Twin Paradox of Relativity
Diagrams and derivations from Relativity Trail
(further down on this page)
From the preface to Relativity Trail:
Any book about relativity that doesn't clearly state that the slowing of clocks and contraction of rigid bodies is a reality which goes beyond our mutual measurements of these things is not only giving the reader half the story, it is giving the reader a confusing story. Without a recognition of this reality, such a book's author is also forcing himself into a mathematically impossible task – that of resolving a clock paradox of his own making. Such impossibility generally does not keep him from trying, and later in this book, we'll check and see how he's fared.
In Relativity Trail, you'll get the full story, with a clear description of relationships of uniform motion. Time-keeping is concretely defined. Time-keeping, distance, and the constancy of the speed of light take on absolute as well as relative meaning. The reader will find unique and clear answers as to the why of absolute clock rate slowing, mutually measured clock rate slowing, absolute length contraction, mutually measured length contraction, the time differential between reunited clocks, consistent light speed measure, mass increase (including mutuality of measure), and E = mc2.
The exact process by which parties take measure of each other is fully described, using only the familiar diagrams and arithmetic of a customary stationary reference frame, the physical existence of which we will demonstrate. Only straight line, uniform motion is considered. No aether or other sort of immutable reference frame is incorporated; nor is variable light speed. Additionally, gone is the long, difficult and abstruse derivation of the Lorentz transformations. These equations make their appearance in a most natural manner in the course of the reasoning within Relativity Trail.
In the introduction, we'll further explain the purpose of this book. But don't worry if much of the content of the introduction seems unfamiliar; we'll introduce everything from the ground up in the main body of the book, which begins with a story about the author's independent discovery of time fluctuation, and then proceeds to a concise and clear development of special relativity, completely consistent with the relativity of Einstein, with the effective equivalence of all inertial systems intact.
From the introduction to Relativity Trail:
Since a primary purpose of this book is to establish precisely how clocks, all clocks – mechanical, electrodynamical or biological – keep time in accordance with various states of motion, we'll begin by addressing an issue which has to do with the establishing of clock fluctuation.
You may have read that Einstein's special theory of relativity was long ago shown to be consistent with a modified form of aether theory, in which clock slowing was postulated in order to allow for mutually measured effects. In aether theory (and in Einstein's), clock functioning itself was not defined. Also, in aether theory, length contraction was attributed to a mysterious interaction with the aether.
In Relativity Trail, we will begin at a more fundamental level, adopting a natural, instinctive form of Einstein's postulates. Whereas Einstein formulated his postulates in the context of measures, our postulates will be formulated in an absolute sense, i.e., pertaining to the actual nature behind the measurements. We'll examine the nature of measuring. The first postulate we'll consider will directly imply actual clock slowing, and our next postulate will directly imply actual length contraction.
Contrary to the impression created by the standard accounts of relativity, this absolute approach will produce a pure form of relativity, logically consistent with Einstein's treatment. Our simple approach will reveal precisely what is transpiring behind the scenes of Einstein's treatment.
We'll demonstrate directly, that there is no incompatibility between Einstein's postulates of SR and the existence of a physically defined universal reference frame against which clocks, rods and light beams display their absolute nature. This reference frame is nothing other than a system at rest with the sum total of the cosmos, whether or not the universe actually has an overall Euclidean structure. We'll find that it brings clarity and simplicity to the study of special relativity.
The structure of the universe is evolving. It imparts inertial properties and produces the effects of special relativity. But no meaning can be attached to any movement of its point (to use the term broadly) of departure (origin). That "point" is therefore considered fixed, and is the universal frame of reference. The terms "stationary", "at rest", "absolute" and "God's eye view" are suitable synonyms.
It must not be confused with the term, "preferred" (or "privileged") reference frame, which would imply that such a frame is experimentally detectable.
If you have read other books about special relativity, you have undoubtedly come across statements similar or identical to the following, concerning the relative merits of two or more inertial frames: “There is no truth of the matter.”
On the Relativity Trail, we can phrase it only as follows: “There is no possibility of a person within any inertial frame of determining the truth of the matter regarding his or anyone else's inertial frame using any type of mechanical or electrodynamical experiment.”
