question put to the test by M/M, after a few basic facts about light were gathered up by others (Newton, Fizeau, et al), dealt with the fact that a moving Earth would integrate, in accordance with some well understood analytics, with the frame of space - to which light possessed unique characteristics - to produce a gap between two light rays as these rays cycled inside of a device called an interferometer. This gap would instruct M/M and the physicists of the day as to the "real" velocity of Earth through the vacuum of space - a handy thing to know when it comes to knowing what other items in space are "really" doing. See Scenario 13
It was "really" a first-order surprise when they saw no gap at all.. (see Scenario 14)
Such was the nature of this surprise, that it remained a mystery from 1887 to 1905. Although Fitzgerald and Lorentz produced some enticing empirical conjuctures, it was in 1905 that Einstein proposed his remarkable model to speak to this mystery. But for 18 years the analytical model stood in plain view for all to see, and somehow it failed to produce a proper interpretation for the M/M interferometer. (see details Scenario 15) The round trip for the leading edge (Leg A) just had to take longer than the round trip for the side edge (Leg B). The integration of light and objects was strangely absent.
Of course, if Leg A were to shrink by just a little (SQRT 1-u~2/c~2) then the round trips would harmonize. In the interim, a guy named Fitzgerald proposed just that.
We can see in Scenario 17 what a shrunken frame look like when mathematically smoothed out. At any point on the shrunken arc, the round trip is the same. However it must be said that it is the underlying integration that does the smoothing. The side edge is still taking SQRT 1 - u~2/c~2 for its round trip. The shrinking only helps by removing the mismatch of the cycles - the cycles are still longer due integration, i.e. the real integration of the frame through the space that governs the light.
It was Einstein's Model of Special Relativity that came along and made the integration seem to disappear altogether. By regrouping the elements of the problem, he is able to mask the integration by localizing the implied outcomes of his model: to any observer looking down at the interferometer, he sees no shrinkage; and equally important he cannot see or determine any extended period for the cycle. This observer can see and determine these things on some other frame. He will have to determine that he is at rest in his own frame. It really works out beautifully.
However, it remains true, whether easy to SEE / DETERMINE or not, if relativistic shrinking is going on in ANY frame, it is integration that holds its roots. (see Scenario 18)
It is not a moot question then to ask if the integration is purely relativistic as the SR model would hold, or absolute- meaning all frames possess a unique and real indexed value against light as the root for any integration applicable to frames.
In these concluding frames we look rather directly at the connection between integration and SR. If we shift the Leg B extension down to represent the way a local frame in SR sees it (the ray is always on the leg) and realize that, after shrinking, the Leg A extension is equal to it , we can see how the time expansion in SR is dependent on the integration along Leg B. The time component for a frame (external) alters by SQRT 1-u~2/c~2, the Leg B extension. This is what integration is doing to the Starlapper's clock. (see Scenario 19) By making the round trip longer, the clock slows down.
Even as we see how beautifully things balance out for the tenets of SR, we wonder if integration is wanting to tell a richer story. Below we see how increasing frame velocity would expand the time inherent to that frame (see Scenario 20). Standard algebra can be shown to produce lengthening "time" cycles along the axis of motion, but they are pure extensions, requiring elongation by 1 - u~2/c~2; this due to the fact that all integration components are honored along LEG A. SR extends time by the SQRT of the 1 - u~2/c~2, ascribing such expansion to the cross-axis factor, the LEG B factor.
For these efforts I have written two papers to illustrate the potentials in an alternate view, more properly honoring absolute integration. In the most recent, "Condensed Argument for Case against the Relativistic Convention of Isotropism...©2002," I propose a strict reinterpretation of the M/M experiment, as means to re-address the puzzle it poses. In its close, I propose a test that will implicate integration as the root to a more rigorous analysis of spatial reckoning - frame against inscribed light rays. This test, as I propose it, can be performed in any closed frame. And its result will implicate absolute motion.
Basic calculations for this survey are available here.
Steven Louis Grillo 4/17/08