There is surprising misunderstanding regarding the exact correct truth and logic of what Time Dilation is. Or, more specifically, of what Special Relativity is. I am a Theoretical Physicist who has spent much of my career studying subjects associated with Einstein's Relativity. Two basic facts are clear and unavoidable. First, the person who Observes any of the events of Relativity does not and cannot Experience any of them. That includes Time Dilation and an assortment of speculated issues such as "time travel" and "wormholes" and many other personal assumptions. The ONLY effects which can be detected is for that (Earth) Observer to look (through a telescope) at someone or something which is either wildly accelerating (GR) or traveling at extreme velocity toward or away from the observer (SR). Oddly, even then, Second, the person being Observed does not sense anything unusual! He lives a very mundane life, where he thinks time is moving at the same rate that he has always experienced during his life.
So neither the Observer nor the person or event being observed actually experiences any alteration of the rate of time passing (or a variety of other things like the apparent inertial mass that an observed person or object seems to have or distances or velocities which objects seem to move at). Even though the Observer SEES an effect such as Time Dilation, it is actually a "perceptional" phenomenon and not anything which is actually "real". People are surprised that this is equally true for BOTH of the scientists who are observing each other.
The following discussion and graphics is a decently clear example of the real situation, essentially directly from Albert Einstein's thought experiments. The following animated graphics present two separate views. First is a view of Earth here. An Earth scientist has an accurate school-style wall clock and a very powerful telescope, which he carefully aims exactly at a distant planet, Miri, which he has measured to be receding away from Earth at extreme velocity, that of 0.6 of the speed of light. Neither the Earth nor Miri is accelerating, so there is only a constant velocity detected between them. After we have examined the situation for the Earth scientist (and the Earth in general), we will travel to the observed planet Miri to see what everything looks for from there.
In both cases, we simplify things by focusing entirely on images of the two wall clocks which the scientists have in their Labs, one in a Lab here on Earth (which is a school-style clock) and the other in a distant Lab on a planet we are calling Miri (which has a wooden frame of its wall clock). These two planets and Laboratories are not accelerating. On Earth, we see Miri receding away from us at a tremendous velocity (at 0.6 of the speed of light).
My ultimate intention is to present a clearer presentation of Special
Relativity "for non-Physicists".
Looking at the Earth laboratory first, we see the prominent Earth wall clock at the top that is always timed exactly to GMT. Much of the rest of the image is of a giant telescope, with its eyepiece image of a similar wooden frame wall clock on the other (receding) planet's wall, which we see as moving away from us at 0.6c. The Time Dilation effect we see of all of Miri therefore appears to have their time passing slowly, running at 0.8 the actual GMT time-rate he observes in his Laboratory here on Earth, due to Special Relativity and its Time Dilation. EVERYTHING we observe through our telescope is really convincing that their clock is running only 4/5 as fast as our clocks run. It is less easy to confirm but the Siri scientist also APPEARS to be aging at 4/5 the rate that everything on Earth ages.
The ACTUAL TRUE time we ever detect and experience on Earth is always exactly what we are used to, GMT, given by our school-style wall clock here on Earth. Only the image of Miri seen in the eyepiece of the telescope has the appearance of having time running much slower due to Time Dilation.
This situation is similar to what many people expect of Time Dilation and Special Relativity. A critical difference is that NOTHING in our Earth Laboratory gives any appearance that any time-rate is or could be any different from what we expect. The Earth scientist cannot "alter time" in any way, and his life is destined to continue as he had always experienced.
Now we look at the image of what the Miri scientist sees in his Lab. As on Earth, there is a large (wooden frame) wall clock in his laboratory, timed exactly to his own planetary time. The other portion of the Miri Laboratory image is the view of his giant telescope, and a scientist looking at its eyepiece. He has his telescope aimed exactly at Earth and so he sees in his eyepiece an image of the Earth school-style wall clock on the wall of the Earth scientist we had just examined.
This is very important. The Miri scientist sees and has measured that the Earth is receding from him at very high (constant) velocity of 0.6c. He also has NO sensation that he is moving at all. As a result, he sees the image of the Earth clock appear to running at 4/5 the actual rate of his Miri wall clock, again due to Special Relativity and its Time Dilation. Actually, it is experimentally harder to confirm but he can also clearly see that the Earth scientist is aging exactly 4/5 as fast as he is aging.
Actually, per Einstein, no scientist could ever detect that he was ON Earth or on Miri, as everything would seem exactly the same to both of them.
