The Earth twin does not accelerate so he assumes that he is also not moving (in the
Earth's Inertial Rest Frame of Reference), but he does see that his twin (on Earth2) has a
very high constant radial velocity, receding from the Earth. For our
discussion here, we are going to say that we measured Earth2 as receding from
us at 0.6 the speed of light, really fast! As far as they went, the 1960s
Physicists were correct, and they were right that the Earth twin would see his
rapidly receding brother appear to be aging more slowly than himself, due to
something called Time Dilation, an unavoidable effect of Einstein's Special
Relativity. A simple formula (from Einstein) below, shows that the Earth
twin would see the rapidly receding twin on Earth2 to be aging at exactly 0.8
times as fast as he was aging. This includes wall clocks, which he could see
on both Earth (next to him) and through his excellent telescope, in the room
next to his twin on Earth2.
The part that those Physicists misunderstood was that they never considered
the experiences of the traveling twin! Since his Earth2 is identical to
our Earth, and the Traveling twin (on Earth2) does not accelerate so he assumes
that he is also not moving (in the Earth2's Inertial Rest Frame of Reference). The twin on
Earth2 looks back at the Earth through his excellent telescope and sees that
his twin brother (and the entire Earth) has a high constant velocity, receding
from him (at 0.6 c). Are you following? What he sees is absolutely exactly
identical to what we described that his twin on Earth sees! In other words,
due to the same Time Dilation Effect of Special Relativity, he sees his twin on
Earth to appear to be aging exactly 0.8 as fast as he was aging, due to the
same Time Dilation. He even sees the wall clock next to his twin on Earth
moving exactly 4/5 as fast as he sees his own wall clock moving!
So a more complete (and correct) presentation of the Twins Paradox would
have added that the Spaceship (or Earth2) Twin would also see his rapidly
receding brother (on Earth) appear to be aging more slowly than himself, by the
exact same Time Dilation effect of Einstein's Special Relativity.
There can be no doubt about it, he definitely sees his twin brother on Earth
receding from him at 0.6c velocity, and due to Special Relativity's Time
Dilation, he also sees his brother appear to be aging 4/5 as fast as he himself
was aging!
As weird as that sounds, it would be true that both of the twin brothers
would be simultaneously watching each other age more slowly than himself! Even
to watch each other's clocks clearly run only 4/5 as fast as he watched his own
clock run!
There are practical reasons that this situation could not endure long, but if
they could watch each other for five of their own calendar years (birthdays),
during that they would each see their identical twin brother only
celebrate four birthdays! Really weird, but true! (The reason that is
impractical is that if the observed target was receding at 0.6 of the speed of light,
for five years, they would then be around three light years apart, a really
long distance apart, even for a really good telescope!)
Einstein had made clear that either nonAccelerating Inertial Rest Frame of Reference is
equivalent. The fact that a planet might be large or a spaceship small is
irrelevant. The wrong or incomplete version of the Twins Paradox had correctly
applied the Time Dilation effect of Special Relativity but only from one
perspective, ours here on Earth! They totally incorrectly neglected the SAME
Time Dilation effect of Special Relativity as seen by the traveling twin. The
Earth is not allowed to have a "specially preferred Inertial Rest
Frame," just because we happen to live here!
Continue your thinking of these two views of things and you can see how obvious
this logical flaw is by looking at how the Twins Paradox story ends!
Allegedly, the Earth twin believes that he will be older than his twin brother
when they meet at the end of the story. However, that (twin) Physicist who
would have started out with the perspective of a nonaccelerating spacecraft
(or Earth2), by Einstein, has watched an absolutely identical story unfold, and
so he would also believe that he is the older brother! BOTH CANNOT BE THE
OLDER TWIN WHEN THEY MEET! Either one of them would have to be older or
they are exactly the same age when they meet. The reasoning below establishes
that, logically, this last situation is true and they would meet where they
are exactly the same age. (as long as they were then again in the same
Inertial Rest Frame of Reference when they met.
The result of this absolute demand of (Special) Relativity and Einstein is
that both twins in that hypothetical story constantly watch the other one age
more slowly than himself! The interesting part is what those 1960s
Physicists had overlooked! They totally ignored the other viewpoint, of the
other twin!
Those Physicists only looked at one perspective, that of viewing from
Earth, and they never considered the absolutely equally logical and
correct perspective of the spaceship (or Earth2) traveler. Given that
(extended) story, the correct situation would have to be that both twins
have personally aged twenty years while watching the other only age sixteen
years. This peculiar situation is a necessary result of Special Relativity
and Time Dilation, for each of the brothers!
No Physicist ever considered what the Universe looked like from the
spaceship, then or since, so the only perspective that was considered has
been from here on Earth, which was dreadfully wrong regarding logic and
Relativity.
The true situation actually makes perfect sense! It necessarily includes
an issue which those 1960s Physicists never even considered, that of General
Relativity and its own timerate effect applying during the (neglected)
Acceleration and Deceleration portions of the trip.
The true story would need to be corrected from some of its flaws, for an actual
result to be determined. One main part of that correction is that the
constant velocity portion of a space trip, where Special Relativity would
apply, is actually only a small portion of the entire trip. Much of the
trip is taken up during acceleration and later, deceleration, where the General
Relativity timerate effect applies. In fact, as fully described and discussed
below, an entire round trip involves six different (sequential) situations,
three on the way out (acceleration, constant speed cruising, and deceleration)
and the same three on the way back. The different stages are quite different
regarding the rate of passage of time, and an important result is that an
entire trip always takes the exact same total amount of time, although the
six different perceived time rates and periods of passage of time are quite
different for the two twin brothers and also different for their perception of
the life of the other brother.
For example, during a popular version of the Twins trip to a planet orbiting
Alpha Centauri, the traveling twin might initially experience more than two
years of acceleration (under GR circumstances), followed by a few weeks of
constant velocity cruising (under SR circumstances), followed by another two
years of deceleration (and again under GR circumstances), where he personally
experienced an entire outbound trip of around 4.5 years (that is, 2.2 + 0.1 +
2.2 years). The observing brother on Earth might have watched an Earth month
of acceleration, followed by four Earth years of constant velocity cruising,
followed by an Earth month of intense deceleration, where he watched a full
outbound trip of about 4.5 years (that is, 0.1 + 4.3 + 0.1 years). However,
note that the two would describe the trip very differently! The traveling twin
would (naturally) experience two birthdays during the acceleration and another
two birthdays during the deceleration, so he would not sense anything weird. The
Earth twin would see the traveling twin seem to move and age really fast during
that "officially observed and logged Earth month of acceleration" where he
watched two birthdays be celebrated. Then, for the next four Earth years
of observing, he would see Time Dilation of Special Relativity, where he would
log the activities of the traveling twin as occurring really slowly! No
birthdays are seen during those four Earth years of observing from Earth while
four birthdays were celebrated here on Earth! Then he must decelerate, in
order to return to the Inertial Rest Frame, which means very rapid aging as
seen from Earth, while the traveling twin would not experience anything weird
while he had two more birthdays, while the Earth twin logged watching only one
month of apparent really fast activities!
Each brother would celebrate four birthdays and watch his brother
celebrate four birthdays, but just not at the same time!
There are two other perspectives which we can examine. The traveling twin,
looking back at us, watches his Earth twin brother spend about 2.2 years of
aging slightly faster during which he watched the Earth twin age an extra day,
then he would watch his Earth twin age slightly more slowly during the next few
weeks, followed by another 2.2 years of slightly faster aging during
deceleration, which results in him watching the Earth twin age a total of about
4.5 years during his outbound trip. The remaining perspective is that of the
Earth twin regarding his own life during the whole observed trip, where he
would age 4.5 years, as is considered normal in this Inertial Rest Frame of Reference here
on Earth. Note that in all four perspectives, exactly the same total amount of
time passes during the trip, 4.5 years. They each celebrate four birthdays and
see their brother celebrate four birthdays, although not at the same time!
Neither of the twins either experiences or sees the twin age any total amount
of time except the correct 4.5 years. More interesting is that exactly at
the halfway point of the trip, during the constant velocity cruising period,
both twins watch each other age more slowly than himself, exactly per the Time
Dilation of the Special Relativity that applies then. This is the explanation
for how and why they both see the other age more slowly than himself, which IS
an unavoidable consequence of Special Relativity of constant velocity cruising!
Also, during each of the acceleration and deceleration portions of the
trip, both twins see the other age faster than himself. In one case, this
appearance of 'faster living' could be very obvious while in the other case,
the "faster living" would be difficult to detect. The net effects of these
factors exactly cancel out the Special Relativity Dilation effects seen during
the relatively brief constant velocity portion of the trip, so the total length
of the entire trip does not and can not change.
(I apologize for one detail in the above story regarding Alpha Centauri. In
order to keep the logic and math simplest, I used a faster maximum trip
velocity than the 0.6 c referred to above. At a maximum velocity of 0.6 c, the
Earth observer would see the Time Dilation effect of 0.8 so that the constant
velocity portion of the trip appear to last shorter than the 4.3 years
described above and the traveler would experience a much longer time than one
month during that constant velocity portion of the trip. The math to calculate
the exact values is fairly complex Integral Calculus, and so I used simpler
time intervals that would have applied to a much more powerful rocket and a
higher maximum trip velocity.)
That popular (wrong) Twins
Paradox trip to Alpha Centauri also has another enormous logical flaw. Any
school student can calculate that for a spacecraft to rapidly accelerate up to
near the speed of light, the spacecraft would have to accelerate at many Gs for
the whole trip. No human could survive such a trip! If the spaceship
passenger(s) were considered, the acceleration and deceleration during the
entire trip needs to be around 1.0 G. Otherwise the passengers might struggle
around as though they weighed two times or ten times their normal weight, where
their muscles would struggle. Their hearts would also not be able to pump
blood to their brains at G = 3 or higher without passing out and dying! At a
continual acceleration of 1.0 G, which is 9.8 meter/second^{2}, such an
acceleration would obviously take 15 million seconds (about onehalf year of
acceleration) before it could get up to even a velocity of half the speed of
light (or 1.5 * 10^{8} meters/second). If the spacecraft had an
acceleration of just 2 Gs, the passengers' bodies would have to endure three
months of severe stress on their hearts and circulatory system, where none
might survive such a lengthy stress. Then two times the stress again, for
another three months, during the deceleration of the trip out and again for
both the acceleration and deceleration of the return trip to Earth. Where
StarTrek zips around the Universe at conveniently enormous accelerations, no
humans could survive such episodes!
Einstein clearly said that if there is no acceleration, what he called Special Relativity, and Time Dilation can occur, both ways! The Earth cannot be treated as a special reference source. So when a Twin who remained on Earth (who does not detect any acceleration on Earth) looked at his traveling twin in a spacecraft, he does see that the traveler appears to be aging more slowly than himself, so that claim of the Twins Paradox is true. But it is also true that the Twin in the spacecraft does not detect any acceleration of the spacecraft and so he also considers himself to be in a Rest Frame of Reference, and so he also sees his Twin on Earth to seem to be aging more slowly than himself. They both see the other as aging more slowly than themselves! (briefly, and only during constant velocity travel!)
