Twin Paradox in Relativity
Explained and Diagrammed
in Absolute Terms
with a link to a twin paradox animation
and links to diagrams and analyses of measuring processes
Cite the book, Relativity Trail
Luebeck, R. Relativity Trail. Mpls: L B Writ Publishing, (2008)
Cite this web page
At the end of the kinematical section of his paper on special relativity, Einstein noted what he called a "peculiar consequence" stemming from his analysis - that of a time differential which would occur between reunited clocks.
Over the course of his remaining fifty years, Einstein offered no explanation for that time differential (which came to be known as the twin paradox).
Einstein's treatment was limited to assigning symmetrical measures across inertial frames, such that his two postulates of measure would be satisfied.
Einstein's paper was published in 1905. As the years passed, a point of view took hold among many commentators that these symmetrical measured effects of relativity were confirmation that uniform motion is purely relative, and that there is therefore no meaning to be attached to absolute uniform motion, and therefore of course, to actual differences in clock rates, etc.
The actual time differential between reunited clocks, a physical reality which necessarily favors one party over the other, does not fit with that interpretation.
As noted even by John A. Wheeler (a promoter of the purely relative approach), special relativity developed in absolute terms is completely consistent with Einstein's purely relative development, specifically including symmetrical measures across inertial frames and the consistent measure of light speed in all inertial frames. Wheeler notes that there is no physical experiment which can distinguish the absolute treatment from Einstein's purely relative treatment. [Wheeler, J., Taylor, E. (1992). Spacetime Physics, second edition. W. H. Freeman: New York, p. 88.]
As noted above, the actual time differential between reunited clocks, a physical reality which necessarily favors one party over the other, does not fit with a purely relative interpretation.
Yet, countless commentators on special relativity have claimed to explain the time differential without acknowledging an actual difference in clock rates.
Every one of those explanations falls into one of two categories, both incorrect:
1. The "inertial force" or "acceleration" explanation.
For starters, neither force, nor identically acceleration,
is allowed in special relativity, which addresses purely
uniform linear motion, as seen in Einstein's paper, as
well as in all subsequent derivations of the Lorentz
transformations. The time differential is deduced through
purely linear uniform motion considerations.
Also, no force could possibly affect the simple act of
starting a clock as an inbound astronaut passes an
outbound astronaut, causing hundreds of years to
disappear in a twinkling.
2. The "lines of simultaneity" space-time explanation.
This is identical to the "space-time diagram", "kink in
space-time", "jump in time", "misperception", and "lattice
of clocks" explanations. They are all one and the same
These commentators are not aware that the construct
known as space-time is dependent on Einstein's clock
synchronization, a clock synchronization that is not
required to deduce the results of special relativity.
Identically, relativity of simultaneity is dependent
on Einstein's clock synchronization.
All the results of special relativity can be deduced
independent of Einstein's clock synchronization.
Space-time is a construct, entirely dependent for its existence on Einstein's particular (and entirely optional) method of clock synchronization. The best one can do with a space-time argument in the context of the twin paradox is to note that one party, when changing inertial frames, will observe a jump in the reading of the clock time of the other party, using the 'lattice of clocks' method dictated by Einstein's particular clock synchronization.
But no such jump in anyone's aging (or identically on anyone's clock) actually occurs.
You should re-read the previous two paragraphs, carefully.
In contrast with the space-time argument: When the two parties check the status of each other's clock with the regular sending of radio pulses, they note an incrementally increasing time differential, first noted at the turn-around point. That real-world experiment reveals the real incremental increase of time differential. [William Geraint Vaughan Rosser (1991). Introductory Special Relativity, Taylor & Francis Inc. USA, pp. 67-68.]
Space-time is a construct, dependent on an optional clock synchronization method, and generates a false jump in time reading.
Space itself is not a construct, nor is time-keeping. And a real-world experiment reveals a real-world incremental increase in the time differential.
Why should we expect anything different? The 'stay-at-home' twin does not age several years in a twinkling just because some distant inbound astronaut starts his watch as he passes an outbound astronaut.
It is precisely ones motion with respect to the universe that dictates the actual clock rate, resulting in the actual time differential upon reuniting with the other party.