The ACTUAL TRUE time that the Miri scientist ever detects is what he is used to, forever given by his reliable wooden frame wall clock of his Miri Lab. Only the image he sees (of Earth and its clock) through the telescope's eyepiece has the APPEARANCE of running much slower due to Time Dilation.
NO acceleration is present anywhere so BOTH Laboratories are certain that their own planet is motionless (what Physicists call an Inertial Rest Frame of Reference and that their own wall clock is perfectly reliable and accurate in their own labs). However they EACH see the "other" wall clock (as seen through their telescope's eyepiece) to be running clearly slowly and that other scientist as aging only 4/5 as fast as himself.
The point is that the "Time Dilation effect" is not actually a physically true phenomenon, but each clearly sees the "moving clock in the telescope eyepiece" appear to run at 0.8 times the rate of their own clock.
This is definitely obvious to witness, for both. However, regarding any "physical sensation in the other Laboratory", no, there would be none.
The distant Miri observer might THINK that you are aging at 4/5 the rate which he is aging at due to his calculation of Time Dilation, but YOU know that your birthdays are exactly as far apart as you have always celebrated. You also THINK that he is aging 4/5 as fast as you do. These are OBSERVED perceptions and they are NOT ACTUALLY REAL.
The same situation has always been true of humans here and decaying Muons high in our atmosphere. By around 1941, in our laboratories here on the surface of the Earth, we were accurately measuring the average lifetime of a Muon, 0.000 001 52 second. Even if they moved at nearly the speed of light, they must decay within about 0.28 mile or so of where they were created. However, research labs on the surface of the Earth, many miles below, easily detect those Muons which they know are created by Cosmic Rays when they crash into atomic nuclei high in our atmosphere (often 50 miles high). The only possible explanation for this is due to the Time Dilation effect of Special Relativity. This was actually the first strict experimental confirmation of Einstein's Special Relativity. For the Muons, their enormous (constant) velocity (around 0.954c) causes (Time Dilation) time to appear 'normal' to them but they see US to APPEAR to age only around 1/100th as fast for the Muons (themselves), much more slowly. In their very slow-moving environment, they see it as easy to travel the entire 50 miles down to a laboratory on Earth, and it SEEMS very logical to them because they appear to see the entire atmosphere to only be a fraction of a mile thick. It is absolutely logical to an Muon scientist that they make it to a lab on Earth. We Earth scientists see those Muons (and their tiny Muon scientists) to APPEAR to be aging only about 1/100th of their actual ages, and so those Muons actually have plenty of (earth-) time to travel the distance of many miles down to the labs.
To the Muons, everything is pretty expected but our atmosphere looks like
it is only less than a half-mile thick. The distance they need to travel
to get all the way down through it to Laboratories on the ground is only
1/100 as far, in their perspective, therefore less than 0.28
mile thick, and so it is totally logical for some of the Muons to be able
to get all the way to the surface, without disobeying the speed of light
being the fastest possible. There have been countless other experimental
confirmations of the Time Dilation effects of Special Relativity.
A tiny Muon scientist sees our atmosphere as only less than half a mile thick, so he sees no logical problem in getting all the way down through what we call 50 miles. But through HIS own (Muon) "telescope" he sees really, really slow moving Earth scientists moving one-hundredth as fast, and so everything makes perfect sense to everyone.
Therefore, no one senses any quirky effects, just stuff that is "normal". Whether it is an Earth scientist or a Miri scientist or a Muon scientist, time passes exactly as it is expected, but when they look at the situation in the different (non-accelerating) location, the effects of Time Dilation are clearly observed. The time-dilation effect is OBSERVED by BOTH SIDES, but neither side EXPERIENCES anything weird.
The graphics above are meant to show that "life is perfectly normal" for each of the scientists, but BOTH see the other as being in a peculiar environment, where time seems to run slowly, that (radial) distances seem to be shorter, that the other appears to age more slowly, where masses seem to be greater. The important point is that these observations are entirely appearances and BOTH see the other clock as running clearly slowly, while his own clock is behaving exactly as he expects.
All those odd Relativity effects make it so that either scientist can do or witness any experiment that the other scientist might do, and logic is never violated, as Einstein had assured us. They each could watch the other shine a flashlight "directly ahead" and everything would be fine. The "speed of light" of that flashlight beam still turns out to ALWAYS be EXACTLY c to each of them. Speed still adds like we all know it does. For one of the observers, the speed he SEES is not "c + c" or 2c, but rather two "Time Dilated" numbers which total exactly c. The other scientist, who actually did the experiment, simply sees that c is the speed of light. For an observer, the speed of a bullet or an accelerating rocket is also just the sum of two "Time Dilated" velocities which results in exactly the correct logical bullet or rocket speed. To that (Relativistic) Observer, the bullet or rocket APPEARS to accelerate more slowly due to it having greater inertial mass (to him), and also because the DISTANCE being seen as being covered is LESS due to the same Dilation formula (again, to him).