Einstein had died in 1955, so he was not around to correct that logical blunders of the Twins Paradox Storytellers! The Physicists who dreamed up the Twins Paradox in the early 1960s were somehow ignorant of these basic requirements of Einstein's Relativity, and so they came up with an idea that is totally illogical and impossible. In the following fifty years, no Physicist or anyone else has seemed to notice this really obvious blunder, and the world seems to universally accept the clearly fallacious Twins Paradox!
Several fields of modern Astrophysics are even completely built on this wrong idea! Time Travel, Wormholes, and maybe even Black Holes have never actually been detected, because they are impossible and illogical ideas that were dreamed up based on foolish assumptions.
The entire principle on which Relativity is based is that two observers in different circumstances in the Universe must see a Universe which makes logical sense to each of them and that they must also agree on basic things when they would ever meet again after being out of contact.
The central assumption of the very popular (wrong) Twins Paradox entirely violates both of these requirements!
The people who dreamed up the Twins Paradox
(in the early 1960s) had made a drastic error in only considering
everything from the perspective of the Earth. And in never even
considering the timerate effects of Einstein's General Relativity.
The claim that a space ship moving away from
(or toward) the Earth (or any other viewer) at extremely high constant
velocity would show an Earth observer that the spaceship occupant was
apparently moving in slow motion is correct.
It does not have to be a spaceship,
but it could equally well be an entire planet (but it still has
to be moving away or toward you at constant velocity.)
However, the Earth is not a
Special Reference Frame! If it had been that you lived on that
other planet (Earth2), and saw the Earth moving toward you or away from
you at constant velocity, you would see everything on the Earth
appear to be moving and aging more slowly than yourself.
The interesting fact is that both observers would certainly
see the other as moving more slowly and aging more slowly than
he personally experiences. No one in Physics seems to have
noticed this obvious and unavoidable fact! (blunder!)
So you have two observers watching
each other where they both observe what is called Time Dilation.
This sounds pretty bizarre, but it is a certain fact! The presentation
below explains how this can be possible, and even that it is
required in order for the Universe to be logical to everyone!
The math equations to support this peculiar situation are also
presented below.
This situation is even true if
the two planets are moving toward each other or away from each
other at a velocity near the speed of light, where each might clearly
see the other as aging 1/2 or 1/10 as fast as himself! Both
might watch the other celebrate ten birthdays while he only
celebrated one! This probably has you doubting Physics or me
or both, but if you read and understand the following presentation,
you will see why this has to happen, and you will even see why
it is beautifully logical in every detail!
Consider this situation: You, in
your laboratory on Earth, have a tremendous telescope, which you train
on a man in a laboratory on a different planet (Earth2), which you have
determined to be moving away from Earth at very high (constant) speed.
Your telescope is so excellent
that you can even see the second hand on the clock on the wall next
to that other scientist, and you certainly see that his clock's second
hand appears to be moving more slowly than the clock which is on your wall
in your laboratory. You can even calculate the difference in the
rates of the two clocks, based on a simple formula (below) and it
IS the correct effect of Special Relativity which is called
Time Dilation. This is the basis on which the Twins Paradox
had been built. It is correct in as far as it goes, but it is
extremely incomplete and horrible science!
What those 1960s Physicists neglected
to ever consider is the view from that man you are observing!
HE also has an excellent telescope and he has it trained on you!
He has no sensation of velocity on his planet (Earth2), but he certainly
sees the Earth receding from him at extremely high velocity.
He also has that same simple formula, with which he calculates the effect
of Time Dilation which he would see on the rapidly receding Earth, with his
excellent telescope. He then confirms that he sees the second hand on the
clock on your wall appear to be moving more slowly than his own clock is
moving.
Where those Physicists of the 1960s
had only considered onehalf of the actual situation, you now see
the complete situation, where both of the two scientists are observing the
Time Dilation effect at the same time! They both can see the other
as moving and aging more slowly than himself!
I realize that this sounds impossible,
but we will show below at both how and why it actually occurs, and
even that simple formula that both scientists use to calculate the
slowing effect they each see!
The people who dreamed up the Twins
Paradox had not considered an entire planet but instead a small
spacecraft. Apparently, their lack of knowledge of Physics caused
them to think that the Laws of Physics sometimes do not apply for small
spacecraft!!
In other words, if those
Physicists of around 1960 happened to be in such a spacecraft (which had no
sensation of any movement due to the constant velocity of whichever one was
actually moving) and they looked toward a rapidly receding Earth, they might
have made the same error they did, but from the (single again) opposite
perspective! Where the traditional Twins Paradox story has the youthful
traveler meeting an elderly (Earth) twin brother at the end of the story, in
this case they would have had an elderly twin in the spacecraft meeting a
youthful twin brother who stayed on Earth! YOU CAN'T HAVE BOTH OF THEM BE
OLDER THAN THEIR TWIN WHEN THEY WOULD MEET!
Neither the traveler nor anyone on the Earth
would have any sensation of velocity! (After all, we are presently spinning at
roughly 1000 mph, orbiting around the Sun at around 66,000 mph and being
carried through the Galaxy with the Sun at around 45,000 mph, and you have
never been aware of any of those movements!) And so both viewers will
certainly consider themselves as stationary! They each see the other as
moving away at high speed and therefore we have a bizarre situation where two
people watching each other must both see the other as aging more slowly
than he is aging himself! This would actually be the truth!
The correct description of the Twins
story is quite different than has always been incorrectly assumed to be true.
More than that, an extremely important lesson comes out of the correct
analysis, which has implications throughout modern Physics.
The popular but wrong
Twins Paradox also has another obvious flaw!
According to that story, the traveler leaves the Earth
(knowing that Alpha Centauri is 4.3 light years away, and since nothing can
travel faster than light, it clearly requires more than 4.3 years for any
spacecraft to get there from Earth). However, supposedly, after only three
weeks of traveling, he arrives at Alpha Centauri. There are a lot of
wrong assumptions which were made to create this preposterous claim!
Relativity has an absolute limit of nothing ever being able to travel faster
than the speed of light. But that story would have the traveler be able to go
into Court to testify that he had had lunch with his twin brother on Earth and
then three weeks later, had lunch at a cafe on a planet of Alpha Centauri. In
other words, he had just traveled at 70 times faster than the speed of light
(going 4.3 light years distance in three weeks of time!). Einstein made
clear that NO such violation was ever possible! You will notice below that
this (correct) description requires the traveler to take, at best, around 4.5
years to travel the 4.3 light years distance, in other words, with no
perception by anyone of ever violating the speed of light.
There is an obvious example to use to show how wrong the Twins Paradox is, and you don't even need to be a Physicist to understand it! Say that both of the twin brothers fell into comas at the restaurant on Earth, where they each were not aware of whatever was going on. One stayed on Earth while aberrant Doctors decided to put the other in a spacecraft and aim it at a planet near Alpha Centauri (a star). Both remain in their comas while the spaceship is accelerating, but after the engines shut off and it then will continue to coast at constant velocity, they both wake up. They each remember nothing of previous events, due to their comas, but each is obviously concerned about his brother, and they get their telescopes out and point them at each other! The Earth brother sees that the spaceship brother is traveling away from him at very high constant speed, and due to the wellproven phenomenon of Special Relativity, he notices a Time Dilation, where his brother appears to be aging only 4/5 as fast as himself. His telescope is so good that he can even see the clock in the spaceship which appears to be clearly running at exactly 4/5 as fast as his own clock is running. Being a good scientist, he determines how fast the spacecraft is receding and finds it to be exactly 3/5 of the velocity of the speed of light (which we shall call 0.6c here).
(Time Dilation factor for Special Relativity and Constant Velocity)
He gets out his pen and paper and does the simple calculation (shown here) regarding the Time Dilation effect and determines that he should see the spaceship's clocks to be running at exactly 4/5 as fast as his clock on Earth shows. This is the standard story of the Twins Paradox. It IS true! (However, this is only one side of the story, and then only briefly true, since it only considers the view from the Earth and only considers the situation during the constant velocity portion of the trip!)
Now consider what the spaceship brother sees. He feels no acceleration,
so he rightfully assumes that he is stationary. (Just like us
on Earth regarding all those high velocities that the Earth is doing
now). In other words, he has no reason to believe that he has ever
moved or accelerated (due to his lack of memory due to the coma.) He looks
out and sees the Earth hurtling away from him at a very high constant velocity
of 0.6c. His telescope is as good as his brother's, and he can clearly see
that the clock next to his brother (on Earth) appears to be running at exactly
4/5 as fast as his own clock is running, and he also notices that his
brother seems to be moving in slow motion. (There is nothing different
in describing his perceptions than for describing his brother's
perceptions). He gets out his pen and paper and calculates that due to Time
Dilation, he should see his Earth brother appear to be moving in slow motion,
and he mathematically confirms what he is seeing, that the second hand on the
Earth clock is moving at exactly 4/5 as fast as on the clock on his
spaceship shows him!
They both see the other brother as seeming to be living in slow motion,
at exactly 4/5 as fast as he knows is true for himself! As weird as that
sounds, (Special) Relativity requires that to be true. And, below,
we will see the proof of why that can and is true, for both of them,
and even the math to show it!
The people who dreamed up the Twins
Paradox neglected to even consider the situation of the twin on the spaceship,
and by doing that, absolutely fouled up everything! They had also made some
poor assumptions regarding some very difficult math problems, which we
will see cause to correct here. The popular conclusion is
dead wrong! And you even see why that is true! The Twins Paradox
breaks a basic Law of Relativity, that there can be no preferred perspective!
How could Physicists have made such a bonehead error? And how could
all the hundreds of thousands of Physicists since then have simply accepted
the claimed statements, without noting that a bonehead assumption
was so obvious? (As a Physicist, I am ashamed for all of us.)
A critically important fact is that such
views of each other cannot be forever and are in fact only
temporary. This will be explained below. (That happens to be another
incorrect assumption of the Twins Paradox!)
The popular Twins Paradox totally contradicts this
and requires (wrongly!) that when the traveler looked back at Earth, he
would see everything on Earth going faster, exactly the opposite of what
he must actually see, according to Special Relativity! (That was instrumental
in why they claimed that the Earth brother would be older when they later met.)
Both of them necessarily have to see the
same effect (known as Time Dilation) (but only during constant relative
velocity portion of the trip), where the other one appears to be
moving in slow motion, because neither can possibly know who is
actually moving (at constant speed) and who is motionless!
This might seem impossible, as much of
Relativity often does, but it is not. This will be clarified and fully
explained below. As peculiar as it sounds, when the Twins would be
looking at each other during the constant velocity travel of the spaceship,
they truly both see the other as appearing to
move and age more slowly than he does himself!. Actually, that is a
basic requirement of Special Relativity, which was ignored when the Twins
Paradox was dreamed up!