You should re-read the previous paragraphs, carefully.
Einstein's clock synchronization method creates a pseudo reality for two inertial frames, whereby the parties of different inertial frames are free to reach opposing conclusions about what distant events are simultaneous. In reality, no one should conclude anything about what is simultaneous, as we know that our perceptions of distant events are constrained by the speed of light.
Einstein's clock synchronization (tB - tA = t'A - tB) has the "jump in time" built in for any situation involving a change of inertial frame.
It is in a space-time diagram that we find the infamous sudden shift of a line of simultaneity. The further away (or the higher the overall speed) the "inbound" and "outbound" astronauts are from the "stay-at-home" when the transfer of clock information occurs, the greater the magnitude of the "jump in time" - the so called "misperception" explanation of space-time diagrams.
Space-time, being dependent on Einstein's clock synchronization, is nothing but a geometrical construct that has no physical reality. Identically, there is no actual "world line" along which someone travels. Space-time, identically with Einstein's clock synchronization, is limited to the symmetrical observations made across inertial frames. It cannot address the time differential between reunited clocks.
Note that the purely relative approach incorporates a hopelessly circular definition of inertial frame.
In Spacetime & Electromagnetism, Lucas and Hodgson, using the space-time paradigm, wrestle with the twin paradox for fifteen pages, and claim no resolution.
In a footnote on page 73, they write: "Is it fair to give the Earth-bound twin the vertical world line? Does not that beg the question in his favour? Why not draw another diagram with his world line set at an angle to the vertical, and his lines of simultaneity correspondingly inclined (but at a contrary angle) to the horizontal?" [J.R. Lucas & P.E. Hodgson, Spacetime & Electromagnetism, p 70-84.]
That alternative diagramming of the situation is identical to the consideration that one might just as well consider that it is the Earth, along with the entire cosmos, that changes inertial frames. That, in fact, is something we hear often from commentators on the twin paradox. Of course, that simply makes the twin paradox unresolvable, as either party can then lay equal claim to being the party that ages the least. This is demonstration that one must consider the entirety of the cosmos, the imparter of inertial properties, to be the judge of the matter regarding actual motion.
As another example of the hopelessness of using the space-time argument, consider that the renowned phycisist, John A. Wheeler, in his book Spacetime Physics, made three attempts to resolve the twin paradox and failed three times.
After claiming to have solved it on page 131 of his book, he writes on page 170 that he will "finally! solve it". But he gets hopelessly stuck again.
He never stood a chance, as he limited himself to Einstein's narrow interpretation - that of simply assuming symmetrical measures across inertial frames. Wheeler, in his book, mocks his own second failed attempt:
Wheeler has his astronaut proclaim – “as I turned around, a whole bunch of Earth clock ticks went from my future to my past. This accounts for the larger number of total ticks on the Earth clock.”
Wheeler continues: “The astronaut renounces her profession and becomes a stand-up comedian.”
Wheeler was attempting to claim that the simple act of starting a watch as an inbound astronaut passes an outbound astronaut could make 162 years disappear on a distant planet (Earth).
Realizing he has failed again, and in an attempt to salvage the discussion, Wheeler, in a footnote, refers his readers to an old journal article which he purports does solve the riddle of the time differential, obviously hoping no-one would actually look up the article.
The article is "The Clock Paradox in Relativity Theory", American Mathematical Monthly, January 1959. See pages 9 and 10 of that article. Even though its author, Alfred Schild, in his informal and incorrect verbiage, feels compelled to claim "acceleration" for the returning twin, Schild in fact, in his diagram and math, specifies an instantaneous turnaround "at the event C", and of course cannot provide any explanation for the ultimate time differential. It is simply the precise nonsense that Wheeler had just twice engaged in and was defeated by.
Until one acknowledges that a clock's rate is dependent on its actual (albeit experimentally indiscernible) state of motion, one cannot explain the time differential between reunited clocks. Only an actual difference in clock rates can bring about the actual difference in clock readings which exists between reunited clocks.