A critical detail for Special Relativity is that both the OBSERVER and that person or event being observed is in what Physicists call an Inertial Rest Frame of Reference, which means that NO ACCELERATION is occurring. On Earth, we only have two locations available which precisely comply with that, the North Pole and the South Pole. So we take a scientist, give him a heavy coat, and stand him at the North Pole. We give him a really good telescope and a wall clock and a precise Cesium clock, too.
We have found him a planet, Miri, any distance away from us which we have found to be traveling exactly radially away from us, at exactly 0.6c, an enormous speed. Either that planet is not spinning at all, or if it is, a scientist has been sent so he is standing exactly at its North Pole. He also has a heavy coat and a really good telescope and a wall clock and a precise Cesium clock.
The Earth scientist and the Miri scientist are each looking through their really good telescopes toward each other. Their telescopes are far better than any telescopes we have ever made and they can see each other smiling and waving at each other. Even the clocks on their walls are visible through each of their telescopes (as the graphics above have shown)
Note that the Miri scientist does not feel any movement or acceleration (any more than we do on Earth) and he sees the Earth moving radially away from him at exactly 0.6c.
We will now visit the scientist at our North Pole. His clock and his Cesium clock are always exactly synchronized. But when he looks at the Miri clock he sees through the telescope, he sees that wall clock appears to be running much more slowly. (The Cesium clock there also exactly is synchronized with THAT wall clock, which WE see as running much more slowly than our clocks here on Earth.
An observer who is not moving at all, such as at the North Pole, would see such a rapidly moving (at 0.6c) (receding) man appear to have time pass much more slowly. Hendrik Lorentz and George FitzGerald discovered the well-known formula which is the Time Dilation time-rate factor ▀ (which is ) ) that gives us 0.800 (which is less than 1.000 so it is an observed time-slowing effect) as the Time Dilation time-rate (slowing) factor which you (at the North Pole) OBSERVE for that person rapidly receding due to Special Relativity due to recessional SPEED (not "experienced" by him but only as seen by you, a motionless observer at the North Pole.)
We will now visit the scientist on that other planet, Miri. His clock and his Cesium clock are always exactly synchronized. But when he looks at the clock he sees (on Earth) through his telescope, he sees that Earth wall clock appear to be running much more slowly. (The Cesium clock there (on Earth) also exactly is synchronized with that Earth wall clock, which HE sees as running much more slowly than his clocks on his planet.Einstein made clear that EITHER one of our non-accelerating scientists has a perfectly logical perspective and neither is "better".
This conversation so far has entirely been about Special Relativity, mostly because the math is decently simple. It turns out that similar reasoning can be done for our two scientists if they were "Relatively Accelerating" from each other, but then General Relativity would need to be examined and the math involved needs to be the math of "curved space", that is Riemannian Non-Euclidean math and space and geometry.
For that same man, we can also calculate the precise time-rate effect of Einstein's General Relativity, also to better than eighteen-digit precision. This results in the Relativistic Equivalency Factor ( ) being 1.000 000 000 001 203 440. That is then the General Relativity time-rate (speeding) factor on the person standing at the Equator being 1.000 000 000 001 203 440 (which is more than 1.000 so it is a time- speeding effect) (due to acceleration).
General Relativity has exactly the opposite time-rate effect from what all Physicists believe to be true. It does not have a "time slowing effect" of Time Dilation, but instead has a "time speeding effect" which totally changes everything in many fields of modern Physics. A very precise math proof follows. All "time travel" speculations are certainly impossible. The popular Twins Paradox story is also impossible. Many entire fields of modern Physics are based on this wrong assumption. NASA has sadly tried many experiments which ignored General Relativity which turned out to be total failures, including the rather famous Hafele-Keating experiment in October 1971. The famous Kip Thorne believes that speculative "wormholes" exist. The famous Stephen Hawking spent half an hour in his recent "Genius" TV series in trying to prove a wrong assumption of his about Cesium clocks on a mountain. Time passes on the surface of the Moon slightly more slowly than it does here on Earth. (A Moon clock would run about one ninetieth of a second slower per year than an identical clock here on Earth). The math to prove this is quite simple, and you can even confirm that this is the truth. The following math is indisputable and precise.