The correct resolution of
this odd situation is that an entire trip of constant velocity is impossible if
they are to ever meet again! There must be periods when the spaceship is
accelerating and later decelerating. A complete trip therefore needs to be
examined as three separate stages. The early part of the trip involves heavy
acceleration, during which General Relativity necessarily applies. Once
the acceleration ends, then the familiar constant velocity portion of the trip
occurs, where very different Special Relativity circumstances
necessarily apply. Finally, there must be a portion of heavy deceleration,
during which General Relativity again necessarily applies.
To have assumed that Time Dilation occurs during all three stages was another horrific error by those 1960s Physicists, and it was not even based on any actual calculations! Einstein's actual equations of General Relativity are so complex that no one has yet fully solved them! Those 1960s Physicists simply guessed that Time Dilation must also occur then, which turns out to be another error on their parts!
The necessary reality is that an opposite effect of Time 'speeding' must occur during those General Relativity portions of the trip. Importantly, during those acceleration and deceleration portions of the trip, they must both see their brother appear to live faster than themselves! The well publicized effect seen from Earth during constant velocity is certainly true, but only during the Special Relativity constant velocity part of the trip. For an entire round trip, there are times of views of faster living which each will see that exactly counteracts the slower living that everyone knows about!
The specific views of the two twins who
spend the entire trip duration staring at each other are rather strange
and different, but, for each, the total effect of an entire trip is such
that the faster and slower perceptions of clocks and living exactly cancel
each other out, when the entire round trip is considered.
This result is required because the two could only ever again actually
meet if they have no relative velocity in what is called an Inertial
Rest Frame of Reference. If either should neglect the necessary acceleration or
deceleration, they might be able to whiz past each other at enormous
speed, and under such conditions, they would not share an Inertial
Rest Frame to actually visit, and their ages might then seem bizarre.
But as long as a complete trip is examined, it is easy to see that
they are the exact same age once they again meet (and also when the
traveler is in the cafe on the planet near Alpha Centauri which shares
our (Earth's) Inertial Reference Frame of Reference.
So, not only is the Twins Paradox wrong regarding the cumulative effect
as seen from Earth, but when the two brothers again meet back on Earth,
they are both exactly the same age and they are exactly the age that
everyone would have expected them to be! (and that they would both
be if the one had never left Earth) Even an observer on some other
planet (which shares our Inertial Reference Frame of Reference) would agree about that.
All Physicists have long known that there are necessarily two extremely
different Relativistic effects which occur during the entire trip.
The spaceship starts out accelerating from Earth, so General Relativity
effects apply. Eventually he shuts off the rocket engine and then constant
velocity travel would occur, and the well known Special Relativity effects
would then apply. Finally, he would have to decelerate in order to be able
to actually visit on Alpha Centauri's planet. And the same three stages
occur on the return trip. Actually, there are only three brief moments
during the entire round trip when the twin brothers could correctly say
that they were of the exact same age (plus at the end of the trip when
they are back together)! Other than that, they would each have portions of
the trip where each was definitely older than his brother, as well as
other portions where each was definitely younger. Their wristwatches
and clocks would show this. Relativity has some strange
effects, and this certainly seems peculiar, but the central point of
Relativity is that it always has to be ultimately logical for each
of them and both of them.
This actually points out another major error of assumption
that was involved in the speculating on a Twins Paradox! That assumption
was that time dilation occurs in both Special Relativity and General
Relativity. The GR mathematical equations of Einstein are immensely
complex, and no one has yet fully solved them, in more than 90 years of
trying! But around 1960, many assumptions were speculated and applied to
those equations, to make them far simpler to solve, and as a result,
it was assumed that time dilation occurs under those conditions, where
the reality is exactly the opposite!
This error is extremely obvious, as described
above, and if Einstein had not died several years before
the Twins Paradox was suggested, he would certainly have
quickly provided the correct explanation. In any case, no matter
who would make such a trip, when they would meet again, they would
be exactly the same age! While they were separated, yes, some very
strange things would seem to be seen to occur regarding time! At various times during a
complete trip, each of them would sometimes believe they are older than
their brother and sometimes believe they are younger than their twin
brother! But Einstein was right after all! When the trip was entirely
done and they were back together again, they would certainly be
exactly the same age!
It is disappointing just how many logical errors were, and still are, made
regarding the timerate effects of Relativity. In the discussion and
explanation of another one of these, I provide the entire precise math
which proves the errors and their correct understandings.
That example is perhaps an even more important error, which has amazingly still not been corrected more than fifty years later, that of the (incorrect) assumption that we on Earth live in a (nonaccelerating) Inertial Rest Frame of Reference. Several entire fields of modern Physics are totally based on this incorrect assumption. For the precise math of the proof regarding that, please see http://mbsoft.com/public4/dilation.html for the Article about a General Relativity Time Dilation Logical Error.
That Article discusses the amazingly superficial thinking of NASA where they recognized that Special Relativity and its Time Dilation certainly exists for us on Earth, due to our rapid daily spinning of the Earth. NASA even decided to try to perform a rather famous (but wrong) experiment in October 1971 to try to prove that Time Dilation assumption that they had wrongly assumed, by sending sets of four identical Cesium clocks both ways around the Earth on conventional airliners, in the HafeleKeating experiment. That experiment wound up with results which were worthless, well within the Experimental Margin of Error.
However, everyone in NASA was apparently ignorant that we also "ride in a daily circle" in that same process, which means we constantly accelerate (radially downward), so that Einstein's General Relativity also applies. (We call it centripetal acceleration!) These effects are both easy to calculate and it turns out that their timerate consequences are exactly the same net effect, but are opposite each other! Therefore, they always exactly cancel each other's net effects out for us! That statement is equally true for people in their homes, for Astronauts who orbit the Earth in the International Space Station, and even in airliners which circle the Earth.
The (General Relativity) timerate effect due to the orbital acceleration of the ISS is a timerate factor of 1.000 000 000 330 (more than 1.000 and therefore, a time speeding effect).
Since both of these Relativistic effects apply continuously, we must multiply the two timerate factors to find the actual Relativistic timerate effect on us, where the product exactly 1.000 000 000 (actually, 0.999 999 999 998 999 999). Please note that this is proof that the two Relativistic timerate effects exactly cancel each other out (for the ISS), within any conceivable error factor.
The (General Relativity) timerate effect due to the centripetal acceleration of the person at the Equator is a timerate factor of 1.000 000 000 001 203 440 (again, more than 1.000 and therefore a timespeeding effect.)
Since both of these Relativistic effects apply continuously, we must multiply the two timerate factors to find the actual Relativistic timerate effect on us, where the product exactly 1.000 000 000 000 000 (actually, 0.999 999 999 999 999 999 99). Please note that this is mathematical proof that the two Relativistic timerate effects exactly cancel each other out (for the person at the Equator), within any conceivable error factor.
The complete math for the examples referred to above is all presented in the Article
at General Relativity Time Dilation Logical Error http://mbsoft.com/public4/dilation.html
Considering again an actual high speed space trip
IF the rocket engine was incredibly powerful, and the two stared at each other for the entire 4.5 year trip to A.C.(and the Traveling twin could survive the extreme acceleration, which he could not), the Earth brother would see maybe a week go by (Earth time) during the acceleration, but during that Earth week, he would see the accelerating twin celebrate two birthdays! In other words, he would watch his spaceship brother age ferociously faster than he lives here on Earth! Then during the next 4.3 years of watching the constant velocity portion of the trip, he would see no Birthdays! And in the final week of extreme deceleration, he would see another two birthdays celebrated, so he would see the spaceship brother celebrate the correct number of four birthdays during the trip. Just in a very odd way, two really fast, then none for an apparent long time and then two at the very end.
The Journal of the spaceship twin would look vaguely similar, but he
would spend more than two years accelerating (probably at a constant rate)
and during that acceleration, he would therefore obviously celebrate
his two birthdays. He would also witness his Earth brother celebrate
two birthdays, but a few days different (earlier) than his. Since he is
a very careful scientist, he notes that the Earth brother appears to be
living slightly faster than he is (during the entire two years of watching and accelerating). This
results in an interesting consequence that, as the acceleration ends,
they both see their brother as appearing to be older than himself! The
Earth brother thinks more than two years older, and the spaceship brother
thinks just a few days older.
Recapping: At this point of the trip, as the rocket engine is shut
off from the accelerating portion, the Earth brother sees his twin
appear roughly two years older than himself. This is at the same time
the spaceship brother sees his Earth twin appear a few days older than
himself. They both see the other as having aged faster and is now
older than himself.
Continuing the spaceship brother's Log Book:
After the rocket engine is shut off (after more than two years
of acceleration) they both have the Special Relativity situation
of Time Dilation, so they both see the other appear to live more slowly
than himself! The Earth brother watches this happen for more than four
years, while the spaceship brother only actually coasts for a couple weeks!
(This is why no birthdays were seen being celebrated on the spacecraft during
the four years that the Earth brother was watching, because the traveler
actually only aged by a few weeks!)
Recapping again: At this point, the Earth brother has seen the spaceship
brother barely age at all (a few weeks) while he ages roughly four years.
So now the Earth brother sees the spaceship brother appear to be two
years younger than himself (exactly the opposite as at the earlier recapping!
The spaceship brother only spends a few actual weeks in this coasting
portion of the trip, but he clearly sees his brother on Earth appear
to have aged more slowly than himself, but only by a few days. The spaceship
brother now sees his Earth twin as appearing a few days younger than himself.
In other words, they both have watched the brother seem to age more slowly
than himself, during the constant velocity portion of the trip. This is
the resolution to the strange statement made above where they both see
the other aging more slowly than himself (but only during the period
of the constant velocity.)
He then turns the spaceship around and
fires the engine for another two years in severe deceleration. So HE only sees
the Time Dilation effect for those few weeks of coasting at constant
velocity. He would watch as his Earth brother aged a few days less than
he did during those few weeks, but they both therefore watch the other
aging more slowly than himself.
The result of all this is that only at the exact halfway point of the trip,
they would both see them as being exactly the same age. (Remember that
they had each seen the brother age faster, first, during the acceleration,
and then slower during the constant velocity part of the trip, and they
exactly cancel out at that halfway point.)
You can see how and why the second half of the trip has the slower then
faster perception of the other, so that after the spaceship has slowed
down to be able to stop on that planet at Alpha Centauri, they are again
exactly the same age. (for the second time since the start of the trip)
There can be no doubt about this fact! The silly ideas that are rampant
in modern Astrophysics where they talk about time travel and goofy things
like wormholes, are all based on the wrong assumptions back in the early
1960s when the Twins Paradox was dreamed up! And so those alleged
phenomena have no actual basis at all!
The wrong assumptions and erroneous logic used in coming up
with the Twins Paradox has endured! There are several complete fields
of Astrophysics which are based on that wrong reasoning! Even famous
Physicists (including Kip Thorne) believe that the Twins Paradox is true and that there
are therefore things like wormholes, and many other alleged effects
on black holes and quasars and more. Those fields cannot be valid!
A series of TV programs from NOVA (in 2011) still repeated the wrong
assumption regarding Time Dilation occurring during General Relativity!
Even without good logic, exciting ideas seem to die hard! That series,
by the otherwise intelligent Brian Greene, contains hundreds of
wrong assumptions, even including things that Greene knows to be
bad assumptions! I find that very sad.
