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Chaos etc - equations and graphical output.
Eight equations generating exotic behavior,
along with the program code and graphical output.


Special Relativity explained in absolute terms -

eliminates the twin paradox, shows Einstein's clock sychronization
diagrammed in absolute terms, and ends all confusion regarding
relative frames of reference.  Completely compatible with, and in
fact subsumes, Einstein's relativity.  Not Lorentzian relativity.
Reveals what is transpiring behind the scenes of Einstein's


Relativity in Absolute Terms.
My most comprehensive online document.  A concise overview
of why special relativity must be diagrammed in absolute terms.

Twin Paradox Animation on youtube.
Light rays and traveling twins are charted in absolute terms, 
free of the misleading space-time diagram.

Twin Paradox Animation.
Expanded text, and animation of the twin paradox.  (Embedded youtube animation.)

Twin Paradox Explained.
A similar discussion of the failure of spacetime diagrams.  

Twin Paradox Animation.
Alternative text, and animation of the twin paradox.  (Embedded youtube animation.)  

Absolute Frame of Reference
Absolute frame of reference in the physics community.

Free pdf file of the book:

Relativity Trail, free pdf format, with 192 pages, 65 diagrams 
and 75 illustrations, will provide you with complete detailed 
algebraic derivations of all the kinematical effects of special 
relativity.  Everything is charted out in absolute terms against 
a system at rest with respect to the totality of the universe 
for perfect clarity as well as soundness of theoretical basis.  
It is the totality of the universe that imparts the inertial 
properties of clock rates and lengths which generate the effects 
of relativity.  This is explained in detail in Relativity Trail.

Excerpts from the book Relativity Trail with included images.

Einstein explained in excerpts from Relativity Trail.

Diagrams and derivations from the book Relativity Trail.


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end links

A few excerpts from Relativity Trail

page 10
Even though the word dilatation (or dilation) is the term most commonly used for time-keeping fluctuation, it's a poor choice of word for our treatment of relativity. Dilatation is defined as a stretching, and is used in conjunction with the inverse of the equation we derived above, referring to the dilatation of a clock cycle, which in turn implies the slowing of a clock rate.

page 11
When we speak of time fluctuation, we are always referring to the slowing of a clock, whether as an actual slowing or as a measured effect. And our simple equation above is the direct formula for the slowing of a clock rate. We'll try to stick with "clock slowing", "time contraction" or fluctuation, as context dictates.

Note in diagram 5, that we express the velocity of the traveler (clock B) as a ratio of the speed of light. Instead of expressing clock B velocity as c/2, we simply express it as 1/2. This simplifies the form of the equation. After all, c (the velocity of light) is our standard by which all other speeds are defined. And the distance light travels (in our arbitrarily defined universal second) gives us a distance of one light second. The light second then serves as our universal clock. As far as light is concerned, time and distance are the same thing. This will take hold in your mind as we proceed.

When we speak of clocks or travelers, let's consider the gents riding along on the Relativity Trail. These gents are travelers who carry clocks and who age according to the functioning of their biological clocks.

An interesting thing about the time difference that crops up when a gent goes on a trip is that it must be a round trip, or at least a trip during which one party undergoes a change of inertial frames, in order for the gents involved to determine that there is a difference in the registering of time on their clocks. Sending out a light signal from one gent to the other to check on each other's clock will not reveal to the two gents any asymmetry in the situation, so long as they continue to move apart at a uniform speed. Yet we saw, from the perspective of "universal time", that B aged more slowly than A.

This relates to the "clock paradox" (Twin Paradox).

And we must pause here to clear up this terminology. Most writers refer to the "Twins Paradox" as the phenomena whereby a person who goes on a trip returns to find that the twin brother he left behind has aged more than he has.

Apparently, this is considered to be a paradox due to time fluctuation being contrary to our common sense.

In the standard presentations of relativity, the consideration of this "paradox" then leads to a seeming contradiction which goes beyond the "paradox".

page 12
It goes like this: The standard interpretation of special relativity is dismissive of any universal reference frame serving as a baseline from which to analyze uniform motion. Some popular writers on the subject have said, “There is no truth of the matter” concerning reference frames. And so, when two gents part company and then meet up again, how do we know which party has really traveled (or traveled more than the other)?

Either gent might seem to be "at rest" or "traveling" if there is no association by which to analyze motion relative to the overall structure of the universe, and the standard interpretation affords no such analytical association of a party's motion relative to the universe.

This controversy has not gone away after over a century of relativity; nor can it, without acknowledging the universal frame of reference.

The standard interpretation is that we must simply take notice that one of the parties underwent a change of inertial frames during the course of the round trip. But that consideration alone does nothing to relieve us of the need of a structure which has imparted actual clock rate differences, considering that reunited clocks, moving in a straight line, strictly without acceleration, display a time differential of an absolute nature. It is precisely one's inertial change with respect to the universe that dictates the new actual clock rate, resulting in the actual time differential upon reuniting with the other party. And that, of course, is something we'll diagram with clarity in this book.