We live our lives on the surface of the Earth, where we constantly daily rotate at around 1000 mph, so Einstein's Special Relativity clearly applies to us. We also "ride in a daily curved circle" (around the Earth) in that same process, which means that we also constantly accelerate (radially downward), so that Einstein's General Relativity also applies to us. (We call it centripetal acceleration!) (The Handbook of Chemistry and Physics provides the precise centripetal acceleration value for the Equator.) These effects are both easy to calculate and it turns out that mathematically their time-rate consequences are exactly opposite each other! One (SR) is a Dilation (slowing) of apparent time-rate while the other (GR) is a Speeding of apparent time-rate. They always exactly cancel each other's net effects out for us! There is and can be no net Relativistic time-rate effect on us! That is equally true for people in their homes, for Astronauts who orbit the Earth in the ISS, and even in airliners which circle the Earth. The precise math proof of this new perspective follows.
Please note one very important detail which no one seems to realize. In Relativity, the person or event being watched does not actually change in any way. No "time travel" ever occurs to him or is even possible. No goofy experiences ever occur in his life. Only the (usually distant) observer sees any appearance where the rate of time passage seems to have changed.
We place you at the North Pole, where
you are neither moving nor accelerating due to the daily spinning of the
Earth. You will be our "stationary observer" for the following analysis of the
movements of a man who is standing at the Equator of the Earth.
The factors regarding the time-rate effects of both Special and General
Relativity are extremely minimal for us on the Earth, but they are easy to
calculate. To calculate the Time Dilation
Effect of Special Relativity: A person standing at the Equator
"orbits" at precisely 1,674.366 102 km/hr, "orbiting" with
the surface of the Earth every sidereal day.
(Equatorial circumference divided by the seconds in a sidereal day, both precisely
known measurements of the Earth). This is the same as 0.465 101 695
km/sec (v). We know that the speed of light (c) is 299,792.458 km/sec.
Hendrik Lorentz and George FitzGerald discovered the well-known formula which
is the Time Dilation time-rate factor ▀ (which is
that gives us 0.999 999 999 998 796 560 (which is less than 1.000 so it is a time-slowing effect ) as the Time Dilation time-rate (slowing) factor which you (at the North Pole) see for that person standing at the Equator due to Special Relativity due to the rotational speed (not "experienced" by him at the Equator but only as seen by you, a motionless observer).
For the time-rate Effect of General Relativity, for that person at the Equator of the Earth we have a 6,378,137 meter Equatorial Earth radius (d) and his radially downward centripetal acceleration (a) there is 0.033 915 785 m/sec2. This results in the Equivalency Factor ( ) being 1.000 000 000 001 203 440 , which is then the General Relativity time-rate (speeding) factor on the person standing at the Equator being 1.000 000 000 001 203 440 (which is more than 1.000 so it is a time-speeding effect) (due to acceleration).
Since both Relativistic effects constantly exist, we must multiply these two factors to determine the net Relativistic effect. That is: SR Time Dilation factor of 0.999 999 999 998 796 560 times the GR factor of 1.000 000 000 001 203 440 which results in the net Relativistic time-rate effect being exactly 1.000 000 000 000 000 000 (within experimental error). This is a precise mathematical proof that the two Relativistic time-rate effects exactly cancel each other out (for you viewing the person at the Equator).
A dreadful logical blunder occurred in Physics in the early 1960s and it was never fixed. Even NASA believed that an orbiting satellite or even a conventional airliner circling the Earth, only experienced the effect of Einstein's Special Relativity and Time Dilation. That was and is wrong. Well, misleading, because NASA did not seem to realize that both Special Relativity and General Relativity always constantly apply at the same time. NASA had neglected to consider the effects of General Relativity. NASA had (wrongly) assumed that we on Earth are in what we Physicists call an Inertial Rest Frame of Reference (that is, there is no acceleration acting on us.) It turns out easy to mathematically prove that, for an object that is circling a massive planet like Earth, as in the precise math example above, both effects exist, and that they exactly cancel each other out! One is due to the speed of the motion (SR) and the other is due to the acceleration of the motion (GR), as Einstein had clearly explained.
During the 1960s, NASA included Cesium clocks on a number of earth satellites, to try to prove that their assumption about Time Dilation applied, but all those experiments dreadfully failed. NASA even performed a rather famous (but wrong) experiment in October 1971 to try to prove that (wrong assumption) that only Time Dilation was acting, by sending sets of four identical Cesium clocks both ways around the Earth in conventional airliners, in the Hafele-Keating experiment. That experiment wound up with results which were worthless, well within the experimental margin of error.