This concept goes to the heart of Einstein's second postulate of special relativity, and identically, to the issue of being able to provide clear diagrams and descriptions of measurement taking. A crystal clear understanding of the underpinnings of special relativity depends on giving this aforementioned statement correct form, as we've done in the preceding paragraph.
Einstein's first postulate is a restating of the Galilean Principle of Relativity, except to include all electromagnetic phenomena. Thus, his first postulate contends that there is no possible experiment one can conduct to determine that one is in motion relative to a stationary aether.
In his initial wording, his second postulate states that “light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.”
With the word "definite", Einstein seems to imply that light has an absolute (actual) speed in reality. But he doesn't explicitly state that there is a physically defined universal reference frame against which light has this definite velocity. This might not seem to be a problem at first, but when he restates this postulate several paragraphs later, he uses a new wording which changes the meaning considerably. These alternate forms of his second postulate have caused a fair bit of anxiety among students of special relativity.
We will show, when we examine Einstein's kinematical section, that his treatment in fact can be diagrammed against the absolute reference frame of the universe in the same manner as we will do with our treatment. Our treatment is not only consistent with Einstein's, it actually subsumes Einstein's treatment.
As we proceed, we'll show that by not maintaining a conscious connection to the universal reference frame throughout his treatment, Einstein left himself no means for diagramming the process of measurement taking, no means for describing what is generating the properties underlying the assumed measures, and no means for explaining the time differential between reunited clocks.
It is natural to find seduction in the denial of such a reference frame, since it is not necessary to consider such a reference frame when doing computations involving special relativity. In fact, that it cannot be experimentally detected is something we'll show.
It can also be surmised that the theory retains more glamour and mystery if the universal reference frame is denied a say in the matter. Perhaps such appeal has helped to perpetuate such neglect, and not necessarily at a conscious level.
The fact that it cannot be experimentally detected is perhaps a clue as to why Einstein did not overly worry about disregarding it once he got past square one. We'll be looking at some of the possible reasons Einstein proceeded as he did. What we'll find, is that our own simple derivation easily accepts Einstein's constraint, so that we still obtain Einstein's transformation equations.
The value in recognizing the universal frame is that it allows us to see plainly what is generating the phenomena of relativity, beginning with clock functioning, length contraction, mutually measured length contraction, mutually measured clock rate slowing, consistent measure of light speed in all directions, and the time differential between reunited clocks. The price, of course, is that we lose all of the mystery and, therefore, much of the glamour.
Here again, in his original paper on special relativity, On the Electrodynamics of Moving Bodies, submitted to Annalen der Physik in 1905, is Einstein's initial wording of his second postulate:
“Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.”
Three pages later, he states it this way:
“Any ray of light moves in the "stationary system" of coordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.”
Here he replaces "definite" with "determined" and uses quotes around stationary system. Thus he is already hedging on what he means by this postulate. To whatever extent he favors this second wording, he abandons the absolute character of his postulate as initially worded, indicating he is already preparing himself to abandon the very reference frame upon which he implicitly relies at the outset, and which could have brought clarity to his treatment.
Einstein then begins the body of his paper by writing, “Let us take a system of coordinates in which the equations of Newtonian mechanics hold good. In order to render our presentation more precise and to distinguish this system of coordinates verbally from others which will be introduced hereafter, we call it the "stationary system".”
This "stationary system" serves as the baseline for Einstein's analysis of a reference system in motion relative to the "stationary system". From this analysis are born his conclusions which form what is called special relativity.
The electrodynamical portion of Einstein's paper has been described by authors as a brilliant and lucid “tour de force”. We find ourselves mostly lost in his complex and abstruse kinematical section, upon which the dynamical section is based. Of course he manages to come to the necessary correct conclusions regarding the kinematics, thus assuring the integrity of his dynamical interpretations which follow.
In Relativity Trail, we begin with a postulate which corresponds – but in an absolute sense – to Einstein's first postulate, and the original unqualified version of Einstein's second postulate, then immediately embark on a completely different type of analysis; one which the reader will have no trouble following, and which places special relativity on concrete footing.