The time registered on a person's clock is dependent on the combination of their speed relative to light speed and distance covered in absolute terms. The time contraction formula is not linear. Thus, the party that changes inertial frames will be the party whose clock registers the least time over the course of a "round trip". This is seen with clarity when everything is charted out in absolute terms.
Using a rational measuring paradigm, whereby the parties of different inertial frames check each other's clock status with the regular sending of radio signals, a noted asymmetry in time keeping differences between A and B builds incrementally, with the asymmetry being first noted at the moment of B's turn-around.
Actual time-keeping fluctuation, in conjunction with actual length contraction and actual states of uniform linear motion, produces all the effects of special relativity as we know them in Einstein's treatment. Such actualities imply an absolute frame of reference, albeit an experimentally indiscernible one.
One need not, and in fact should not, consider the existance of an immutable aether. Nor should one consider the concept of "absolute space". Rather, the totality of the cosmos, the very thing that imparts inertial properties to objects, serves as the absolute frame of reference. In other words, we need to consider a system at rest with respect to the totality of the universe. (i.e. - a system at rest with respect to the barycenter of the universe.)
It is meaningless to ascribe linear motion to the universe itself, considering that the universe is the baseline by which linear motion acquires meaning.
Although the universe is expanding, special relativity easily holds true at any conceivable scale of distance at which we might test the theory.
The notion that there is no absolute frame of reference or actual differences in clock rates has caused widespread wonderment and confusion among physicists, who regularly state that the mutual measured effects of relativity, along with the "relativity of simultaneity", are confounding, deeply mysterious, and impossible to understand.
Two brief examples:
Physicist Banesh Hoffmann writes in Einstein, Creator and Rebel p. 76-78:
“Though (relativity of simultaneity) may be shocking, we have to learn to live with it. ... its logical consequences are often such as to outrage common sense.”
No shock or outrage to our common sense is to be found in the context of the universal frame of reference, and there is nothing we need to “learn to live with.” We would be shocked only to learn that there is no time differential between reunited clocks.
Without such concept of an absolute frame of reference, one is left with a clock paradox of ones own making, which cannot be resolved in any rational manner. One is also left with no means to account for the symmetrical assessments across inertial frames as we know them in relativity.
The eminent physicist John A. Wheeler, writes in Spacetime Physics that relativity of simultaneity is "strange" and "difficult to understand". Yet, Einstein's definition of simultaneity can be easily diagrammed against the rest frame of the universe.
In Relativity Trail, the mutually measured effects of Einstein's special theory of relativity are explained with simple diagrams and arithmetic arising from an absolute frame of reference - a system at rest with respect to the totality of the universe.
Einstein's entire treatment of special relativity - most notably his clock synchronization method - can easily be diagrammed against that stationary frame of reference from the God's eye view. And as noted, such frame of reference cannot be experimentally detected by entities in the universe.
Einstein's definition of what constitutes a synchronization of two spatially separated clocks of the same inertial frame amounts to a disregard of an analytical incorporation of an absolute frame of reference. The two treatments (Einstein's and Relativity Trail's) vary, in large part, by the manner in which two spatially separated clocks of the same inertial frame are set.
In his treatment, Einstein utilizes inertial frames to which
he arbitrarily assigns the status of "stationary" and "moving".
His definition of what constitutes clock synchronization
makes an absolute frame of reference superfluous.
Hence, Einstein's treatment does not address
the question of which clock is actually running slower or
faster over any interval of the analysis, nor, identically,
the question of which entity's measuring rod is actually
shorter. Symmetric assessments across inertial frames
are assumed, without any diagramming of the process. In the course of Einstein's
derivation, certain measures must simply be assigned to
the entities involved for the sake of satisfying Einstein's
postulates of measure.
In the end, Einstein concludes that there is a time
differential between reunited clocks, but with the
absolute frame of reference neutralized by Einstein's
clock synchronization, he cannot explain the missing time.
Why is it called a paradox? As we noted at the beginning of this document,
Einstein's relativity is almost universally treated as
though it precludes any actual difference of length and
clock rates between different inertial frames. This leads
immediately to the state of mind that "there is no truth
of the matter" concerning inertial frames. That in turn,
creates a seemingly paradoxical situation - two reunited
clocks do show an actual difference in recorded time, as
if there must have been a "truth of the matter" regarding
their clock rates as they moved uniformly.