A popular claim is that if identical twin brothers were born on Earth, and one went to the ISS (International Space Station) for ten years, the brother who remained on Earth would be able to detect a slightly slower aging of his brother on the ISS, due to Time Dilation. The calculated effect of Einstein's Time Dilation would only be a fraction of one second difference in age, but using identical Cesium (atomic) clocks, that would be easy to detect. That claim is wrongly based on the Earth twin being in an Inertial Rest Frame of Reference that was not accelerating, such that only Time Dilation would apply. That claim was also wrongly based on the ISS twin not being in such an Inertial Rest Frame of Reference.
We on Earth (incorrectly) think we are in such a non-accelerating Inertial Rest Frame of Reference! (Of course, for thousands of years we also thought we lived on a flat Earth that was not moving at all!) However, we each revolve around the Earth once every day, at high speed in a curved circular path, which means we are also each constantly accelerating (radially downward) in a circle. And so NASA and everyone else (incorrectly) assumed that such satellite and airliner Time Dilation effects would be experimentally detected.
Here is the problem!
The passengers on the ISS also think they are in a non-accelerating Inertial Rest Frame of Reference, since they also do not sense any acceleration. If we make a slight adjustment to that story of twins, we can see an obvious demonstration of the logical blunder they had done (and which everyone still accepts as correct!) Consider that a family had lived on the ISS and they had twins, and one of those sons decided to take a rocket ship trip to the surface of the Earth (for ten years). The "non-traveling" brother who stayed on the ISS would see that his brother (on Earth) was living more slowly than he was living, due to Time Dilation, which he could prove by a comparison of cesium clocks. If we looked at everything from the perspective of the ISS rather than the Earth, and the ISS twin brother compared his Cesium clock with one he could watch on Earth, Einstein's Time Dilation would (seem to) require that he would also see that the Earth clock was running slower than his own Cesium clock was running.
Because they both (incorrectly) believed they were in non-accelerating Inertial Rest Frames, this must be true, that both brothers would measure that the other one was aging more slowly than himself!
This would be an easy experiment to do, since there are lots of Cesium clocks in Earth laboratories and also several on board the ISS. And it would obviously fail. Only after each of them correctly considered his own acceleration and therefore that General Relativity was acting, would they then resolve the flawed situation.
When the traveling brother got back home, neither would be younger or older than the other! This is true for either version of the story, whether the family lived on Earth or on the ISS.
This situation is vaguely similar to the popular Twins Paradox story, which also happens to be totally wrong due to neglecting the effects of General Relativity, but for slightly different reasons, discussed below.
The logical flaw here is that both clocks and brothers are constantly and continuously accelerating, in the process of the curved paths of circling the center of the Earth.
In an accelerating (non-inertial) Rest Frame of Reference, Einstein showed that a different set of circumstances must exist, that of General Relativity. Physics even has a formula called the Equivalency Principle, which can be used to calculate the Time-rate effects of any acceleration (or 'equivalently' the effect of any gravitational field such as due to the Earth).
We know the orbital speed and altitude of the ISS, and we can easily calculate the strength of the Earth's gravitational field at that orbital distance. The Time Dilation effect due to the orbital speed of the ISS is a factor of 0.999 999 999 669 (calculated below, which is only a difference of less than one part per billion of time). The time-rate effect due to the orbital acceleration of the ISS is 1.000 000 000 330 (also calculated below, which is also a difference of less than one part in a billion of time, but in the opposite direction!) We must multiply these two numbers to account for the net Relativistic effects of both speed and acceleration, and the product is amazingly close to being exactly 1.000 000 000 (actually, 0.999 999 999 998 999 999).
In other words, for an orbiting ISS satellite, the net effect of the speed and acceleration is such that there is no Relativistic time change effect! In other words, NASA in the 1960s and still, is dead wrong about expecting to see Time Dilation effects in satellites or airliners, due to a logical failure of not including the (really obvious) effects of the General Relativity of the orbital acceleration!
There are even still people today (2016) who aggressively claim that the 1971 experiment should have worked and so they present endless claims about why it didn't. But they do not seem to realize that the experiment failed because it was based on faulty logic, that of ignoring General Relativity.
Up above, we also calculated the Precise Time Dilation Effect and the Equivalency Factor for us who are orbiting on the surface of the Earth as it rotates, and that also shows that the two effects exactly counteract each other and there is no net Relativistic time difference effect due to the Earth spinning.
