# Twins Paradox of Relativity Is Absolutely Wrong

### Physicists neglected the effects of General Relativity, which changes EVERYTHING

Around 1960, some Physicists came up with an idea which got a lot of attention but it was dreadfully wrong for having been based on a wrong assumption. They had (correctly) considered Einstein's Special Relativity regarding a Time Dilation effect, but they (wrongly) neglected Einstein's General Relativity and its (opposite) time-rate effect. The "Twins Paradox" story started out with two identical twins, but then treated them very differently! Einstein had made very clear that since each of the twins in that story was not accelerating (during their story), they must each see the entire Universe from what is called an "Inertial Rest Frame of Reference" (one from here on the non-accelerating Earth and the other from the non-accelerating spacecraft). What they had wrongly overlooked was that before their story began, the traveling twin had to accelerate enormously to change from being stationary having lunch on Earth with his brother to receding from the Earth at a speed comparable to the speed of light. During all that acceleration, Einstein's General Relativity must have a time-rate effect of its own, which was all neglected.

The flaw can be more obvious if we put the traveling Twin onto a giant planet, exactly like our Earth. Still not accelerating, just a whole lot bigger than a tiny spaceship! We are going to call it "Earth2." Actually, WE cannot know for sure whether we are actually on Earth or on Earth2 in that story.

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 non-Accelerating 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 non-accelerating 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 time-rate 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 time-rate 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/second2, such an acceleration would obviously take 15 million seconds (about one-half year of acceleration) before it could get up to even a velocity of half the speed of light (or 1.5 * 108 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 time-rate 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 one-half 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 well-proven 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 time-rate 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 (non-accelerating) 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://mb-soft.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 Hafele-Keating 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 time-rate 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.

### Math Example for anyone on the ISS

For anyone in the ISS (International Space Station) which orbits the Earth, the Time Dilation Effect (which is due to Special Relativity and the high speed with which it is orbiting) can easily be calculated to be a time-rate factor of 0.999 999 999 669 (less than 1.000 and therefore, a time slowing effect).

The (General Relativity) time-rate effect due to the orbital acceleration of the ISS is a time-rate 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 time-rate factors to find the actual Relativistic time-rate 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 time-rate effects exactly cancel each other out (for the ISS), within any conceivable error factor.

### Math Example for a Person Standing at the Equator on Earth

For anyone standing at the Equator on Earth who "orbits" the Earth, the Time Dilation time-rate Effect factor (which is due to Special Relativity and the high speed with which he is "orbiting") can easily be calculated to be a time-rate factor of 0.999 999 999 998 796 560 (again, less than 1.000 and therefore a time-slowing effect.)

The (General Relativity) time-rate effect due to the centripetal acceleration of the person at the Equator is a time-rate factor of 1.000 000 000 001 203 440 (again, more than 1.000 and therefore a time-speeding effect.)

Since both of these Relativistic effects apply continuously, we must multiply the two time-rate factors to find the actual Relativistic time-rate 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 time-rate 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://mb-soft.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 half-way 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 half-way 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 web-page 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 earth-based 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 time-rate effect of Special Relativity and the Relativistic time-effect of General Relativity were calculated to be exactly the same amplitude, but in exactly opposite time-rate directions. That was proof that the two Relativistic time-rate 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 soft-land 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.

Consider two small planets that are moving apart or toward each other at a constant relative velocity, measured by both as being exactly 0.6c. (c represents the speed of light.) There is a baby boy born on each planet, at a moment that we will say is simultaneous, although that is not important here. (We might say that an observer, who is moving at a velocity exactly halfway between their planets' velocities and happened to be exactly equal distance from both planets at that moment, witnessed both births at what he considered the same moment. A critically important factor in this is that the observer was traveling at a velocity that was exactly halfway between.)

When each looks at the other, from their own inertial rest-frame, 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 rest-frame 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 time-rate speeding effect would account for a time-rate speeding effect of 700 to one. That is, the baby who would (naturally) age by 60 years during the apparent one-month 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 'mid-point' 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 non-traveling 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 non-accelerating (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 high-velocity-differential 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 points-of-view.

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 so-called 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 well-known effects of Special Relativity, specifically Time Dilation, is counter-balanced 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.