But Einstein's treatment does not preclude such actual
differences of clock rates. In fact, his postulates
demand it, as he should have noted at the conclusion of
his derivation. By extension of logic, the famous
experiments performed around the turn of the century
which drove Einstein's postulates also demanded it. At
the conclusion of Einstein's kinematical section, where
he noticed the "peculiar" time differential
between reunited clocks, he should have realized that
his clock synchronization was obscuring the reality
underlying the symmetrical measures across inertial frames.
That does not mean that Einstein's special relativity
is not valuable. Einstein's treatment succeeds in
predicting observations of electrodynamical
When presenting relativity from the God's eye view, it
is obvious that there is no clock paradox. From the
God's eye view, light is seen to have a constant speed
in an actual, not merely measured, sense. Thus
Einstein's second postulate is so replaced.
An actual difference in clock rates follows immediately
from this postulate of the absolute nature of light,
provided of course, that one assumes that photons are
the maker of every relationship (specifically here,
the regulators of atomic functioning; but also the
carriers of force information and our means for
The consideration of photons being massless particles,
along with the consideration that mass and energy are
interchangable, serves as the basis for postulating
that light has an absolute speed and is also the
limiting speed, with the photon having the property of
existing in the form of pure energy. The preceding
properties of photons and matter were actually known
prior to Einstein's theory. Einstein himself
introduced the notion of light existing in the form
of a quanta of energy.
Einstein's first postulate, which itself pertains
only to measures of properties, is the Principle of Relativity
extended to electromagnetism. As seen in the Michelson-Morley experiment, as
well as in Galilean mechanical demonstrations, the
Principle of Relativity is inextricably bound with synchronization of
motion along different axes. It is replaced, in the
Gods' eye view, with the requirement of actual
synchronization at the base of our physical
structures, for the sake of their stability.
Machian notion of a particle's relationship to
totality, along with absolute light speed, combine
with the need for atomic synchronicity to explain
actual length contraction of rigid bodies. This is
because photons (or virtual photon events) are
considered to be the fundamental agent of
communication within atoms, maintaining the
organized structure of the atom. It is the
equivalent of the Michelson-Morley paradigm, but
on the atomic scale. Implicit here, is that particles
have both a translatory relationship with the universe, and a rotational/orientational
relationship with their translatory path, in the Machian
When light rays and the motions of objects, along with
their photon clocks, are charted out against the
stationary reference frame of the God's eye view, all
the mutual effects of special relativity fall into
place, including of course, consistently measured
light speed in all directions, in all inertial frames.
There is no clock paradox from such a vantage point.
It is seen with clarity, why it is that the clock
whose change of inertial frame brings the clocks back
together is the one that records the lesser time over
the course of the round trip.
A few details from Relativity Trail:
Symmetry of Measuring with diagrams and math.
Twin Paradox Analysis with diagrams and math.
The same links are found at the bottom of this page.
No aether need be considered in such a treatment,
rather a structure which is made up of an evolving
set of relationships between all elements of the
universe, in keeping with the nature of general
relativity. Light is the maker of every relationship
in this structure. The photon (or virtual photon event)
is the agent of communication, both of forces, and of
positioning information. This structure is not fixed,
rather evolving. The sum total of the structure,
identically with its point or points (as in the
inflation case) of departure is regarded as being at
The description of this structure most simply
incorporates the notion of a point of departure (the
Big Bang) within the spatial dimensions of a Euclidean
universe. No meaning can be attached to any movement of
such a point of departure. But in the context of the
notion of a universe of inflation, Euclidean or non-Euclidean, the description is fundamentally the same.
In this latter case, the origin of the universe lies
outside its spatial dimensions, and the membrane of
space expands due to inflation along that other
dimension. There are then any number of points of
departure for motion with respect to the membrane of
space. In this latter case, it is obvious there would
be no single center point of the universe within its
own spatial dimensions. But even in the former simple
case, such an actual center point would be
experimentally undetectable. This is detailed in
The relativity of Relativity Trail is
fundamentally different from the relativity of Lorentz:
1. Lorentz embraced a fixed
structure of space - the aether.