The reader will be happy to learn that certain old absolutes are secure. In fact, without them, there can be no relativity.
From page 6 of Relativity Trail:
Thus, we've already come to a vital and consistent theme in our presentation. We make the observation that whenever we depict the motion of an object on paper, we do so by using a line of some length. This is of course also true of our depiction of the motion of a light ray. We further observe that we, the readers, see this depiction on paper as if we are in a higher dimension, noting the various lengths resulting from various speeds of objects. In fact, all lengths (thus speeds) are established using the length (speed) of light as the base.
The entities in our study appear as dots on the paper. The lines on the paper represent the motions of these entities as well as the motion of light. These entities can perceive things only at the speed of light. Thus we, the gods, are assigning a length to the speed of their perceptions. No such assignment is made to our (the readers') perceptions. The notions of speed, length and time, and the relationships between them are born on the paper.
We're making use of the universal reference frame. Our piece of paper we spoke of represents it well enough. We consider it to not be in motion. From this reference frame, we monitor the activities of the objects in all other reference frames.
When we speak of "universal time", we are referring to arbitrarily defined lengths of light ray travel as viewed by us, the observers of the piece of paper, or equivalently, the universal reference frame. Later, we'll expand the argument we made in the introduction that no meaning can be attached to the overall movement of this reference frame.
We will also show that any arbitrary reference frame might, for all we (the people actually living in the universe) can ever physically determine using light signals, be that "at rest" reference frame.
But the reader should note – we'll show plainly that only totality can impart the specific properties of clock functioning and length contraction to an object. These will be referred to as inertial properties. Thus, the fundamental difference between the universal reference frame and some arbitrary reference frame is that only totality can impart the properties being assessed by any particular frame, be it the universal frame (our analytic perspective) or some arbitrary frame. We will make this plain and revisit it frequently.
From page 10 of Relativity Trail:
In relativity, as we'll later explore, photons are the maker of every relationship – the carrier of force information as well as our means for perceiving events. Thus time-keeping processes are photon dependent.
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.
From page 12 of Relativity Trail:
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.
From page 23 of Relativity Trail:
From page 25 of Relativity Trail:
From page 30 of Relativity Trail:
So length, at the bottom of things, is about positioning. Electrons think they are positioned the same distance from the nucleus when in a parallel orientation as they are when in a perpendicular orientation. We're saying that the need for stability trumps the need for maintaining equal distance, to the meaningless satisfaction of the onlooking universal reference frame.
From page 37 of Relativity Trail:
This diagram, meaningful only against the universal reference frame, explains the fundamental workings of special relativity.
Length contraction need not be considered a separate postulate, rather a necessary condition to satisfy our two postulates of relativity – atomic stability and light as having an absolute and maximum speed.
Note that length must be contracted to the same degree as time-keeping is contracted in order to achieve symmetry of measure across inertial frames.
Time contraction = length contraction = (1 - v^2) ^ 1 / 2
Next, we'll formally derive the length contraction equation in a completely straightforward manner. Not a single step will be baffling or abstruse, as we find in the derivations based solely on relative frames.
From pages 38-39 of Relativity Trail:
From pages 40-41 of Relativity Trail:
From pages 44-45 of Relativity Trail:
MUTUALLY MEASURED LENGTH CONTRACTION
ACROSS INERTIAL FRAMES
One cannot take measure of the length of a rod in motion relative to oneself without first assessing the relative speed of the rod to be measured. This relative speed will of necessity be considered an absolute speed by the party taking the measurement, as he can make no considerations about his own state of motion.
A person who considers himself to be at rest must, of course, consider his length and time to be not contracted. These considerations are used in the process of taking stock of the "moving" rod.
We'll consider two spaceships, A and B, each with a rest length of 1 light second, as established by their laying out of rods. On board ship A are two clocks, CT1 and CT2.
The diagram on the opposite page shows ship A in motion relative to the universe U. Ship B is at rest with respect to U. A is moving at .6 c, thus contracted to .8 ls (light second) from its rest length of 1 ls.