### Case 1 or 2: Either one accelerates to match the Velocity of the other

Either one of them could unilaterally get in a spacecraft and accelerate to reduce the initial velocity differential of 0.6c to 0.0c. We do not yet know whether either of them actually believes that the other was born before or after they were, only that the neutral observer had that perception. In fact, the fact that they have been traveling at a constant velocity differential pretty much ensures that they each do not think that they were born at the same moment! This situation is asymmetric in that only one of them will accelerate, but it necessarily must include a "fast aging" perception due to the acceleration and deceleration! Neither of them would sense anything peculiar in their own lives. But in fact, it barely matters which of them does the acceleration of the trip, as in both cases, the other will see him as aging more rapidly than he does.

An (incorrect) assumption in the Twins Paradox thinking is that the well-known 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 cosmic-ray created Muons in the Earth's high atmosphere. Those Muons are known to have very accurately known rest-lifetimes (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 non-accelerating 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 non-accelerating observer would have seen him seem to be constantly moving in high-speed 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 non-accelerating 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 non-accelerating 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 constant-velocity (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.

The perspective of the one who makes the trip is similar. He starts out the trip having aged 30, and having watched the other person having aged just 24 years during that early time. In order that his perceptions of the two maintain logic, he must experience a six year trip and see the non-accelerating one apparently age twelve years. Therefore, in his perception, they each aged 36 years during the whole story, his own 30 + 6 years and the non-accelerating one's 24 + 12 years.

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.

The non-accelerating 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 double-speed, 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 non-accelerating 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 non-traveler, 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 non-accelerating, 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 counter-act 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!

## The Remaining Case, where both accelerate to a velocity halfway between that of their two planets

Now, with this example, we have the complication of each of them accelerating themselves and each is seeing that the other is accelerating. It certainly shows that there is a "fast aging" effect.

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.

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 Twins Paradox story only examines only one of their viewpoints on all this, and it makes an incorrect assumption by not considering the other viewpoint. That assumption has always been that GR has a Time Dilation effect. The examination of both viewpoints provides the reality necessary to see that it necessarily has the opposite effect.

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 well-known 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!

The Twins Paradox (or Clock Paradox) was first suggested based on the Time Dilation consequence of Special Relativity. No problem there! During the SR portion of such a trip, the commonly accepted description is correct. It was actually only presented in regarding the Special Relativity part of a trip, for which it is entirely true. The GR portions of the trip where acceleration occurs, was simply ignored! However, when the entire Twins Paradox trip is considered, it is easily seen that the Twins Paradox story and conclusions cannot be true, even though SR and TD are absolutely true!

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 pre-conditions, which required Relativity in order for them to be true. They are:

• (1) that anyone in the Universe who does the same experiment, in an (inertial) non-accelerating environment, must get the identical results of any identical experiment of any other inertial environment;
• (2) that no one could ever experience any discontinuity of time;
• (3) that no one could ever experience any discontinuity of space;
• (4) that no one could ever get the experience of traveling faster than the speed of light; and
• (5) the sum of this is that the Universe must be able to "make logical sense" to any person in any environment.

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 re-meets his twin brother, he is now 8.4 years younger than his twin!

That story seems very believable, mostly because the story makes clear that distance he would see was only 3 light-weeks (the distance shortening effect of SR), and so it would "make sense" to him that he could make the trip in around 3 weeks of time, without violating the limitation of the speed of light. This Twins Paradox is apparently SO convincing that 100% of Physicists simply accept it as absolutely true, and even most school textbooks describe it as a solid fact! But even that seemingly obvious statement that A.C. would appear only 3 light-weeks away, is not correct!

Did you notice the huge flaw in the reasoning?

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 light-weeks) 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 three-week 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!

There are many "explanations" of the Twins Paradox story commonly presented, but they each manage to contain incorrect assumptions! For example, one popular version entirely neglects that there is any acceleration ever necessary, and simply considers the (SR) constant velocity part of the trip. (The traveler gets in the rocket and travels at constant velocity to some destination). If that could actually be true, the Twins Paradox could be true. But the part left out is that he first had to accelerate from being in Earth's inertial frame up to his cruising speed and then back down to the inertial frame at the other end. Those "details" entirely change everything! It is interesting that many such "explanations" confidently state that both brothers could not be experiencing the Special Relativity consequence of Time Dilation, "because of previous accelerations." But they really do, and SR says they have to! It is unbelievable that educated Physicists would claim that previous accelerations somehow alter a current situation! When the Earth was first being formed, if there was some acceleration that happened, do we really need to know about it now to solve the equations of motion? In this "explanation" just mentioned, no acceleration is ever mentioned, which establishes that the twin in the spacecraft can certainly consider himself to be in an inertial rest frame, and therefore he would see Time Dilation occurring to his brother on Earth, which appears to be rapidly receding from him. (The current presentation explains how they really can each be experiencing the SR effects of Time Dilation, watching each other age more slowly than themselves, without there being any actual paradox!)