Meanwhile, refer to diagram 6 to see what happens when, in the course of a one way trip by a gent, the two gents involved (stay at home and traveler) try to determine whether there is any difference in the time-keeping of their clocks. Their best tool, of course, is the sending of light signals to each other to relay information about the status of their clocks.

Note the symmetry between case 1 and case 2 of diagram 6. They cannot detect that one is recording time more slowly than the other.

In diagram 7, B "reverses" direction by way of transferring clock information to clock B' (B prime) coming from the opposite direction at the same speed relative to the universe.

Interestingly, even though the two gents cannot agree that one or the other is recording time more slowly as B moves away, they do note a lesser recorded time by B as soon as B begins his return, as shown in diagram 7. This noted time difference builds incrementally as signals are exchanged ever further beyond B's turn-around point. (In adventures of this sort, much of what A and B conclude could also simply rely upon the comparing of notes upon reuniting.)

page 13

page 14
They must conclude that B has changed inertial frames with respect to the universe (as well as with respect to any inertial frame), due to his lesser recorded time passage upon receiving the light ray at the end of that ray's round trip. Yet that is not the same as A or B being able to know anything about their motion in relation to the universe at any point in the adventure, for they did not know what motions they had in relation to the universe to begin with. We'll illustrate with diagrams, this impossibility of knowing, in chapter 4.

(For a comprehensive table of "clock status" data via light signals, i.e., radio pulses, experienced by gents A and B, see the appendix.)

Clearly, no acceleration effects are involved in explaining the time differential. In fact, no acceleration is incorporated in Einstein's derivation, yet the time differential arises from his derivation, just as does the mutuality of measured clock and length distortion across inertial frames. You cannot derive one without the other. Just as clock information is exchanged across inertial frames to effect the observation of mutually measured clock rate slowing, so too does the transfer of clock reading from an outbound traveler to an inbound traveler effect an observed incremental increase in clock rate differences.

page 18
Instead of light behaving as if though it were a wave phenomena in this experiment, it behaved as though it were a particle, capable of having additional speed imparted to it by the motion of its source. For that is what should be required for a tie to occur in such an experiment. Its path seemed to be just what a stationary observer would witness looking down through the glass ceiling of a moving railroad coach car in which mechanical principles were being demonstrated. The impartation of additional speed is what is required for the Galilean Principle of Relativity to hold true.

But of course, physicists had already amply demonstrated through other experiments that this could not happen to light.

But here again is that pivotal point we mentioned in the introduction:

They did not consider light to be of a particle nature because they considered its speed to be not dependent on the motion of its source. Remember, they believed all particles had the mechanical property of additive velocity, dependent on the motion of the emitting body.

page 20
The experimenters were trying to detect the aether wind, based on light having a wave nature. They didn't detect any aether wind, and to explain the results, were forced to assume that an object underwent length contraction in the direction of its motion through the aether. Specifically, they had to assume that the test apparatus shrunk when aligned parallel with its motion through the aether. It was as if the pressure of the aether wind shrunk the object.

Now this assumption came about as a result of assuming a wave nature for light. Yet, since physicists had assumed that the speed of light is constant, they would have needed to assume length contraction even if they considered that light had only a (special) particle nature and that there was no aether. And the amount of contraction they needed to assume is the same in either case.

This strikes us as a curious thing. The following is also curious:

Physicists had three other pertinent concepts available to them at the time of the MM experiment that went unutilized. Einstein had already shown there existed a quantum nature of light, indication of it being a particle. Mass associated with light was considered nil or zero. Studies of radioactive material pointed towards a transformational relationship between matter and energy.

Yet no one made the conceptual leap that therefore light might be a special particle of zero mass and therefore pure energy, which should in turn mean that it possesses the greatest possible speed in the universe, that speed of course being constant, so long as it remained a photon.

Even Einstein did not make this connection. Oddly enough, that connection was made by our sleepy western gent earlier in this book, whom, we point out, had the benefit of doing his contemplating seventy years later.

And that is where Relativity Trail begins. It is the pivotal point.

But to repeat, if a physicist of the era had made such connections, he should still expect asymmetry in the MM results, just as he would if he held to a wave theory of light through the aether. Thus he would still be surprised at the results, which showed only symmetry.

Our sleepy western gent took notice of this as well. Even though a massless photon had provided our gent with his certainty of time-keeping fluctuation, it did not immediately explain away the symmetry of the MM experiment.

page 21
Something we seldom see mentioned about this whole business of trying to detect ones motion through the aether, is that physicists were trying to detect something which really was as untenable a concept as Newton's absolute space. Like Newton's absolute space, the aether was regarded as something which had a one way relationship with the objects it contained, for it was considered to be not only an absolute of the moment, but for all time, fixed.

Furthermore, the hypothesized pressure of the aether used to explain length contraction would necessitate continuous energy expenditure regarding light transmission, just as with our swimmer in the water. Even sound waves peter out due to the resistance of the medium they disturb.

We'll be utilizing the particle nature of light in our treatment of relativity, and like Einstein, disregard the existence of an aether. (If you're familiar with the dual nature of subatomic particles and are wondering why an aether is not needed for the wave nature of light, you might find a satisfactory answer in the description of wave forms as a distribution of the probability of photon location.)

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