This mathematically proves two interesting new insights! First, that the time effect of acceleration is opposite that of speed, creating a Time Speeding General Relativity effect which opposes the known Time Dilation effect of speed and Special Relativity. Second, for any satellite in a perfectly circular orbit around a symmetric planet (or star), the time altering effects of the two different Relativitys of speed and acceleration exactly cancel each other out!
It is sad that, today, 2015 and 2016, famous Physicists still get this wrong, and still it is because they totally ignore the consequences of General Relativity. A currently popular showboating move is to take two Cesium clocks, which are lavishly bragged about as incredibly precise and reliable, and first synchronize them. Then one of those atomic clocks is driven up a mountain, and the (allegedly smart) commentator describes how a Relativity time-rate effect causes one of the clocks to "run slower" than the other due to the difference in altitude. Even the "infallible" Stephen Hawking demonstrates sadly poor logic in promoting this claim. A recent TV series by Hawking (Genius) spent half an hour of valuable TV time in presenting this faulty logic and conclusions! There are a bunch of logical errors there! At a location which is a mile higher on a mountain, yes, the gravitational field of the Earth is a tiny bit weaker, which would have a General Relativity time-rate effect, but the person and clock is also circling the center of the Earth FASTER due to having to travel a longer circumference each day, which has a Time Dilation effect due to the higher speed. Hawking (and the others who have presented such public spectacles) never mention that there are actually two Relativistic time effects going on, due to both General and Special Relativity. Since they never even mention the fact that two effects are happening, no one ever does the math (presented here) to show that the two Relativistic time-rate effects happen to exactly counteract each other! I expect a lot more of Hawking. More than that, ALL modern Physicists (including Hawking) have never yet solved Einstein's original set of ten Tensor Calculus Differential Equations to discover that the time-rate effects of General and Special Relativity happen to be opposite. They all (wrongly) assume they are the same, all Time Dilation effects. Why doesn't someone at least try to do the math? And so the "major conclusion" that Hawking presented in his mountain experiment is simply not true! He did a one-day experiment and his results were that the two (identical) Cesium clocks were then 20 nano-seconds different where he claimed that "the results confirmed the assumption he had started out having made." It is a sad aspect of modern science that researchers first make an assumption and then (selectively) do some experiment which seems to confirm that they are right!
By the way, the mountain experiments of Hawking and others are essentially just variants on a theme of the failed 1971 Hafele-Keating experiment in the jet airliners. But at least then, four Cesium clocks were used instead of one. A valid such mountain experiment should run for a month or a year rather than just a single day. And it should be repeated by others who do not carry the same pre-conceived assumptions.
Since 2006, I have tried to get NASA or the ESA to include a Cesium clock in a soft-landing on the Moon. In 2018, the Japan Space Agency (JAXA) intends to launch another soft-lander. Such a simple experiment, where identical Cesium clocks on the Earth and Moon might be continuously compared, will prove once and for all whether Einstein was right about General Relativity. My calculations, based on Einstein, indicate that the Earth clock should count about 10,976 times more ticks every hour than the identical clock on the Moon, entirely due to the difference in our gravitational fields on the surface of the Earth and Moon, and on Einstein's General Relativity and the Equivalency Principle. Such a simple and inexpensive experiment would instantly confirm that Einstein was right and it would also demonstrate the logical flaws in the popular assumptions regarding Time Dilation, Hafele-Keating, the Twins Paradox, alleged Time Travel, and other fields of modern Physics.
Physicists use a fairly simple equation to calculate the Time Dilation effect that occurs during Special Relativity, that is, when there is no acceleration but a relatively high constant velocity. This can be due to either the observed light source moving toward the observer or away from the observer (the Time Dilation effect is the same, either way).
Here is the Time Dilation formula factor for time-rate: ▀ (which is ) This is the multiplying factor of the rate of time that appears to be passing to an observer. (c) represents the speed of light and (v) represents the velocity. If v is in the range of any velocity that we humans are capable of creating, this factor is extremely minimal. This factor (called ▀) was discovered by Hendrik Lorentz and George FitzGerald around 130 years ago.
Consider the Earth satellite called the ISS (International Space Station) orbiting at 27,743.8 kilometers per hour, orbiting the Earth every hour and a half. This is 7.7066 kilometers per second. In this case, we would have v = 7.7066 and c = 299,792.5 kilometers per second. ▀ is then 0.999 999 999 669, which gives us the Time Dilation proportion effect for the ISS satellite due to Special Relativity due to the orbital speed. This is assuming a perfectly circular orbit such that the speed stayed exactly constant, and an Earth that was perfectly spherical, none of which is quite precisely true.