I have a different webpage regarding a relatively simple and
inexpensive test which I have tried to encourage NASA to do for ten years,
which can and will provide an absolute experimental proof of General Relativity
and whether there is Time Dilation or Time speeding as a result. The
General Relativity 
A Moon Experiment to Confirm It simply has a set
of standard Cesium atomic clocks softly landed on the surface of the
Moon, and a set of identical Cesium clocks in a laboratory here on
Earth. We know the precise frequency of the Cesium atoms on
earthbased atomic clocks to be 9,192,631,770 cycles per second.
The gravitational field on the Moon's surface is only about 1/6 of the
gravitational field we experience here on Earth. In Physics, there is an
Equivalence Principle which is broadly accepted as being an
expression of Einstein's claim that a gravitational field is
indistinguishable from an acceleration. The tiny difference in the
strength of the gravitational fields on the surface of the Earth and
Moon is such that, every second, the atomic clocks on the Moon should be
counting numbers of oscillations of the Cesium atoms which are several
different from the exact number presented above. Every hour, the Earth
atomic clock should count about 10,976 more ticks, a really obvious
experimental result.
Over a period of an entire year, the clocks should be different
by an obvious amount, a good portion of a second. I am certain that the
Earth clocks will be running faster, in agreement with a Time
Speeding effect, but even if that 1960s assumption of Time Dilation
occurring turned out to be correct, the clocks would still be easily measurably
different in the rate of time passage. General Relativity would finally
actually be experimentally confirmed!
The correct logic contained here is a
confirmation of some of Einstein's Thought Experiments, specifically those
which related to a stationary rocket sitting on the Earth's surface with an
identical rocket in deep space which is accelerating at 32 ft/sec/sec.
Einstein pointed out that the residents of both would experience
exactly identical environments, where one was due to acceleration due
to a rocket's motor and the other one was due to gravitation of the planet Earth.
Above, we gave the exact math for a person standing at the Equator of the
Earth, where the Relativistic timerate effect of Special Relativity and the
Relativistic timeeffect of General Relativity were calculated to be
exactly the same amplitude, but in exactly opposite timerate directions.
That was proof that the two Relativistic timerate effects are opposite
each other.
Somehow, Physicists discarded all of that logic in creating the Twins
Paradox! But clearly, the rate of time passage is being affected
by General Relativity, which means that it is also affected by the gravitational
field of any planet. The linked page above has suggested to NASA for several
years to send some precise atomic clock up to softland on the surface of the
Moon, while an identical atomic clock would remain here on the surface of the
Earth. The two would be regularly checked regarding whether they were
keeping the same time! They would not, due to GR!
That simple experiment would prove Einstein's General Relativity,
and even give specific precise numbers to the effects, as we have for the
effects of Special Relativity! My linked page includes my calculations
of Einstein's set of General Relativity equations, where I find that
we are all older than we would be if we had lived our lives on the
surface of the Moon. In my personal case, as an old man now, I am
probably only by about 0.73 second older having lived my life here
on the surface of the Earth!
General Relativity  A Moon Experiment to Confirm It.
The basic logic of the reasoning below is very
basic and simple, but the complexity of nearly any Relativistic subject can
make full understanding somewhat more difficult. This presentation was
composed with the intention of providing as much clarity as is possible.
A set of simple equations are provided which
can predict the accurate experiences and viewings of either or both of the
two brothers, and for any maximum velocity trip, of any length, and
of any power rocket engine. Just those three variables establish
all the accelerations, velocities, and locations at all instants during
a trip.
Frighteningly, there is such universal acceptance of this very
wrong concept of the Twins Paradox that many fields of modern Astrophysics
are entirely dependent on it being valid, and so those fields are in grave
doubt regarding them maintaining any credibility!
This Research and analysis was done during 1997 through
2004. This presentation was first placed on the Internet in
August 2004.
When each looks at the other, from their own inertial restframe, they each have no sensation of motion. Therefore, they see the other planet as moving at 0.6c and expect Special Relativity (SR) effect of Time Dilation, due to what they each see as the other planet moving toward or away from them. For this velocity of 0.6c, we can easily calculate that this is a factor of 0.8 regarding time passage. As each grows up, they therefore each see the other as aging more slowly (0.8 times as fast) than they age. We are going to momentarily neglect the fact that the two might see each the other as having been born before or after themselves, and only consider the interval while they constantly watch each other. During an interval when they each live 30 (Earth) years, they each see that the other has only lived 24 (Earth) years!. This is true of both of them! There can be no doubt of this because each of them is in a restframe coordinate system which is not accelerating, and which each therefore considers to be "stationary", such as we tend to do here on Earth.
This alone negates any possibility that the popular Twins Paradox could be valid! They both see the other person and planet as having aged slower than themselves! The Twins Paradox was incorrectly built upon the premise that requires that only one (who happens to be on Earth) would sense this, when a viewed traveler was moving at a high but constant velocity away from or toward the Earth. The people who promote the Twins Paradox simply ignore the fact that the traveler on the spaceship is traveling at constant velocity and therefore would see the same Time Dilation effect when looking at the Earth. Imagine that he lived on a planet instead of a spaceship, where you might see it as much more obvious that he would be the one who felt he was on a stationary planet. This simple example shows that the (constant velocity) traveler must experience the same perception, where the Earth twin (and everything else on Earth) MUST be perceived by him to be happening more slowly than in his own life! The Twins Paradox cannot be credible, as the reality is clearly that both have to experience the same sensation!
We will return to our example momentarily, but we wanted to note here that a similar situation could exist where the relative velocity of the two planets was 0.999999 of the speed of light. The Time Dilation factor is 700 to one for this relative speed. The result would be that baby A would age to 60 years while he watched Baby B only age one month! OK, you might be willing to swallow that, but at the same time, Baby B was also constantly looking at Baby A. He would only see Baby A age by about one month in his entire life of 60 years. Both of these situations could actually be true, but the necessary circumstances are very bizarre. And they could never decelerate to the same speed to ever meet, without one or both of them dying of old age first! Incidentally, before this bizarre Special Relativity circumstance, IF they had been born as twins, one of them would have to accelerate spectacularly, where the General Relativity timerate speeding effect would account for a timerate speeding effect of 700 to one. That is, the baby who would (naturally) age by 60 years during the apparent onemonth aging of the other during the Special Relativity Time Dilation would have watched as the 'accelerating traveler' would have been seen to age 60 years during one month of his own (natural) aging. As with all these other situations, at the 'midpoint' of any actual trip (where acceleration, cruising and deceleration necessarily occur) both observers would be 60 years and one month older and see the other as also being 60 years and one month older. They would just record it in extremely different Observational Journals. The nontraveling one would NOT have any of his own birthdays during the first month while he watched the ferocious acceleration, while he would watch two birthdays be celebrated on the traveler's ship every day. Then, he would see no more birthdays be celebrated on the traveling ship during the following sixty years of watching. Etc.
Back to our example described above: This is all completely true, because each is in a nonaccelerating (Inertial) Rest Frame of Reference, and so SR applies. We now have both of them seeing the other as having aged six years less than they did! (These comments and this analysis are not going to include an additional complicating factor of the fact that the light image of events on either planet takes some time to propagate the distance they happen to then be apart. In principle, we are going to consider the specific situation in each case where an event occurred as the highvelocitydifferential planets happen to pass each other very closely at that very high velocity, so that the distance apart is small enough such that the propagation time of the light images is very small.)
On that 30th birthday which they each celebrated while watching each other celebrate their 24th, they had previously decided to get to a situation where they are both moving at the same velocity. They realize that once they accomplish that, all SR and GR effects will end, and they will then age at exactly the same rate as each other, from both their pointsofview.
They realize that there are three obvious different ways they could do this: (1) A gets in a spaceship and accelerates to the velocity of B; (2) B gets in a spaceship and accelerates to the velocity of A; or (3) each get in spaceships and accelerate halfway toward the other's velocity. Whether they are next to each other or far apart does not then matter, because once they have no relative velocity to each other, no relativistic effects of either SR or GR would apply. To confirm, once at that common velocity, they will then both see the other as aging exactly at the same rate they do themselves. That is the only situation where they could actually establish whether one is older than the other! In any situation where they are either moving at relative velocities to each other (SR) or accelerating (GR), they will appear to age at different rates to each other, and no such valid comparison of ages would be possible.
Remember that both of them watched the other as having aged six years less than themselves aged! How could this be resolved if they should manage to meet? They cannot both think that the other is younger then!
This situation has been set up with maximum symmetry, such that when they would meet, or equivalently, be traveling at the same relative velocity, they must certainly find themselves to have corrected those perceptions regarding aging. That eliminates any socalled paradoxes!
This specific situation is one of many similar situations which show that the popular ideas of what is called the Twins Paradox is simply wrong! That premise only considers the situation from one of the two perspectives, and totally ignores the other, and then applies one very incorrect assumption, to arrive at an alleged way where it is presented that one could travel vast distances across space without aging much! It is such an attractive concept that it has become universally accepted as valid, even being included in a huge number of textbooks. But it is simply not true! That description is also quite illogical, as we have already started to see above, but there is a perfectly logical explanation for all the actual experiences both SEE and experience, as well as what any outside observer would see of each of them. It is presented here.
A very interesting conclusion exists in all of this. Say that you wanted to go to a planet circling Alpha Centauri. Traditional thinking is that you would want to accelerate as fast as you could, so you could spend the bulk of your time traveling at the highest possible velocity, to get there quickest. However, that is not the case here! Speed does matter, but in a rather odd way. It is only important the fastest you had traveled during the trip, and any concept of "average velocity" does not have any meaning! We will show below that you could have a rocket with minimal power, where it was only capable of less than one G of acceleration, your rocket would get up to a maximum velocity of 0.6c or 0.8c by the halfway point of the trip, after which you immediately start decelerating all the rest of the way. Alternately, you could have a rocket with a ferociously powerful engine, where it accelerated you very quickly to that desired 0.6c or 0.8c, at which point you would coast for most of the trip, and then rapidly decelerate near the end. In both of these cases, you would get to that planet in exactly the same total amount of time! (This will be thoroughly discussed later).
This discussion actually has a very simple conclusion, that the wellknown effects of Special Relativity, specifically Time Dilation, is counterbalanced by opposite effects which occur during the acceleration and deceleration required to get to that differential relative velocity. And, interestingly, where both the Earth and the traveler has the SR Time Dilation perception that the other is aging more slowly, it is also true that both have the opposite perception during acceleration (of either of them) where they see the other as aging faster than they are!
We can consider each of the three possibilities that our two men might do, and think about their individual experiences! The first two situations are actually identical, from logic perspectives, and we will examine them first. They are also simpler to conceptualize.
An (incorrect) assumption in the Twins Paradox thinking is that the wellknown Time Dilation ("a perception of slow aging") of Special Relativity (constant relative velocity) would also occur due to the acceleration effect (which is General Relativity).