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 non-relativistic 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.

If the above is not sufficient, a simple variation could be added! Using eyedrops, so the traveling twin would never have to blink, he faces Alpha Centauri from the moment that he leaves his brother. He clearly sees that it is then 4.3 light years away. He never takes his eyes (or instruments) off of A.C., but, somehow, an hour later, he sees it as being very close, only 3 light-weeks away. A fair question would be, did the distance reduce from 4.3 light years to 0.06 light year, instantly, or did it happen gradually over that hour? It obviously had to happen, if the Twins Paradox is then to be able to have the usual story! Every Physicist alive would love to hear how someone could be constantly staring at a distant star, to have it, immediately or quickly, appear extremely close! Relativity cannot explain bizarre things like that!

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!

I have recently been told by a man who informed me that he is one of the foremost experts on Special Relativity, that he sees nothing wrong in what is described in the paragraph above! He even told me that if he measured the distance to A.C. at 4.3 light years, and then an hour (of his experience) later, saw that it was 3 light-weeks away, he would see nothing wrong with that! He then explained to me that I am simply too stupid to understand any of Relativity, rather insulting the University of Chicago which gave me my Degree in Physics! But I guess he is free to have his own opinions, including about my native intelligence! But I wonder if he might someday be walking down the street in Kansas City and suddenly be in front of the Eiffel Tower in Paris! He apparently would see nothing odd in that! By the way, he chose never to identify who he is, only that he is a "foremost expert on Special Relativity"!

There is a resolution of all this, and it definitely includes well-proven Time Dilation during the constant-velocity (Special Relativity) part of the trip. Even more, it shows how, during that SR part of the trip, both brothers would see the other as apparently aging more slowly! (which is the actual main paradox that exists, but it is not actually a paradox at all!)

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 constant-velocity 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 rest-reference 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.

One central reason why such incorrect assumptions have been made regarding the GR aspects of the trip is that the equations of GR are immensely complex. A commonly known anecdote was about the famous British astronomer Sir Arthur Eddington, one of the first to fully understand Einstein's General Relativity theory in detail, who was once asked if it were true that only three people in the world understood general relativity. He is said to have replied, "who is the third?"

### A Detailed Trip

We will consider an alleged Twins Paradox trip that would appear to him to take ten weeks, instead of the three used above. We will examine the trip from all four possible separate perspectives, of the Observed actions of the other person (traveler or Earth-bound) and of their own lives during the trip.

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 light-weeks) 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, constant-velocity 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.

Log Book - Trip from Earth to Alpha Centauri
(His)
Date
Dist.
to go
Dist.
Done
Week
Avg.
Speed
Start on Earth4.30 ly0.00 ly -
Week One3.87 ly0.43 ly23 * c
Week Two3.44 ly0.86 ly23 * c
Week Three3.01 ly1.29 ly23 * c
Week Four2.58 ly1.72 ly23 * c
Week Five2.15 ly2.15 ly23 * c
Week Six1.72 ly2.58 ly23 * c
Week Seven1.29 ly3.01 ly23 * c
Week Eight0.86 ly3.44 ly23 * c
Week Nine0.43 ly3.87 ly23 * c
Week Ten0.00 ly4.30 ly23 * 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,

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 light-weeks 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!

Log Book - Trip from Earth to Alpha Centauri w/ Time Dilation Effects
(His)
Date
Dist.
to go
Dist.
Done
Week
Avg.
Speed
Start on Earth4.30 ly0.00 ly -
Week One
Acceleration
2.23 ly2.07 ly115 * c
Week Two2.21 ly2.09 ly~ c
Week Three2.19 ly2.11 ly~ c
Week Four2.17 ly2.13 ly~ c
Week Five2.15 ly2.15 ly~ c
Week Six2.13 ly2.17 ly ~ c
Week Seven2.11 ly2.19 ly~ c
Week Eight2.09 ly2.21 ly~ c
Week Nine2.07 ly2.23 ly~ c
Week Ten
Deceleration
0.00 ly4.30 ly115 * 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
week) There is no violation of
the speed of light during those
SR weeks. But notice that we
now have really spectacular
problems with his experience
during the acceleration and
deceleration weeks, when he
would see his speed as over
100 times the speed of light!