So for an observer on the Earth's surface the satellite might seem to show time passing slightly more slowly than we would measure with our Cesium (atomic) clocks in a lab on Earth. NASA tried to do this experiment during the 1960s by putting an identical Cesium clock inside an orbiting satellite, and compared the time with that of their "Standard" Cesium clock in Washington, DC. Those experiments were all failures, where they always showed greater variation between the various Cesium clocks than the expected Dilation effect!
You might notice that the apparent difference is really minimal! In an entire year, the two clocks (in the ISS as compared to an atomic clock in a Lab on the surface of the Earth) would only have become different by about 1/100 of a second.
The Hafele-Keating experiment, which had Cesium clocks inside conventional airliners, actually had slower speeds (around 900 k/h) than our example of a person standing at the Equator, so we will pass on adding those calculations here, although any reader could easily do the math, to see why that NASA experiment failed so dreadfully.
Such experiments did seem to detect slight differences in the rate of time of the Cesium clocks, but the expected Dilation differences were so small that they never exceeded experimental errors! There were two reasons for that. One is that even though Cesium clocks are extremely accurate and reliable, trying to consistently find a difference of less than one part per billion of time is near the limit to the capability of such atomic clocks. áThe other reason was that NASA seemed to be ignorant of the continuous effects of General Relativity.
In October 1971, NASA installed sets of four Cesium clocks in two passenger airplanes, and sent one of the airplanes westward around the Earth while the other airplane was sent eastward. This experiment was considered very significant, and the Hafele-Keating experiment is still cited as being something important. However, that experiment was really just a sad joke, demonstrating amazing ignorance! The most prominent actual experimental result might have been the significant variation between the set of four allegedly identical Cesium clocks on each aircraft! Given the speed of passenger airliners, the (expected) effect of Time Dilation would have only been around 1/50,000 of a second in a year. And yet the experimental results of the Hafele-Keating experiment showed experimental errors between the individual Cesium clocks which were far greater than what they hoped to find. The 1971 NASA experiment was a total failure!
The other reason is actually far more important. In order for a satellite (or an airliner) to orbit, it must constantly radially accelerate due to the pull of Earth's gravity, to create the curved circular orbital path. You do this when you try to drive around a tight circle, and you feel an acceleration which seems to be pushing you sideways in the seat. We call it centripetal acceleration. Einstein had showed that an entirely different set of Relativity equations are necessary for when accelerations are present, which he called General Relativity. The Special Relativity equations are mathematically simple. The General Relativity equations are horrifically complicated. Einstein presented a set of ten complex Tensor Calculus Differential equations, which must be simultaneously solved to get an answer. So far, in a hundred years of countless mathematicians and physicists trying to solve those Tensor Calculus equations, no one yet has solved them!
Einstein died in 1955. Around 1960, various groups of physicists decided that they were free to apply speculative assumptions to simplify Einstein's equations, and then they could solve their simplified problem. Unfortunately, the various groups of mathematicians and physicists had made different assumptions (none of which were solidly based on logic!) and so they each got different answers! There is a strong likelihood that ALL of those resulting answers are wrong!
We want to go back to that orbiting ISS satellite and its Cesium clock, to now try to examine the time-rate effects of the centripetal orbital acceleration, which is the time-rate effect of General Relativity.
Here is the factor called the Equivalency Principle in physics. (c) represents the speed of light, 299,792.458 km/s2. (a) represents the local acceleration due to a gravitational field or other source of acceleration. And (d) represents a distance, which is the orbital radius distance in that gravitational field.
Note that the Equivalency Principle is a dimensionless parameter, [m/s2] * [m] / [m2/s2]
It is also true that both the acceleration and the distance happen to be Vector quantities, but both are always along the exact same direction, and so their Vector Product is the same as for Scalar numbers, so their Vector nature is irrelevant here.
The ISS satellite is at 342.2 km altitude plus 6378 km Earth radius or 6720.2 km from the Earth's center and the gravitational acceleration it is subjected to is around 8.827 meters/sec2. This results in the Equivalency Factor for the ISS being 1 + 0.000 000 000 330
This then results in the (perceived) General Relativity time-rate effect on the ISS (as seen from the Earth's surface as) being 1.000 000 000 330 (this is more than 1.0000 so therefore it is a time-rate speeding effect).