CT2 sends a light ray towards CT1 as CT2 lines up with point Y of ship B. CT1 is triggered by the reception of this ray, and ship A must necessarily allow that 1 second was required to effect the triggering. CT1 then ticks off .67 second during the interval in which CT1 proceeds to point Y of ship B.
Thus A calculates B's velocity as follows:
distance = velocity * time
1 = v * 1.67 which implies v = .6
(Remember, A considers its own length to be 1 ls.)
To calculate B's length, A uses the fact that CT2 has ticked off 1.33 during the interval in which CT2 travels from Y to X. (Ut during this interval is 1.67 seconds.) d = vt yields .6(1.33) = .8 ls.
Ut = universal time
We can see from this same diagram that A will measure the rate of a clock which B has placed at point Y to have a rate of 0.8 times its own:
As clock Y passes from CT2 to CT1, where clock readings are exchanged, it registers a change of 1.33 seconds, the same as Ut. This is the interval during which A determines its own time passage to be 1.67 seconds.
So A regards B's clock to be slowed.
From pages 44-45 of Relativity Trail:
MUTUALLY MEASURED LENGTH CONTRACTION
ACROSS INERTIAL FRAMES (continued)
In diagram 17, we showed that even though A is in motion relative to U, A still measures B's length as contracted, and to the same extent as its own contracted length as seen by U.
In diagram 18 on the opposite page, we again have A in motion relative to U, with B at rest relative to U.
Now let's have B take stock of A.
As point X of ship A lines up with CT3, CT3 sends a light ray towards CT4. CT4 is triggered by the reception of this ray, and B considers 1 second to have passed. CT4 then ticks off .67 seconds during the interval in which point X travels from CT3 to CT4.
Thus B calculates A's velocity as follows:
distance = velocity * time
1 = v * 1.67 which implies v = .6
To calculate A's length, B uses the fact that CT3 has ticked off 1.33 during the interval in which CT3 travels from X to Y. d = vt yields .6(1.33) = .8 ls.
In these calculations, we have been making use of the fact that the party doing clock triggering always regards his measuring rods (the very rods he uses to position his two clocks) to be non-contracted.
The reader can verify for himself or herself that these relationships hold even when both ships are assigned a non-zero speed.
We can see from this same diagram that B will measure the rate of a clock which A has placed at point X to have a rate of 0.8 times its own:
As clock X passes from CT3 to CT4, where clock readings are exchanged, it registers a change of 1.33 seconds, in keeping with its time contraction of .8 Ut. This is the interval during which B determines its own time passage to be 1.67 seconds.
So B regards A's clock to be slowed.
This establishes the mutuality of measured clock rate slowing.
From page51 of Relativity Trail:
The point of the following analysis is to further demonstrate that there is no possibility of determining one's motion with respect to the universe. A simpler analysis is offered in chapter 6.
From page 53 of Relativity Trail:
From page 55 of Relativity Trail:
From pages 56-57 of Relativity Trail:
Now let's apply A's and B's actual velocities to our time contraction equation to learn what their clock times were for the out and in trips.
From page 58 of Relativity Trail:
We would like to pause here to congratulate any trail rider who actually followed along throughout the course of the preceding diagrams and calculations, of which it must have seemed there would be no end.
The clock paradox is put to rest by the preceding result. As we mentioned, a simpler analysis is offered in chapter 6.
From pages 62-65 of Relativity Trail:
A and B are in the same inertial frame, separated by one light second, as established by the laying out of measuring rods. If A triggers B's clock by sending a light ray to B, A and B will always assume B is one second behind A. They will be right. And of course, if B triggers A, then A will always be one second behind B.
The times tA and tB in the diagrams are as perceived instantaneously by the universal reference frame.
Again, one light second is established by the laying out of measuring rods, but this time of contracted length. A and B cannot know this, so again assume B's clock is always one second behind A's, though it actually lags 1.6 seconds behind.
If B triggers A, then B will always be .4 second ahead of A. This we have not diagrammed here. You might check it for yourself.