Notice also that he would not
see A.C. as "a few light weeks
away" during that SR Time
Dilation part of the trip, but
around half as close as when on
Earth. It is another error of
the Twins Paradox. And when he
turned around, he would see the
Earth and Sun around half as far
away as A.C. actually is.

Obviously, something very strange
has to occur during the GR
acceleration and decelerations!

We could now consider a LogBook that was fully compatible with the usual presentation of the Twins Paradox story. We consider (in the traveler's sensation of time) one week of extreme acceleration, followed by eight weeks of travel at constant velocity, and a final one week of extreme deceleration. This results in a LogBook that is different from the example above. Notice that, once the initial acceleration was completed, everything looks fine, where the "observed speed" by the traveler is essentially the speed of light, so there are no violations that seem to exist. Except for the first week. Actually, except for the last week, too, because of the extreme deceleration. As already noted the first week would therefore present a discontinuity of space, violating Relativity. The last week necessarily has the same problem. This results in an actual LogBook as presented below, which includes a correctly described eight weeks of constant velocity, as appearing to require traveling at slightly below the speed of light.

Log Book - The TRUE Trip from Earth to Alpha Centauri w/ Time Dilation Effects
(His)
Date
Dist.
to go
Dist.
Done
Week
Avg.
Speed
Start on Earth4.30 ly0.00 ly -
He records 115 weeks of acceleration
Week 115
Acceleration
2.23 ly2.07 lyrise to
~ c
Week 1162.21 ly2.09 ly~ c
Week 1172.19 ly2.11 ly~ c
Week 1182.17 ly2.13 ly~ c
Week 1192.15 ly2.15 ly~ c
Week 1202.13 ly2.17 ly ~ c
Week 1212.11 ly2.19 ly~ c
Week 1222.09 ly2.21 ly~ c
Week 1232.07 ly2.23 ly~ c
He records 115 weeks of deceleration
Week 238
Deceleration
0.00 ly4.30 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 light-week/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
A.C. as "a few light weeks away"
during that SR Time Dilation part
of the trip, but around half as
close as when on Earth. It is
another error of the Twins Paradox.

### The Traveler's Personal Experience

This LogBook is the actual one that would present his real experiences, all of which comply with Relativity and Special Relativity. This particular trip involved his accelerating at a rate of around 55 g's (528.2 m/sec2) for a period that he carefully measures to be 2.07 years. A complete LogBook of the entire acceleration period is here. At no point does he ever believe that he has exceeded the speed of light, so there is no violation there. He simply experiences his rocket engine constantly and reliably firing and accelerating him at a constant 528.2 m/sec2. He doesn't really have any good way of accurately determining his cumulative velocity. It is important to note that he has to have the sensation of accelerating for over two years (115 weeks), during which HE would see that he traveled nearly half the distance to Alpha Centauri. Then 8 weeks of the well-known Special Relativity constant velocity, where he will see that he travels another few light-weeks closer. And then another 115 weeks of deceleration, during which he would experience traveling nearly the other half of the distance. He would therefore not experience any discontinuities of space or time or any other violations of Relativity. He would record in his LogBook taking 238 total weeks to get the 4.30 light year from Earth to Alpha Centauri.

### View from the Earth Observer

From Earth, the perception would be rather different. If the Earth brother had also kept a LogBook of his observations, it would be as follows: One ferocious week of acceleration would be observed and recorded, during which he would watch his twin brother age by 115 weeks, in just that single week of observation. The traveling twin would now be seen as 114 weeks older than the Earth twin. Then the Earth twin would record watching for the next 236 weeks of constant velocity (SR) travel, during which he would watch his brother only age 8 weeks. (That is the popular part that is so famous!) During this period, the Earth observer would (naturally) age more than four years and would therefore become older than the traveling twin brother, eventually becoming 114 weeks older. And then the Earth brother would observe and record one week of extreme deceleration, during which again he would watch his brother age another 115 weeks. This last causes the traveler's age to catch up to the Earth-bound twin's age, where they are now exactly the same total age.

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 rest-frame 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 one-week-older 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.