There is no question that SR and Time Dilation occurs, as it is easily confirmed by simple experiments, such as some first performed in the 1920s regarding cosmicray created Muons in the Earth's high atmosphere. Those Muons are known to have very accurately known restlifetimes (0.000 002 212 second), where they are capable of traveling less than half a mile (at the maximum of 186,000 miles per second) before they disintegrate into other particles. Yet, enormous numbers of such Muons manage to make it all the way down to the Earth's surface and to laboratories where it is easy to confirm that they are making it distances of 20 to 60 miles! Time Dilation due to the extreme high velocities of those Muons is the only explanation for such experimental results of Muon detection in laboratories.
But virtually all Physicists have simply assumed that whatever happens in SR also occur in GR. That is absolutely wrong! The math of GR is immensely involved, and few people have even attempted to solve the GR equations, and those few people have claimed that GR causes Time Dilation, after incompletely solving the equations. It is clear that they are wrong about that!
Keep in mind that as he starts the trip, they each have seen the other having aged 6 years less than themselves, which obviously cannot still be true when they meet or even get to the same relative velocity. In one way or another, logic requires that they then agree on whatever the reality is. They will not be able to both believe that the other is younger than themselves!
What necessarily occurs is this: The nonaccelerating observer would have a normal life where he aged six years (from age 30 to 36) while the other was traveling. But he would see the other seem to experience an opposite effect to Time Dilation! In fact, he would see the traveler experience a total of twelve years (from age 24 to 36) during that (acceleration) trip! The nonaccelerating observer would have seen him seem to be constantly moving in highspeed motion for the whole trip, seeming to live life at double the normal speed!
The traveler would not have noticed anything unusual, and would have instead have experienced a trip that took him six normal years (from age 30 to 36) to complete, with a constant acceleration for the entire time. He would have watched the nonaccelerating one appear to age twelve years (from age 24 to 36) during that trip.
Once he stops accelerating, that effect would end, and they would see each other as then aging at exactly the same rate. The nonaccelerating one would have seen each age a total of 36 years during the whole story, either his own 30 + 6 years or the traveler's 24 + 12 years. He would see neither as having had any Twins Paradox advantage regarding the rate of passage of time. Yes, he would have seen that effect during the constantvelocity (SR) years, but then he would have seen it counteracted during the acceleration years (GR), where the total of his perception would be the same for himself and for the traveler.
This results in both of them not having any illogical results, and in fact, they even both would agree that a total of 36 years had gone by while they had observed each other. The primary difference would be that they each saw a different distribution of years before and during that trip.
No "paradox" exists at all!
The nonaccelerating one therefore starts out seeing that the other start the trip as being six years younger, but then watches as the traveler accelerates and ages at essentially doublespeed, to age 12 years during those perceived six years of travel.
The traveler therefore actually spends 6 years of his life during that journey, but the nonaccelerating one sees him age 12 years during that journey.
There are alternate ways of describing this, but they are all actually the same.
The end result of this is that the acceleration would cause each of them to see the other one as experiencing fast aging, aging 12 years while his own clock shows the passage of six years. This is exactly the opposite effect of SR regarding the Time Dilation perception by either inertial nontraveler, whether on Earth or in a constant velocity rocket. Therefore, they agree upon meeting that there is nothing peculiar about their ages!
There are some slight adjustments to this, primarily due to the relative velocity that exists during all parts of this trip, so certain, generally minor, SR Time Dilation effects are also present.
It is important to note that no previous history of accelerations has any effect in this. They had each lived their entire lives on their nonaccelerating, inertial, planets. They each start out at the trip beginning as seeing the other as having aged six years less than themselves. But when they meet, only a General Relativity acceleration has occurred during this trip and a direct result is that an effect of fast aging resulted. This is therefore actually equally valid for any of the three possibilities of their meeting which are possible.
Note that the "perception of age saving" due to the SR period of travel can be entirely and perfectly cancelled out by a "perception of rapid aging" during the GR acceleration. But the SR result is cumulative over however many years they choose to watch each other, and the GR result has a peculiar cumulative effect over the period of the constant acceleration. If, for example, they had waited 60 years instead of 30, then the age differential would have become twelve years instead of six. Depending on the rate of acceleration during the trip, it may or may not have an effect to counteract six years of effect or twelve years of effect.
During SR, we all know that an apparent slowing of time occurs, Time Dilation. But this reasoning establishes that, during GR, Relativistic acceleration, there is necessarily an apparent speeding of time that occurs. Traditional Physics has denied that such an apparent "speeding of time" ever happens, insisting instead that Time Dilation also applies during GR acceleration. But the example above clearly shows that if GR caused TD, the two would then each meet a person who was more than six years younger than themselves, which is obviously impossible. They can be the same age, or one of them can be older when they meet, they cannot be both older than the other!
We have the same initial situation, where both have seen the other age six years less than they have, which must be resolved. If we assume that they both have identical rockets, in other words, identical accelerations, then from above, we have a situation where the effect of the other's GR acceleration will cause an apparent rapid aging by three years in the other. His own acceleration has a separate GR effect of causing a separate three years of aging in the other. This results in the other seeming to age an extra six years, three plus three, during the pair of trips. This counteracts the initial fact that the other was seen as six years younger than the observer, which then results in the situation where they are both the same age when they are both traveling at the midway velocity of 0.3c from both planets. They would both agree that this is the case, and in fact, it also matches the initial observation of the impartial observer who had initially been traveling at the 0.3c and had seen what he considered simultaneous births on the two planets. The fact that they now meet, at that velocity, and they are exactly the same age agrees with that.
Again, no paradoxes whatever!
We have chosen some of the simplest possible situations to try to make this all most understandable. It is actually not necessary that the cumulative effect of the GR time speeding is identical to the cumulative effect of the SR time slowing, except for very specific situations. Most specifically, if an "entire round trip" is considered, then they must necessarily exactly balance each other out, where there is no net advantage or disadvantage regarding time. (Examples are shown below) (Exactly opposite what the Twins Paradox says occurs!) Trips are also limited by the total trip time required having to be greater than that of the speed of light. For most other real situations, the GR and SR cumulative effects could be different.
The consequences of this are enormous. When this is all carefully and thoroughly analyzed, an entire trip (acceleration, cruising and deceleration to the initial inertial rest frame) results in a very different conclusion than if only the SR portion of a trip is considered. In that second case, a Twins Paradox story actually seems to be reasonable, where they would meet being many years different in age. However, a correct explanation of such a real trip must necessarily include periods of apparent rapid aging (during GR acceleration) as well as the very well known period of slowed aging (during SR cruising). An entire trip then necessarily involves (as seen from the initial location, i.e., Earth) first a perceived rapid aging during acceleration, then the wellknown perceived slowed aging during the SR constant velocity part of the trip, and finally another perceived rapid aging during deceleration. A careful analysis of the entire trip always results in the total trip taking the "correct" total amount of time. There IS no "time benefit" due to taking fast trips and coming back. The "Twins Paradox" story would therefore actually result in the twins being exactly the same age when they meet again!
IF the entire trip occurred in the conditions of SR, of constant velocity, then the Twins Paradox could be true. But that is certainly not the case, as enormous accelerations and decelerations are necessary. And that changes everything!
When Einstein first proposed Relativity, he gave several preconditions, which required Relativity in order for them to be true. They are:
The Twins Paradox story violates at least one of these!
The easiest way to see this is to temporarily set aside Physics and Relativity, and simply consider the actual experiences of the traveling twin. Here is the standard Twins Paradox story, with the addition of a single introductory sentence!
Twin brothers have lunch together on Earth, while carefully determining the actual distance to Alpha Centauri, where they each get a result of a distance of 4.3 light years away.
One of the twins immediately gets in a spacecraft and accelerates extremely rapidly. He has a forward window in the spacecraft, and he sees that Alpha Centauri is a distance of three light weeks away from him. So, it then makes complete sense to him that his entire trip takes only three weeks of his life! Once on a planet around Alpha Centauri, he is then about 4.2 years younger than his twin on Earth. He says hello and goodbye, and travels back to Earth. Again, he "gains" another 4.2 years, so when he remeets his twin brother, he is now 8.4 years younger than his twin!
He and his brother carefully measured the distance to A.C. as being 4.3 light years, but then an hour later, HE saw A.C. as being only 0.06 light year (3 lightweeks) away! That is not possible! It represents what Einstein called a discontinuity in space, and such things violate Physics and Relativity. It also requires the traveler to have the personal experience of having traveled nearly 4.3 light years of actual distance in just a threeweek trip, meaning that he could allegedly then go into Court to Testify that he had just personally traveled around 75 times faster than light travels! In fact, Relativity was developed specifically so that no person could ever experience such a discontinuity of space, or of time, or of believing that he/she was traveling faster than the speed of light. The Twins Paradox involves a required assumption, where at least one of these is violated. Therefore, even though Relativity, including Special Relativity and General Relativity, and Time Dilation and all the other consequent effects are true, the Twins Paradox definitely is not true!
Some of the attempted explanations invoke some very peculiar ideas! One claims that the Twins traveler actually would be the 8.6 years younger than his brother when them meet again but that all of his body processes would still have gotten "older" and he would die just as if before! Where could someone come up with such a silly idea? Such things are easily shown to be silly if a trip to a star 100 light years away was involved. His earth brother would therefore have to be over 200 years old when he arrived back on Earth, while he might only be 25. Is he supposed to instantly die of old age at 25 on returning? Silly!
A major reason for such errors is that it seems often true that the SR and GR portions of the trip are freely interchanged, and that results in the wrong conclusions. One popular claim is that the two twins do not both experience the effects of Time Dilation because they are "actually not in symmetrical situations, because the one had gone through accelerations". That statement is foolish, because if only the constant velocity portion of the trip is considered, as though there is no "memory", the two actually are in perfectly symmetrical situations, each feeling that he is in an inertial rest frame with the other rapidly moving. It is an error to invoke that an acceleration that occurred at some previous time could or would alter experiences during SR! (The very first example presented here shows that. It would not matter whatever accelerations those two planets experienced prior to the babies' births, only their SR lifetimes have effect.)
There have been countless experiments, such as the muon experiments at CERN in 1966, which confirm that Time Dilation occurs for constant velocity motion near the speed of light. Time Dilation is certainly true for SR conditions. But those muons disintegrated while still at that relativistic velocity, and not after the muons were slowed to nonrelativistic velocities. So no conclusions regarding GR can be gleaned from such experiments, only regarding SR. I must admit, though, that the fact that those muons were traveling in circular paths means that they actually were under constant (lateral) acceleration seems to me to indicate that an additional complication must have existed for those analyses.
Notice that no assumptions have been applied here, and only the personal experiences of the individuals have been considered. When Physicists have neglected that first sentence (which I added), then they always immediately start citing equations on why the Twins Paradox is true. But they invariably neglect to consider the discontinuity in space that would therefore be required, and which my added paragraph makes clear. Then it is appropriate to start doing the math!
This certainly sounds very peculiar! How could two people be looking at each other and both see that the other has clocks that are advancing more slowly than his own? But it is certainly a reality of the situation. That is actually pretty easy to see, too. The "symmetric situation" that this presentation started with certainly shows it to be an unavoidable fact.