I find it fascinating that for the ISS in a circular orbit, the Time Dilation factor for the ISS due to the speed is 0.999 999 999 669, while the Time Speeding factor due to the acceleration, per General Relativity is 1.000 000 000 330
These two time-rate Relativistic effects are both constantly in effect so the two time-rate factors have to be multiplied and they have opposite time-rate effects to each other, where their product is incredibly close to experimentally being exactly 1.000 000 000 net time-rate effect!
Please note that this is a mathematical proof that the two Relativistic time-rate effects exactly cancel each other out (for the person on the ISS), within any conceivable error factor.
We might try to claim that the passengers on the ISS do not detect any acceleration and so we could claim that they were in an Inertial Rest Frame of Reference.
Since the ISS passengers do not feel any acceleration, and they therefore feel that they are in an Inertial Rest Frame of Reference, what do they see when they look down at Earth? They would see the surface of the Earth constantly zooming past their (apparently, to their, stationary) satellite. By all the laws of science, they would see an apparent Time Dilation effect where everyone on Earth would seem to be living and moving slower than they would in their spacecraft!
The Earthbound scientists and their Cesium clocks would detect the apparent Time Dilation time-rate effect on the ISS satellite and its occupants and its clock.
Simultaneously, the scientists on board the ISS and their Cesium clock would detect the apparent Time Dilation time-rate effect on the Earth's surface and its billions of occupants and its Cesium clocks!
If there were not any acceleration due to the orbiting or the rotation of the Earth then, yes, they actually could both see the others as apparently living more slowly than they live! That sounds impossible but it is certainly possible. Special Relativity and its effects are only for straight line flight, no acceleration, where the duration of an encounter would necessarily be rather brief.
For a satellite, which is orbiting and therefore constantly accelerating, that is not possible. In fact, this reasoning proves that the time effects of General Relativity is opposite that of Special Relativity, and the calculations done above for a man standing at the Equator and for a satellite even show that the two opposing time effects exactly cancel each other out. Because of the two time-rate effects, and the fact that they are opposite, the satellite and its passengers could orbit for as long as desired without becoming older or younger than "twin brothers left on Earth!"
That is, for the ISS, the SR Dilation time-rate effect of speed of 0.999 999 999 669 times the GR Speeding time-rate effect of acceleration of 1.000 000 000 330 is exactly unity!
We already showed the math that for a person at the Equator of the Earth, the SR Dilation time-rate effect of speed of 0.999 999 999 998 796 560 times the GR Speeding time-rate effect of acceleration of 1.000 000 000 001 203 440 is exactly 1.000 000 000 000 000 000 within experimental measurement error!
NASA has sent the ISS into a higher orbit recently, so the numbers cited here from NASA are now out-of-date.
Reference was made to the Twins Paradox story of around 1962, which is similar to the example mentioned above. The Twins Paradox is (wrongly) similar in that it assumes no acceleration and therefore, no General Relativity, in order for the Time Dilation effect to seem to be accurate and correct. There were two huge logical errors in the setup of that story. The really obvious one is that they only considered their story from the rest frame of a non-accelerating Earth. The Earth brother would see the spaceship brother appear to be aging more slowly than him. But they overlooked an important fact that Einstein made clear, that anyone in an inertial rest-frame can equally look at the Universe! They overlooked that everything can be looked at from the rest frame of the non-accelerating spaceship. Per Einstein's Special Relativity and Time Dilation, he would certainly see the earthbound brother appear to be aging more slowly than himself. Both of these situations would be occurring simultaneously! They both would see their twin brother as aging more slowly than him! Relativity has some peculiar aspects to it, and this is one of them!
The second huge logical error is that the Twins Paradox story totally ignores the fact that there has to be massive acceleration and deceleration occurring during much, most of such a trip. Given any acceleration rate that humans could survive, the process of getting up to any velocity close to the speed of light, takes a long time, many months, and then many months of deceleration to slow down to an Inertial Rest Frame of Reference on any destination planet. The analysis above shows that the effect of all that acceleration is substantial and it is opposite the Time Dilation effect which does famously appear to occur during the "coasting" (Special Relativity) part of the trip. A very thorough examination of all the true effects show that the Time Speeding effects during the acceleration and deceleration exactly cancel out the Time Dilation effects of the well known Time Dilation time-rate effect during the coasting phase. Any such entire trip (beginning from and ending in the same Inertial Rest Frame of Reference) necessarily has no time paradox at all! A thorough presentation of that analysis is at http://mb-soft.com/public2/twinspar.html The Twins Paradox is Wrong