### The Earth Observer's Experience

We can easily see that he simply experiences 238 weeks of normal existence while he is doing these observations of his brother in the rocket.

### The Traveler's View of the Earth Twin Brother

We might now consider what the traveling twin would see regarding his brother on Earth. During the very long period of acceleration (115 weeks), he would not notice much different, but the Earth twin would seem to him to be aging very slightly faster than he did. (This is while the Earth twin sees him apparently aging very fast, around 115 times normal aging rate!) They are both seeing their twin brother appear to age faster than they themselves are aging! This is necessarily true, because of the basic premise of Relativity, that neither viewpoint would have any preferential situation, and the consequences of Relativity must apply to both viewpoints.

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 Earth-bound 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 one-thirtieth 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 rest-frame with us, his Earth brother will appear to him to have aged (119-weeks + 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 constant-velocity 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 Earth-bound 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 inter-relationship 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!

There are various ways in which a long space flight might be made. There could be immensely fast acceleration for a brief period and then a long cruising period at constant velocity, and then fast deceleration again. That could be done with many different target top speeds, from nearly the speed of light down to speeds that we are already capable of today. That acceleration could be gradual, spread out over up to half of the entire trip, immediately followed by gradual deceleration.

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 one-way 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.

 Accelerate Cruise Decelerate Total Trip with Minimal Rocket Motor - 0.15 g acceleration Traveler Lives 1308.8 DaysAccel=1.6 m/s2 Distance=2.15 ly 0 DaysAccel=0.0 m/s2 Distance=0.000 ly 1308.8 DaysAccel=-1.6 m/s2Distance=2.15 ly 2617.6 DaysDist=4.3 ly Earth Watches 1308.8 Days Accel=1.6 m/s2Distance=2.15 ly 0 Days Accel=0.0 m/s2Distance=0.000 ly 1308.8 Days Accel=-1.6 m/s2Distance=2.15 ly 2617.6 DaysDist=4.3 ly

 Accelerate Cruise Decelerate Total Trip with Stronger Rocket Motor - 0.5 g acceleration Traveler Lives 425.1 Days Accel=4.9 m/s2Distance=0.698 ly 1767.4 Days Accel=0.0 m/s2Distance=2.903 ly 425.1 DaysAccel=-4.9 m/s2Distance=0.698 ly 2617.6 DaysDist=4.3 ly Earth Watches 204.2 Days Accel=10.2 m/s2Distance=0.335 ly 2209.3 Days Accel=0.0 m/s2Distance=3.629 ly 204.2 Days Accel=-10.2 m/s2Distance=0.335 ly 2617.6 DaysDist=4.3 ly

 Accelerate Cruise Decelerate Total Trip with Powerful Rocket Motor - 0.8 g acceleration Traveler Lives 262.8 DaysAccel=7.9 m/s2 Distance=0.432 ly 2092.0 Days Accel=0.0 m/s2Distance=3.437 ly 262.8 Days Accel=-7.9 m/s2Distance=0.432 ly 2617.6 DaysDist=4.3 ly Earth Watches 1.3 Days Accel=1591 m/s2Distance=0.002 ly 2615.0 Days Accel=0.0 m/s2Distance=4.296 ly 1.3 Days Accel=-1591 m/s2Distance=0.002 ly 2617.6 DaysDist=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.

 Accelerate Cruise Decelerate Total Trip with Extreme Rocket Motor - 0.8 g acceleration Traveler Lives 261.7 DaysAccel=7.95 m/s2 Distance=0.430 ly 2094.0 Days Accel=0.0 m/s2Distance=3.440 ly 261.7 Days Accel=-7.95 m/s2Distance=0.430 ly 2617.6 DaysDist=4.3 ly Earth Watches 0.01 DayAccel=159100 m/s2 Distance=0.000 ly 2617.6 Days Accel=0.0 m/s2Distance=4.300 ly 0.01 Day Accel=-159100 m/s2Distance=0.000 ly 2617.6 DaysDist=4.3 ly

Slow Trip to Alpha Centauri, maximum velocity 0.1c

Forty years, with maximum velocity around 30,000 km/second

 Accelerate Cruise Decelerate Total Trip with Minimal Rocket Motor - 0.04 g acceleration Traveler Lives 7853 Days Accel=0.04 m/s2Distance=2.15 ly 0 Days Accel=0.0 m/s2Distance=0.000 ly 7853 Days Accel=-0.04 m/s2Distance=2.15 ly 15,706 DaysDist=4.3 ly Earth Watches 7853 Days Accel=0.04 m/s2Distance=2.15 ly 0 Days Accel=0.0 m/s2Distance=0.000 ly 7853 Days Accel=-0.04 m/s2Distance=2.15 ly 15,706 DaysDist=4.3 ly