In Special Relativity, we have a situation where one person (our traveler) is moving at very high speed, away from or toward the other. The Twins Paradox proponents have always considered the Earthbound twin as experiencing "normal time" and all descriptions are based from that perspective. However, that is not the only available perspective! Say the Twins Paradox traveler does not have any memory and simply wakes up, or is born (on an extremely long trip) and looks out the window, to see the Earth hurtling away at a constant extremely high speed. From his perspective, and his experience, he is "experiencing normal time". But he sees his brother and the Earth receding at very high speed, so his view of his brother clearly shows (by Special Relativity) that his brother (and everything else on the Earth) is experiencing "slower time". Since the two brothers have been constantly staring at each other for the whole trip, that means that they each must see the other as aging more slowly than he does, during the entire constant velocity, Special Relativity portion of the trip. This has to be true! Otherwise, he would be required to have experiences that violate Relativity.
This is required because neither twin is then experiencing any acceleration. They each therefore see themselves as being in the "Inertial Rest Frame", while the other is therefore rapidly moving away from or toward them.
As odd as it sounds, that does not actually violate anything, because when the entire trip is considered, there is a perfectly logical and mathematical explanation and description, from both viewpoints, and without having to invoke any weird assumptions!
The actuality of the situation is then necessarily that, yes, during the Special Relativity constantvelocity parts of the trip, there IS the Time Dilation that we all can easily calculate. However, when the entire trip is considered, it turns out that he necessarily lives a total of exactly the same amount of time that his twin brother does on Earth. (At different points of the trip, one or the other brother is older or younger, because of the differential aging effects of Time Dilation of Special Relativity and this "Time Speeding" effect of General Relativity, but whenever they are both in the same restreference frame, they are exactly the same age! So, when he gets back to Earth, he and his brother meet and are exactly the same (correct) age!
A basic premise, on which several fields of Astrophysics are based, is therefore incorrect! This is troublesome, but certainly true.
As an experienced traveler, the traveling twin carries a LogBook. While he is still with his twin brother on Earth, about to get in the spaceship, he makes the first Entry, where he and his brother each confirm that Alpha Centauri is 4.3 light years away. For clarity and simplicity, I enable him to never need to sleep during the entire journey to A.C.. The Twins Paradox would claim that he would arrive at A.C. maybe 10 weeks later (of his watch's time), the entire point of the Twins Paradox! So he would later write in his LogBook his arrival at A.C., ten weeks after leaving Earth. He has (allegedly) now arrived at a location 4.3 light years (225 lightweeks) from where he was 10 weeks earlier, and he has not slept, so nothing weird could have happened to him. In his personal experience, he could then confirm that he just covered that distance at around 22.5 times the speed of light! And he even has a LogBook to prove it! If he was taken into Court, he could even testify to those facts! And he could even submit his LogBook as evidence! He (allegedly) experienced ten weeks of life (due to the Special Relativity effect of Time Dilation) and is now, provably, is over 22 times as far away as even light could have gotten! (This is obviously impossible, and is another proof of why the Twins Paradox story is wrong, even though the Time Dilation on which it is based is absolutely true.)
Physicists seem to want to completely ignore the parts of the trip other than the Special Relativity, constantvelocity part! And, if only that part is considered, yes, he would essentially experience what the Twins Paradox would claim. But that is not the whole trip! There is also another assumption that Physicists seem to ready to make, regarding whether the Time Dilation of SR also applies to GR (General Relativity) during acceleration. That assumption is clearly wrong, as demonstrated in the examples above, where the exact opposite effect is seen.
(His) Date  Dist. to go  Dist. Done  Week Avg. Speed 

Start on Earth  4.30 ly  0.00 ly   
Week One  3.87 ly  0.43 ly  23 * c 
Week Two  3.44 ly  0.86 ly  23 * c 
Week Three  3.01 ly  1.29 ly  23 * c 
Week Four  2.58 ly  1.72 ly  23 * c 
Week Five  2.15 ly  2.15 ly  23 * c 
Week Six  1.72 ly  2.58 ly  23 * c 
Week Seven  1.29 ly  3.01 ly  23 * c 
Week Eight  0.86 ly  3.44 ly  23 * c 
Week Nine  0.43 ly  3.87 ly  23 * c 
Week Ten  0.00 ly  4.30 ly  23 * c 
Notice that every week in this Log Book, his average speed is seen (by him) as being 23 times the speed of light! (He measures a movement of 0.43 ly/week, or about 23 lightweeks/week) 
Not even considering any Physics yet, try writing any Log Book entries, going from 4.3 light years to zero remaining distance, in ten weeks of recordings. If you did it "equally", (as in the example Weekly Log Book shown at the right) then during each of his weeks, he would see the remaining distance decrease by around 0.43 light year. (or around 23 lightweeks per week), a clear violation of exceeding the speed of light! If you try to tweak the numbers, yes, you could have part of the trip appear to him to comply with the speed of light, but then other parts have to be even greater violations! (as seen below)
As long as he does not sleep, and does not go insane, there is no possible set of LogBook entries that could show how he could go (in his experience) 4.3 light years distance in just ten weeks! A discontinuity of either time or space would be required. But Physics and Relativity do not allow such discontinuities, or exceeding the speed of light, in ANY reference frame. Therefore, he cannot possibly arrive at A.C. in ten weeks of his time as the Twins Paradox insists. In fact, he cannot arrive there, in any possible way, in less than 4.3 years total time (for the one way trip). This proves that the Twins Paradox is entirely wrong (except for its references regarding Time Dilation during the Special Relativity portion of the trip, which IS true!)
It is interesting that the only time that the Twins Paradox argument makes sense is during the Time Dilation, Special Relativity portion of the trip! But for the entire trip, it cannot make sense!
(His) Date  Dist. to go  Dist. Done  Week Avg. Speed 

Start on Earth  4.30 ly  0.00 ly   
Week One Acceleration  2.23 ly  2.07 ly  115 * c 
Week Two  2.21 ly  2.09 ly  ~ c 
Week Three  2.19 ly  2.11 ly  ~ c 
Week Four  2.17 ly  2.13 ly  ~ c 
Week Five  2.15 ly  2.15 ly  ~ c 
Week Six  2.13 ly  2.17 ly  ~ c 
Week Seven  2.11 ly  2.19 ly  ~ c 
Week Eight  2.09 ly  2.21 ly  ~ c 
Week Nine  2.07 ly  2.23 ly  ~ c 
Week Ten Deceleration  0.00 ly  4.30 ly  115 * c 
Notice that during the SR weeks in this Log Book, his average speed is seen as being essentially the speed of light (due to time dilation effects).
That would make sense to him.
Notice also that he would not
Obviously, something very strange 
(His) Date  Dist. to go  Dist. Done  Week Avg. Speed 

Start on Earth  4.30 ly  0.00 ly   
He records 115 weeks of acceleration  
Week 115 Acceleration  2.23 ly  2.07 ly  rise to ~ c 
Week 116  2.21 ly  2.09 ly  ~ c 
Week 117  2.19 ly  2.11 ly  ~ c 
Week 118  2.17 ly  2.13 ly  ~ c 
Week 119  2.15 ly  2.15 ly  ~ c 
Week 120  2.13 ly  2.17 ly  ~ c 
Week 121  2.11 ly  2.19 ly  ~ c 
Week 122  2.09 ly  2.21 ly  ~ c 
Week 123  2.07 ly  2.23 ly  ~ c 
He records 115 weeks of deceleration  
Week 238 Deceleration  0.00 ly  4.30 ly  ~ c 
Notice that during the SR weeks in this Log Book, his average speed is seen as being essentially the speed of light (due to time dilation effects). That would make sense to him. (He measures a movement of 0.02 ly/week, or about 1 lightweek/week) Notice also that we have logical experiences for him during the many acceleration and deceleration weeks, when he would see his speed increase from zero to nearly the speed of light and then drop back down to zero.
Notice also that he would not see

The total trip, as seen from Earth, would therefore be seen as taking (1 + 236 + 1) or a total of 238 weeks. The Earth observer would personally age 238 weeks during the entire period of observation, the same total interval. But he would see the peculiar fact that he would see the traveler apparently experience extremely rapid aging for one week, then very little aging for the next four years, and then another week of extreme rapid aging.
Both of them would therefore agree that the traveling twin aged a total of 238 weeks during the entire trip, but only once the traveling twin had slowed back down to the initial earth velocity, to be in the restframe of the Earth, even though the traveler was then at Alpha Centauri.
Notice that during the SR portion of the trip, the traveler experiences 8 weeks of life, while the Earth would see his brother travel for 236 weeks during that time potion of the trip, which is exactly what the Time Dilation consequence of SR says. During that part of the trip, yes, the Earth twin therefore ages 236 weeks while the traveling twin ages only 8 weeks. But when the GR consequences are included regarding the acceleration and deceleration, we find that both age a total of 238 weeks during the entire trip!
Actually, the Earth twin would see the traveling twin appear to age incredibly rapidly during that week of acceleration, then being 115 weeks older rather than the oneweekolder the Earth twin has become! So, at the moment of the end of the acceleration, the Earth twin would see his brother as being 114 weeks older than he was. But then the traveling twin is seen as aging only 8 weeks while the Earth twin experiences 236 weeks of life (the very well known part!). This results in the Earth twin then seeing his traveling twin as having become 114 weeks younger than himself, just as deceleration began. By the way, at the exact midpoint of the trip, they would see each other as exactly the same age! But the traveling twin would be seen to again appear to age extremely fast during the deceleration, actually aging 115 weeks while the watching Earth twin ages only a week. Again, the net result is that both of them live through 238 weeks of existence during the entire trip.
Note that they are in complete agreement that the traveling twin aged 115 weeks during acceleration, then 8 weeks more during the cruising, and then 115 weeks more during the deceleration. They both therefore absolutely agree that the traveling twin ages a total of 238 weeks during the one way trip.
By the end of the acceleration (which he measures as taking 115 weeks of his time), he will have seen his twin brother on Earth appear to age about one day short of 119 weeks. So, at that moment, he would see the Earth twin as roughly 4 weeks older than himself.
Notice an odd effect here. At the end of the acceleration phase, both of them see their twin brother as then being older than themselves! The Earth twin sees his brother as then 114 weeks older than himself, while the traveler sees his brother as then being about 4 weeks older than himself.
During the next eight weeks of his time, the Traveler travels at constant velocity, in SR conditions. With a constant velocity, there is no way to claim that one or the other is the one that is actually moving, and so Special Relativity must apply similarly from both viewpoints. So the Traveler sees a similar Time Dilation effect occurring to his brother on Earth as his brother sees of him! During those next 8 weeks of the Traveler's time, he would watch his Earthbound brother only age about two days. They each would see their twin brother aging far slower than themselves. In fact, with the specifications we have made for this trip, they each would see their twin brother aging only around 1/30 as fast as they were aging themselves! The traveler would see the Earth twin age about two days during eight weeks of observations, and the Earth twin would see the Traveler age about eight weeks during 236 weeks of observation. (This is the explanation of how both brothers could see each the other as aging about onethirtieth as fast as themselves, during the entire SR portion of the trip, in complete compliance with what Time Dilation formulas indicate.) Note that they each saw their brother as being older than they were at the start of the cruising (SR) portion of the trip; that at the exact halfway point of the trip, they also both see that they are momentarily the exact same age; and then they both see the other brother as having become younger than they are just prior to the deceleration. During his deceleration, the traveler would again see his Earth brother age just shy of 119 weeks, while he ages 115 weeks. As he measures it, once he is at Alpha Centauri and in a restframe with us, his Earth brother will appear to him to have aged (119weeks + 2 days + 119 weeks) a total of 238 weeks, just like all other perceptions also give.
Note that both of them see a "fast aging" effect during the GR accelerations and the Time Dilation "slowing" during the SR constantvelocity cruising. No one else seems to have ever noticed and described this necessary situation.
Whether it is his actual transit time, the amount of time the Earth twin experiences, or either brother's record of the other's life, 238 weeks would have passed during the entire trip! There is no "Twins Paradox" in the popular sense!
These four different perspectives seem extremely different from each other, but the main fact is that they each represent exactly the same total amount of time for the entire trip, and they also each record the same basic facts.
This explanation fully agrees with all the Special Relativity effects of Time Dilation, as can be seen by either of the two twins during the period of no acceleration. (All other attempts at description can only apply to the view from the Earthbound observer, and they all violate Relativity when the view from the traveler is considered.) However, it requires a change in the assumption regarding what happens during GR, the accelerations and decelerations. It has always been assumed that Time Dilation occurs during GR just like it does during SR. But this reasoning shows that during GR, there must necessarily be an opposite effect from Time Dilation, which I guess could be called Time Speeding! The equations of GR are extremely complex, and mathematically proving that was harder than I had expected! But the solution has been found and confirmed, and the equation which describes this Time Speeding is actually very similar to that of Time Dilation.
An interesting consequence of this is that there must always therefore be an identifiable interrelationship between rate of acceleration, the interval of that acceleration, and the maximum relative velocity, such that an SR effect of Time Dilation is always exactly canceled out by an equal and opposite GR effect of "Time Speeding" during the necessary acceleration and deceleration. A strict mathematical treatment identifies exactly what that relationship between SR velocity and GR acceleration is. g, the gravitational constant, appears to be significant in it, as somehow causing the rate that we see time pass!
It might seem as though such choices might create an immense number of resulting effects, but that is not the case. In fact, regarding the amount of total time for an entire trip, the only factors which are important are the known distance of the trip (such as in light years) and the maximum velocity, at any point along the trip! It turns out that there is no advantage in using the most powerful rocket engine to get the spaceship up to that maximum speed quickly, with the possible expectation that an entire trip at a high speed might require less total time. Some examples below show that is not the case! In fact, the most efficient trip is one with the smallest possible rocket engine where the maximum desired speed is achieved at the exact halfway point, where deceleration is then immediately begun.
Here are some various ways of making a trip to Alpha Centauri. This identifies one of the three required parameters, 4.3 light years as the total distance of the oneway trip. Most of these are practical examples which do not require getting right near the speed of light, and they generally also require acceleration rates that the human body can withstand.
The first set are specifically selected for having a maximum velocity of 0.6 c, as recorded from the Earth. Each of these trips therefore would show the expected Time Dilation during Cruising of a factor of 0.80. These examples show the effect of different rocket engine power and therefore acceleration rates. Note that really powerful rocket engines could be operated for a short period of time, which would then allow cruising at constant velocity for much of the trip. For each situation, we provide a description of the entire trip from first an Earth observer, and then a different description of the entire trip as recorded by the Traveler himself. The time interval of the acceleration, and then the cruising and then the deceleration is different from the two perspectives. So is the distance covered in each segment and the acceleration which would be measured. However, the total trip is always exactly the same as seen by both, regarding the total distance covered and the total time taken for the trip.
Trip to Alpha Centauri, maximum velocity 0.6c
Taking about seven years, with maximum speed just above half the speed of light.
Trip with Minimal Rocket Motor  0.15 g acceleration  
Accelerate  Cruise  Decelerate  Total  

Traveler Lives  1308.8 Days Accel=1.6 m/s^{2} Distance=2.15 ly  0 Days Accel=0.0 m/s^{2} Distance=0.000 ly  1308.8 Days Accel=1.6 m/s^{2} Distance=2.15 ly  2617.6 Days Dist=4.3 ly 
Earth Watches  1308.8 Days Accel=1.6 m/s^{2} Distance=2.15 ly  0 Days Accel=0.0 m/s^{2} Distance=0.000 ly  1308.8 Days Accel=1.6 m/s^{2} Distance=2.15 ly  2617.6 Days Dist=4.3 ly 
Trip with Stronger Rocket Motor  0.5 g acceleration  
Accelerate  Cruise  Decelerate  Total  

Traveler Lives  425.1 Days Accel=4.9 m/s^{2} Distance=0.698 ly  1767.4 Days Accel=0.0 m/s^{2} Distance=2.903 ly  425.1 Days Accel=4.9 m/s^{2} Distance=0.698 ly  2617.6 Days Dist=4.3 ly 
Earth Watches  204.2 Days Accel=10.2 m/s^{2} Distance=0.335 ly  2209.3 Days Accel=0.0 m/s^{2} Distance=3.629 ly  204.2 Days Accel=10.2 m/s^{2} Distance=0.335 ly  2617.6 Days Dist=4.3 ly 
Trip with Powerful Rocket Motor  0.8 g acceleration  
Accelerate  Cruise  Decelerate  Total  

Traveler Lives  262.8 Days Accel=7.9 m/s^{2} Distance=0.432 ly  2092.0 Days Accel=0.0 m/s^{2} Distance=3.437 ly  262.8 Days Accel=7.9 m/s^{2} Distance=0.432 ly  2617.6 Days Dist=4.3 ly 
Earth Watches  1.3 Days Accel=1591 m/s^{2} Distance=0.002 ly  2615.0 Days Accel=0.0 m/s^{2} Distance=4.296 ly  1.3 Days Accel=1591 m/s^{2} Distance=0.002 ly  2617.6 Days Dist=4.3 ly 
Notice that the Earth observer would see apparent accelerations that no human could withstand, but that the Traveler actually experiences very reasonable accelerations. Notice also that the Earth observer would see the entire acceleration occur in just over one day, while the Traveler would actually experience nearly nine months of acceleration.
Trip with Extreme Rocket Motor  0.8 g acceleration  
Accelerate  Cruise  Decelerate  Total  

Traveler Lives  261.7 Days Accel=7.95 m/s^{2} Distance=0.430 ly  2094.0 Days Accel=0.0 m/s^{2} Distance=3.440 ly  261.7 Days Accel=7.95 m/s^{2} Distance=0.430 ly  2617.6 Days Dist=4.3 ly 
Earth Watches  0.01 Day Accel=159100 m/s^{2} Distance=0.000 ly  2617.6 Days Accel=0.0 m/s^{2} Distance=4.300 ly  0.01 Day Accel=159100 m/s^{2} Distance=0.000 ly  2617.6 Days Dist=4.3 ly 
Slow Trip to Alpha Centauri, maximum velocity 0.1c
Forty years, with maximum velocity around 30,000 km/second
Trip with Minimal Rocket Motor  0.04 g acceleration  
Accelerate  Cruise  Decelerate  Total  

Traveler Lives  7853 Days Accel=0.04 m/s^{2} Distance=2.15 ly  0 Days Accel=0.0 m/s^{2} Distance=0.000 ly  7853 Days Accel=0.04 m/s^{2} Distance=2.15 ly  15,706 Days Dist=4.3 ly 
Earth Watches  7853 Days Accel=0.04 m/s^{2} Distance=2.15 ly  0 Days Accel=0.0 m/s^{2} Distance=0.000 ly  7853 Days Accel=0.04 m/s^{2} Distance=2.15 ly  15,706 Days Dist=4.3 ly 
Trip with Stronger Rocket Motor  0.4 g acceleration  
Accelerate  Cruise  Decelerate  Total  

Traveler Lives  81.2 Days Accel=4.3 m/s^{2} Distance=0.022 ly  15,544 Days Accel=0.0 m/s^{2} Distance=4.256 ly  81.2 Days Accel=4.3 m/s^{2} Distance=0.022 ly  15706 Dist=4.3 ly 
Earth Watches  42.0 Days Accel=8.3 m/s^{2} Distance=0.011 ly  15,622 Days Accel=0.0 m/s^{2} Distance=4.277 ly  42.0 Days Accel=8.3 m/s^{2} Distance=0.011 ly  15706 Days Dist=4.3 ly 
Fast Trip to Alpha Centauri, maximum velocity 0.9999c
Taking about 4.3 years, with extreme maximum velocity!
Trip with Minimal Rocket Motor  0.4 g acceleration  
Accelerate  Cruise  Decelerate  Total  

Traveler Lives  785.4 Days Accel=4.4 m/s^{2} Distance=2.15 ly  0 Days Accel=0.0 m/s^{2} Distance=0.000 ly  785.4 Days Accel=4.4 m/s^{2} Distance=2.15 ly  1570.7 Days Dist=4.3 ly 
Earth Watches  785.4 Days Accel=4.4 m/s^{2} Distance=2.15 ly  0 Days Accel=0.0 m/s^{2} Distance=0.000 ly  785.4 Days Accel=4.4 m/s^{2} Distance=2.15 ly  1570.7 Days Dist=4.3 ly 
Trip with Extreme Rocket Motor  0.5 g acceleration  
Accelerate  Cruise  Decelerate  Total  

Traveler Lives  774.3 Days Accel=4.48 m/s^{2} Distance=2.119 ly  22.1 Days Accel=0.0 m/s^{2} Distance=0.061 ly  774.3 Days Accel=4.48 m/s^{2} Distance=2.119 ly  1570.7 Dist=4.3 ly 
Earth Watches  0.01 Day Accel=442000 m/s^{2} Distance=0.000 ly  1570.7 Days Accel=0.0 m/s^{2} Distance=4.300 ly  0.01 Day Accel=442000 m/s^{2} Distance=0.000 ly  1570.7 Dist=4.3 ly 
Notice that in only around 10 minutes of watching, the Earth observer would see two birthday parties!
He would record more of the trip as having occurred during acceleration and deceleration, while the Earth observer would record more of it as having occurred during the cruising phase.
We can also see that he arrives at Alpha Centauri 2617.6 days after leaving Earth, which is easily predicted by using the maximum velocity as though it was always the velocity, or 1567 lightdays (same as 4.3 lightyears) divided by 0.6, getting 2617.6 days for the trip.
There appears to be a "basic acceleration" which is 4.42968 meters/second^{2}, which is somehow the basis for all other accelerations! Motion at that acceleration may somehow cause time to pass at a basic rate! Equivalently, per Einstein, being in a gravitational field which has that value for the gravitational acceleration might somehow provide a basic rate of time passage. The fact that Earth has an acceleration due to gravity of more than double that might mean that time passes for us at a rate different, and calculable!