Having "synchronized" their clocks by way of A sending a light ray to B, A and B now engage themselves in measuring how long it takes for a passing light ray to travel from A to B. As trivial as this is, in light of the fact that A and B have based their assumption of time lag on the speed of light, we still wish to illustrate the symmetrical results obtained when measuring a passing ray of light.
Clock B will read 1.6 when it receives the light ray, but for a reason different from what A and B can assume, since they can't know that they are not at rest.
The results obtained are the same as in the case of an inertial frame at rest with respect to the universal frame.
The same is, of course, also true when B has triggered A's clock and the two then engage in timing a passing ray of light, as illustrated above.
These simple diagrams can be vexingly deceptive. It matters not whether there are observers stationed at clocks A and B. Such observers will always have the same assumptions about what the time reading of the clock at the receiving end must have been when the light ray was sent, based on what they know about their previous triggering process.
When the light ray goes from the clock that had previously initiated the triggering to the triggered clock, the clocks register the same reading as each other at the moment of light ray passing for each. When the ray goes from triggered clock to triggering clock, the triggered clock shows a reading 2 seconds less than that registered by the triggering clock, just as the parties would expect, based on what they know about their triggering process.
Caution and thoroughness of consideration of these diagrams is recommended.
From page 69 of Relativity Trail:
From page 70 of Relativity Trail:
Now imagine we accelerate the clock AB system from absolute rest in a straight line motion, maintaining an orientation of the clocks that is parallel to their motion through the universe.
As illustrated below, as long as a force is applied at either end of the system, there will be, just as in the case of a mechanical spring, greater contraction at the trailing end of the system than at the leading end. This is due to the fact that the communication of force throughout the system cannot exceed the speed of light. In turn, this means that the trailing clock, A, will keep faster time than the leading clock, B, during the acceleration phase.
From pages 88-89 of Relativity Trail:
What we'll do now, is to examine Einstein's assignment of tA - tB = tB - t'A in the context of the universal frame.
Consider the following situation in the context of the universal frame:
Clock B is in the positive direction of the AB motion from clock A, the AB system has an absolute velocity of .6, and A and B have a rest spatial separation of 1 ls (.8 contracted) as seen against the universal reference frame:
Einstein's definition of what constitutes a synchronization of those two clocks dictates that B's reading will be .6 second less than A's reading as seen against the universal frame, .6 being the velocity of AB. See the appendix for the formal derivation of this.
(Keep in mind that Einstein had no awareness of this superimposition onto the universal frame.)
Using this convention (the assignment of tA - tB = tB - t'A) amounts to a disregard for an analytical incorporation of an absolute frame of reference. It is in keeping with Einstein's notion of simultaneity, wherein he elevates a direct observation of a distant event to a psuedo-reality of time passage between the event and the moving observer.
In his 1916 book, Relativity, Einstein introduces the concept as follows:
He considers lightning strikes at points A and B, with observers M (stationary) and M' (in motion) accordingly observing simultaneous and non-simultaneous events, concluding that “Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash A took place earlier than the lightning flash B.”
Of course, in the context of a universal frame of reference in which light moves at a constant speed, they must conclude no such thing. In fact, they would conclude, after comparing notes with the people on the embankment, that the train had motion relative to the embankment. And this is not the same as saying they could determine whose motion was zero or even closer to zero. So the POR is safe.
What Einstein does here, is call synchronous whatever appears synchronous to an observer, adopting a utilitarian paradigm for his treatment, where light is the messenger of moments.
He thus leads himself to his definition of clock synchronization, in which time passage for light travel is predefined as 1 second for an entity of a given inertial frame who has separated his clocks a distance of his particular contracted (such as .8) light second. (The most obvious limitation of this approach is that it cannot explain where the missing time has gone in round trip situations, as we'll examine on pages 103 - 125.)
From page 99 of Relativity Trail:
We can obtain Einstein's transformation equations as follows:
Suppose we want to know x' according to the moving scale at some point x between A and B on the rest scale. According to the moving scale, A' = 0 and B' = 1 permanently. Thus the ratio of (x - A ) / ( B - A) = x' / 1 , the left side of this equation being the rest ratio and the right side being the moving ratio.