 Accelerate Cruise Decelerate Total Trip with Stronger Rocket Motor - 0.4 g acceleration Traveler Lives 81.2 Days Accel=4.3 m/s2Distance=0.022 ly 15,544 Days Accel=0.0 m/s2Distance=4.256 ly 81.2 Days Accel=-4.3 m/s2Distance=0.022 ly 15706Dist=4.3 ly Earth Watches 42.0 Days Accel=8.3 m/s2Distance=0.011 ly 15,622 Days Accel=0.0 m/s2Distance=4.277 ly 42.0 Days Accel=-8.3 m/s2Distance=0.011 ly 15706 DaysDist=4.3 ly

Fast Trip to Alpha Centauri, maximum velocity 0.9999c

Taking about 4.3 years, with extreme maximum velocity!

 Accelerate Cruise Decelerate Total Trip with Minimal Rocket Motor - 0.4 g acceleration Traveler Lives 785.4 Days Accel=4.4 m/s2Distance=2.15 ly 0 Days Accel=0.0 m/s2Distance=0.000 ly 785.4 Days Accel=-4.4 m/s2Distance=2.15 ly 1570.7 DaysDist=4.3 ly Earth Watches 785.4 Days Accel=4.4 m/s2Distance=2.15 ly 0 Days Accel=0.0 m/s2Distance=0.000 ly 785.4 Days Accel=-4.4 m/s2Distance=2.15 ly 1570.7 DaysDist=4.3 ly

 Accelerate Cruise Decelerate Total Trip with Extreme Rocket Motor - 0.5 g acceleration Traveler Lives 774.3 Days Accel=4.48 m/s2Distance=2.119 ly 22.1 Days Accel=0.0 m/s2Distance=0.061 ly 774.3 Days Accel=-4.48 m/s2Distance=2.119 ly 1570.7Dist=4.3 ly Earth Watches 0.01 Day Accel=442000 m/s2Distance=0.000 ly 1570.7 Days Accel=0.0 m/s2Distance=4.300 ly 0.01 Day Accel=-442000 m/s2Distance=0.000 ly 1570.7Dist=4.3 ly

Notice that in only around 10 minutes of watching, the Earth observer would see two birthday parties!

We can see during the usual long Cruising phase, that we on Earth would see him appear to age noticeably slower than we were (sometimes more than a year gain!) That is simply the effect of the Time Dilation effect of Special Relativity, and the fact that our likely designed Cruising is at 0.6 of the speed of light, which gives a Time Dilation factor of 0.8. During both the acceleration and deceleration, we would see him appear to age faster by around half a year each. The result is that the total number of days that the Earth sees the trip take (such as 2618, or around 7 years) is exactly the same as the number of days that he personally experienced and recorded on board the spaceship. This is true since Alpha Centauri and its planet are not moving at any significant velocity from us on Earth.

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 light-days (same as 4.3 light-years) divided by 0.6, getting 2617.6 days for the trip.

The implications of all this are even more interesting! It means that the rate of passage of time, as we understand it, may be unique to us on Earth!

There appears to be a "basic acceleration" which is 4.42968 meters/second2, 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 space-time 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:

• trip is the actual distance of the trip, in light years
• vmm is the maximum velocity during the entire trip, in meters/second
• proportion is the portion of the trip which is during acceleration

The equations:

• Earth seen acceleration time: trip * 365.25 / vmm * proportion / 2
• Earth seen distance during acceleration: trip * proportion / 2
• Earth seen acceleration measured: (vmm2 * c) / (trip * 365.25 * 86400 * proportion / 2)
• Earth seen cruise time: trip * 365.25 / vmm * (1 - proportion)
• Earth seen cruise distance: trip * (1 - proportion)
• Earth seen cruise (maximum) velocity: vmm
• Traveler measured acceleration time: trip * 365.25 / vmm * (1 - (1 - proportion) * SQR(1 - vmm ^ 2)) / 2
• Traveler measured acceleration distance: (trip * (1 - (1 - proportion) * SQR(1 - vmm ^ 2))) / 2
• Traveler measured acceleration rate: (vmm * c) / (trip * 365.25 * 86400 / vmm * ((1 - (1 - proportion) * SQR(1 - vmm ^ 2)) / 2)
• Traveler measured cruise time: trip * 365.25 / vmm * (1 - proportion) * SQR(1 - vmm ^ 2)
• Traveler measured cruise distance: (trip * (1 - proportion)) * SQR(1 - vmm ^ 2)
• Traveler measured cruise (maximum) velocity: vmm
• Total Trip time: trip * 365.25 / vmm
• Total trip distance: trip

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.