On planets with stronger or weaker g, the intrinsic rate of time passage might be different than for us! The implications of that seem somewhat frightening! But continuing the theme, it might suggest that SR and GR happen to be "special cases" of a single larger set of equations! This seems to have potential importance regarding Minkowski's spacetime concepts. The traditional Minkowski description is not compatible with the existence of gravitational attractions, and this might enable some future compatibility there. I find that intriguing!
I composed a separate presentation which suggests that NASA send an atomic clock to be placed on the surface of the Moon (or Mars). Atomic clocks have been in orbit many times, but apparently none have ever been placed on the surface of either the Moon or Mars. It seems like a very important experiment to do. Einstein emphasized the equivalency of the acceleration due to gravitation and the acceleration due to rocket propulsion. Therefore, whatever the effect of General Relativity on the rate of passage of time, there should be clear and measurable differences between the surface of the Earth and the surface of the Moon, due to a factor of about six in the surface gravity. In fact, calculations seem to show that the rate of time should be significantly different on the Moon or on Mars than on Earth, by about 1/13 of a second difference per year! It is difficult to comprehend the implications of such findings!
The reason why the theories regarding the effects during General Relativity appear to all fail in having Time Dilation seems to be related to slight flaws in the Metric Theory behind General Relativity. A number of mathematical simplifications and assumptions were applied in order to make the set of ten General Relativity equations more soluble. These simplifications were known then to cause slight approximations in General Relativity concepts. Interestingly, those simplifications were primarily initiated in the early 1960s, virtually concurrently with the rise in popularity of the Twins Paradox story! Many alternative Metrics have been presented since Einstein presented his, and I am unaware if any have ever been carefully examined regarding whether they could have opposite consequences from Time Dilation during General Relativity. It seems like an area worth investigating. I have a feeling that General Relativity will win out, but that some flaw in those assumptions and approximations is responsible for the "inverse conclusion" regarding GR and TD.
I have determined that, for a full trip where the initial and final velocities are identical, the equations used above, for both Special and General Relativity are given by the following equations:
Three basic parameters must be chosen for them:
The equations:
These equations are all simplified versions of the Integral Calculus equations which actually apply. Their forms are obvious, and would be needed for any situations where the rate of acceleration changes during a trip. These simplified forms are given here because most real future space journeys are likely to have constant acceleration during powered portions of a trip.
These analyses are all based on complete trips, where an observer begins with no significant velocity or acceleration from an observed object, then there is a period of acceleration, followed by a period of coasting/cruising, followed by a period of deceleration down to the initial conditions of velocity and acceleration. Most phenomena detected tend to only have had the Special Relativity, constant velocity portion of such movement analyzed. These equations can be applied to a particle that began (as far as we knew) with massive velocity or acceleration, but then the treatment of the analysis has to be somewhat different.
Possibly some of the great difficulties of Physics dealing with Quasars, Pulsars and the like might turn out to be far simpler to resolve. We have generally assumed that if an object sends us the radiation equivalent to 10 million stars, it must be quite huge, but when we see it have brightness variations on the period of months, we see a great dilemma because that implies that it is small. If the actual object was actually only 1/100,000 as bright as we perceive it to be and if the months we see during a variation are actually many years, many of the serious problems of Astrophysics might find some fairly simple resolutions.
But, of course, that would depend on whether GR effects actually apply for Relativistic radial acceleration. It is merely mentioned here as one of many possible implications of this new perspective.
Imagine that there were two planets near opposite sides of the Universe, each headed toward the other at extremely high constant speed. That means that Special Relativity should apply and therefore Time Dilation. Say within two meters per second of the speed of light, where the Time Speeding effect would be more than a thousand trillion to one in both directions. There is an accurate atomic clock on both planets and they each have amazing telescopes to be able to see the clock on the other planet at any time. See the problem? While planet 1 sees exactly one year pass on planet 2, planet 1 actually would experience a thousand trillion years, longer than anything has ever existed. This then requires that planet 1 is immensely old. Now look from planet 2, and see the similar situation, where only one year on planet 1 would pass while planet 2 experiences a thousand trillion years, but we just saw that planet 1 necessarily existed a thousand trillion years, each of which would have to match up with a thousand trillion years on planet 2. As Special Relativity and Time Dilation is currently understood, both planets would have to exist for impossible periods of time. That indicates that the current theory must somehow be wrong. But Time Dilation is considered to be a simple and obvious consequence of Special Relativity.
The explanation of this bizarre situation is equally bizarre! In order that two objects get to a relative constant velocity of just two m/s less than the velocity of light requires the one which had done all that accelerating to have "aged" astoundingly fast, due to the Time Speeding effect of acceleration (in General Relativity). In other words, the one that did the acceleration would have to have already experienced those thousands of trillions of years of acceleration, before the situation that we now are considering. The point being made here is that, since the Universe appears to be about 13 billion years old, neither object had enough time to accelerate to that great a relative constant velocity, which means that the example we have been speculating about could not have been possible! Even though Relativity often seems very peculiar, it still has to comply with the Laws of science!
The equations above make clear that there is an intimate relationship between the Time Dilation of Special Relativity and the Time Speeding of General Relativity. A trip can only make sense once it is completed, that is that the observer and traveler are both again in the same inertial reference frame.
That indicates that acceleration is a necessary precondition for Time Dilation as an earlier Time Speeding due to acceleration, at least for any Static Reference Frame. More, whatever the total cumulative effect of that (previous) Time Speeding might then become available for a later Time Dilation effect being witnessed. Actually, both the Time Speeding of the acceleration and then the later deceleration, must necessarily exactly match the total observed Time Dilation. The final result is that the trip always takes the "correct" amount of total time, from both of their perspectives as well as from the perspective of any other observer of their interactions. No matter who is watching, when they get back to the same Static Reference Frame, they will be exactly the same total age (and will again appear as Twins!)
Conservation of Angular Momentum  An Exception or Violation (Sept 2006)
Galaxy Spiral Arms Stability and Dynamics A purely Newtonian gravitational explanation (Nov 1997, Aug 1998)
Twins Paradox of Relativity Is Absolutely Wrong (research 19972004, published Aug 2004)
Perturbation Theory. Gravitational Theory and Resonance (Aug 2001, Dec 2001)
Origin of the Earth. Planetary Gravitational Resonances (Dec 2001)
Rotation of the Sun (Jan 2000)
Origin of the Universe. Cosmogony  Cosmology (more logical than the Big Bang) (devised 1960, internet 1998)
Time Passes Faster Here on Earth than on the Moon (but only a fraction of a second per year!) (Jan 2009)
Globular Clusters. All Globulars Must Regularly Pass Through the cluttered Galaxy Plane, which would be very disruptive to their pristine form. (Nov 1997, Aug 1998)
Existence of Photons. A Hubble Experiment to Confirm the Existence of Individual Photons (experimental proof of quanta) (Feb 2000)
Origin of the Moon  A New Theory (June 2000)
Planetary Rotation of Jupiter, Saturn, and the Earth (Jupiter has a lot of gaseous turbulence which should have slowed down its rapid rotation over billions of years) (March 1998)
Cepheid Variable Stars. Velocity Graph Analysis (Feb 2003)
Compton Effect of Astrophysics. A Possible New Compton Effect (Mar 2003)
Olbers Paradox Regarding Neutrinos (Oct 2004)
Kepler and Newton. Calculations (2006)
Pulsars. Pulsars May Be Quite Different than we have Assumed (June 2008)
Sun and Stars  How the Sun Works  Nuclear Fusion in Creating Light and Heat (Aug 2006)
Stars  How They Work  Nuclear Fusion. Lives of Stars and You (Aug 2004)
Sundial Time Correction  Equation of Time. Sundial to ClockTime Correction Factor (Jan 2009)
General Relativity  A Moon Experiment to Confirm It. Confirming General Relativity with a simple experiment. (Jan 2009)
General Relativity and Time Dilation. Does Time Dilation Result? (Jan 2009)
Geysers on Io. Source of Driving Energy (June 1998)
Mass Extinction, a New Explanation. A New Explanation for Apparent Periodicity of Mass Extinctions (May 1998, August 2001)
Precession of Gyroscopes and of the Earth. Gyroscope Precession and Precession of the Earth's Equinoxes (Apr 1998)
Ocean Tides  The Physics and Logic. Mathematical Explanation of Tides (Jan 2002)
Earth's Spinning  Perfect Energy Source (1990, Dec. 2009)
Earth's Magnetic Field  Source and Logic. Complex nature of the magnetic field and its source (March 1996)
Earth Spinning Energy  Perfect Energy Source From the Earth's Spinning (1990, Nov. 2002)
Nuclear or Atomic Physics Related Subjects:
Nuclear Physics  Statistical Analysis of Isotope Masses Nuclear Structure. (research 19962003, published Nov 2003)
Quantum Defect is NOT a Mathematical Defect It Can Be Calculated The Quantum Defect is a Physical Quantity and not a Fudge Factor(July 2007)
Atomic Physics  NIST Atomic Ionization Data Patterns Surprising Patterns in the NIST Data Regarding Atomic Ionization (June 2007)
Nuclear Physics  Logical Inconsistencies (August 2007)
Neutrinos  Where Did they all Come From? (August 2004)
Neutrinos  Olbers Paradox Means Neutrinos from Everywhere (Oct 2004)
Quantum Nuclear Physics. A Possible Alternative (Aug 2001, Dec 2001, Jan 2004)
Quantum Physics  Quantum Dynamics. A Potential Improvement (2006)
Quantum Physics is Compatible with the Standard Model (2002, Sept 2006, Oct 2010)
Quantum Dynamics (March 2008)
Ionization Potential  NIST Data Patterns. Surprising patterns among different elements (March 2003)
Mass Defect Chart. (calculation, formula) (research 19962003, published Nov 2003)
Assorted other Physics Subjects:
Precession of Gyroscopes and of the Earth. Gyroscope Precession and Precession of the Earth's Equinoxes (Apr 1998)
Earth's Magnetic Field  Source and Logic. Complex nature of the magnetic field and its source (March 1996)
Earth Spinning Energy  Perfect Energy Source (1990, Nov. 2002)
Earth Energy Flow Rates due to Precessional Effects (63,000 MegaWatts) (Sept 2006)
Accurate Mass of the Earth. Gravitational Constant  An Important Gravitation Experiment. (Feb 2004)
Tornadoes  The Physics of How They Operate, including How they Form. Solar Energy, an Immense Source of Energy, Far Greater than all Fossil Fuels (Feb 2000, Feb 2006, May 2009)
Radiometric Age Dating  Carbon14 Age Determination. Carbon14, C14 (Dec 1998)
Mass Extinction, an Old Explanation. An Old Explanation for Apparent Periodicity of Mass Extinctions (Aug 2003)
Hurricanes, the Physics and Analysis A Credible Approach to Hurricane Reduction (Feb 2001)
Sundial Time Correction  Equation of Time. Sundial to ClockTime Correction Factor (Jan 2009)
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