2. Lorentz did not define time-keeping
(clock functioning). (Neither did Einstein.)
3. Lorentz provided no clear basis for why clock
rates should slow, or why rods should
contract, let alone the needed amount to
satisfy the Principle of Relativity.
4. Lorentz provided no diagrams of the measuring
Additional reading for stubborn mules who maintain that
there is no absolute frame of reference by which motion
can be defined:
Although the twin paradox is strictly an effect
of special relativity, where no inertial force is
involved, consider that A. P. French writes on page
150 of Special Relativity: "Note, though, that
we are appealing to the reality of A's acceleration,
and to the observability of the inertial forces
associated with it. Would such effects as the twin
paradox exist if the framework of fixed stars and
distant galaxies were not there? Most physicists
would say no. Our ultimate definition of an
inertial frame may indeed be that it is a frame
having zero acceleration with respect to the
matter of the universe at large."
In Mach's Principle, an object is affected by a
change in motion relative to the matter of the
universe at large. But such an effect cannot occur
unless the object is in a relationship with the
matter of the universe at large regarding its
initial state of motion to begin with.
Remember, an effect due to a change in motion is
not simply an "either or" effect, rather it is an
effect of degree based on "degree of change". No
effect of degree stemming from a degree of change
can occur unless there is an effect based both on
initial state of motion and final state of motion.
Put another way, a change in motion in the context
of the universe is not meaningful without an initial
and final state of motion in the context of the
universe, and both those states are necessarily
uniform states of motion.
Mach himself regarded the matter of the universe
at large to be an actuality, and the effect on the
object to be actual. We can't have it both ways.
If the matter of the universe at large is a reality
which has an actual relationship with an object
concerning a change in motion, then so too is it a
reality which has an actual relationship with an
object concerning motion itself, i.e. - the initial and final state of motion, related to each
other by the act of change in inertial motion.
And again, our
observations (measures) are something fundamentally
different from the underlying reality - a reality
which generates our observations (measures),
which can occur only at light speed.
In SR, Einstein was able to make an absolute
frame of reference superfluous by postulating
only the constant measured speed of light.
In GR, he made Mach's Principle (somewhat) superfluous by
again holding to only a postulated measured
speed of light.
A spatial seperation between two objects can
actually be achieved, rather than merely seem
to be achieved.
Thus, there actually is relative motion.
One can relocate oneself relative to ones house
can one not, by traveling from the living room
to the kitchen? Can one not relocate oneself
relative to the universe by traveling through it?
Thus, there actually is motion relative to the
My house actually exists, even though it is not
perfectly rigid. I can actually move relative
to my house, despite the fact that all the sub-atomic
elements of which my house is comprised are
moving about relative to the overall structure
of my house. I can actually travel from one
end of my house to the other end. And in so
doing, I affect the motion, relative to my
house, of each individual element of my house.
The universe does actually exist, rather than
merely seems to exist.
Thus, there actually is motion relative to the
After all, how can you agree that there is
actually relative motion, agree that there
actually is a universe, but not agree that
there is motion relative to the universe.
All the elements of the universe are moving
about relative to the overall universe, just
as with my house.
Also, one obviously does not need to be at
the barycenter of the universe to be at rest
with the sum total of the universe. You don't
need to be at the center of your house to be
at rest with the sum total of your house, do you?
Light, being massless, has a constant and
maximum speed as it moves through the universe,
and all objects move at some fraction of that
speed, the speed by which all other speeds must
be defined. If the speed of light is real in a
real universe, then the speeds of objects are
real in a real universe.
Screen shots of the two animations:
Other documents which are recommended
reading before reading the book:
Twin Paradox Animation on youtube. Similar textual content as found on current page.
Twin Paradox Explained. A similar discussion of the failure of space-time diagrams.
Twin Paradox Animation. Embedded youtube animation, with textual description.
Symmetry of Measuring. Diagrams and equations demonstrating the symmetry of clock rate and length measures across inertial frames.
Twin Paradox Analysis. Diagrams and equations demonstrating the symmetry of measures and time differential regarding case 1 and case 2.
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.
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