### Reasons for the Previous Errors regarding GR and Time Dilation

Human experience does not involve any situations where General Relativity has noticeable time rate effects, and rarely even has any interactions with phenomena that have Special Relativity Time Dilation effects. So it might have been reasonable to assume things which were not true. Specifically, we see things like muons in the high Earth atmosphere which we see arrive at surface laboratories while we know that their rest-lifetime is so short that they could not travel even a kilometer before disintegrating. It might be that the muon experiences extremely severe Time Speeding effects as it is accelerated by the cosmic ray which created it in the atmosphere, where it "aged" by extreme amounts, prior to our awareness. So the General Relativity effect of Time Speeding might never have come to our attention. Many other phenomena that we are aware of are similar to that in that we do not have any awareness of much of what had happened before the particle decays and shows the effects of Time Dilation.

### Implications

The implications of this might be huge in Astrophysics. Consider that this presentation started off with a non-accelerating observer possibly aging one hour of his life, as he watched the accelerating one appear to age 12 years. Such an example represents a perceived time rate about 100,000 times faster than he was personally experiencing. This would only occur under the conditions of General Relativity, where there was acceleration involved. We saw that in the most extreme (last) of the calculation examples given, where the Traveler lived 774.3 days while the Earth observer would have only aged around ten minutes. That is a factor of more than 80,000 as a time factor. You might also notice that the situation presented there was not even a very extreme one. In a situation discussed below of material which is falling into an immensely strong gravity well, the acceleration would constantly increase, and the Time Speeding factor would continue to rise at immensely quick rates.

### Pulsars and other very rapidly perceived phenomena

It is not absolutely clear whether the GR effects apply when the Relativistic acceleration is a (radial) central acceleration. But if they do, that means the same effects would occur when we on Earth observe rapidly (Relativistic) rotating objects in space. Say we look at a (rapidly rotating) pulsar, or a quasar or an accretion disk. We see that it seems to be rotating one hundred times per second. And so Physicists devise all kinds of peculiar ideas and speculations regarding how to explain this ultra-rapid rotation. However, if this "TC" effect applies for circular motion and central Relativistic acceleration, it seems credible that the reality is that the object may be actually be rotating only 1/100,000 as fast as it appears to us! A rotation of once per thousand seconds is still really fast, but it then does not require all the exotic logic currently being applied to try to explain lightning fast rotation!

### Black Holes, Accretion Disks, etc

The reasoning presented here might even show that such a concept as a black hole might not even be very possible! The general assumption is that an object which would fall into a black hole would accelerate due to the pull of gravity, and have ever increasing acceleration. Fine, that means that GR would apply. Consider now the perception of time! Shortly above, we discussed how a non-accelerating observer might see an accelerating space traveler appear to age 12 years during a single hour of observing. Now consider that the acceleration associated with a black hole would ever increase. An hour of observing might quickly include a thousand years, or a million years or a billion years of actual time experienced by the object that was being accelerated. The point being made here is that IF there are actual black holes, the forever accelerating situation that we assume might mean that the object has been actually falling in for millions or billions of years, not yet to have even reached the actual location of the black hole itself! Yes, we might see (in our non-accelerating rest-frame) something appear to fall inward to an unseen destination very quickly, but the reality might be extremely different than we think we see!

### Perceived Brightness

Note also that there would have been 12 years worth of radiated energy that would have arrived here in a period of one hour! This suggests that the object would appear to us to be 100,000 times brighter than it actually is.

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.

Relativity has many aspects which are hard to understand or to see where logic exists. I see one that seems especially troublesome.

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 pre-condition 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!)

This presentation was first placed on the Internet in August 2004.

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Carl Johnson
Theoretical Physicist,
Degree in Physics, University of Chicago, '67

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