Since A = vt and B - A = L , we have x' = (x - vt) / L. That is Einstein's coordinate transformation for x.
Next, we use Einstein's handy clock setting for B, which is t'B = t'A - v, to obtain t' at the location x' :
We apply the same ratio, (x - vt) /L , to the time difference t'B - t'A (which is -v) , adding that result to the time passage t'A. Since t'A = L t , we have : t' = [(x - vt) /L ] [ -v] + L t .
By writing L in the form (1 - v2)1/2 and performing some simple algebraic manipulations, we acquire t' = (t - vx) / L , which is Einstein's transformation for t' at x'.
From pages 106-108 of Relativity Trail:
ADDITION OF VELOCITIES AND THE CLOCK PARADOX
To check on the clock paradox more simply, let's use the situation as described on page 84, comparing the situation of clock A changing from a state of rest in the form of A' to catch B with the situation of a traveling B reversing motion in the form of B' to return to A.
What we should find is that A and B will regard each other's velocities, as well as their prime operatives (A' and B'), in the same manner in either case.
Assuming a relative velocity of .6 between A and B (as measured by each of them) for both the out and in trip, we'll present two possibilities for the round trip:
In case 1, B passes by A, continues 1 light second, then reverses direction by virtue of transfering clock information to B'.
tB = 1.333 at reversal. Final tB = 2.667. Final tA = 3.333.
This means that in case 2, tA must equal 1.333 at the moment it transfers its clock reading to A', with A' then chasing down B.
tB = 1.067 at that moment, since it has velocity of .6.
Final tB must equal 3.333, meaning tB during the chase interval equals 2.267. This implies Ut during chase interval is 2.833.
dB during chase interval thus equals 1.7 ls.
We also note that tA = Ut = 1.333 at transfer implies B has traveled .8 ls.
Thus dA = dB + .8 = 2.5. This implies actual vA' = .88235.
Since A was at rest when A' passed it, A measured vA' as .88235.
We must now check to see what value was obtained by B for vB' as B' passed by B:
B makes timing of B' with clock towers .8 ls apart as established by the laying out of rods, so that B regards the towers to be 1 ls apart.
We know actual vB' = .6, so we can read the familiar numbers off the diagram. CT2 meets B' at Ut = .167, CT2 time of .133.
B adds 1 second for triggering of CT2 (initiated by CT1) to yield total time tB = 1.133.
B thus calculates vB' = 1 ls / 1.113 s = .88235.
We'll dispense with the trivial considerations of A and B taking stock of each others velocity, as we've shown this elsewhere.
We do note however, that A' has contraction of .4706. Thus A' took stock of B's velocity in accordance with the diagram below:
.6 t + .4706 = .88235 t , yielding t = 1.667. Since A' considers d = 1, A' uses d = vt to calculate velocity of B as .6 .
What we have just done on these pages is to verify the addition of velocities formula for relativity:
B + A'(B) B - B'
A' = --------------------- B(B') = ------------
1 + B * [A'(B)] 1 - B * B'
where A' is velocity of A', A'(B) is the velocity of B according to A', B is velocity of B, B' is the velocity of B', and B(B') is the velocity of B' according to B.
From page 146 of Relativity Trail:
From page 149 of Relativity Trail:
From page 157 of Relativity Trail:
If the universe were actually Euclidean, we could never know that we were at the edge of the universe. At the edge of this flat universe, all gravitational source is on one side of the person at the edge. Thus his line of sight is always bent in towards the universe, as is any translatory physical path he might venture along.
As we discussed on pages 134-137, this is part of the reason such a universe would appear roughly the same in all directions, giving any observer the impression that they are at the center point of the universe.
From the appendix of Relativity Trail:
We can recreate diagram 7 of page 13, but with gent B traveling outbound until his clock reads 10 seconds, at which point he hands off his clock information to B' coming from the opposite direction at the same speed, and have gent A and gent B (and ultimately B') send radio pulses to each other every one second as registered on their respective clocks. We obtain the following table:
From the appendix of Relativity Trail: