NASA intentionally neglected General Relativity when they decided that they were going to try to prove Special Relativity with their Hafele-Keating experiment in October 1971. By ignoring the effect of General Relativity, NASA guaranteed that that experiment would be a dismal failure, and it was.
The results of that experiment showed that the "error factors" regarding the sets of four "identical Cesium clocks" were much greater than the Time Dilation effect which they expected to prove! The Hafele-Keating experiment was massively promoted to the public by NASA, but it was scientifically a total failure as an experiment. Sadly, some people today (2020) still brag about the Hafele-Keating experiment as though it had been a success, but any analysis of the (still existing) NASA data shows that it was instead a dismal failure as an experiment (even though it seemed to the public like it had been impressive, with their sets of four [assumed] identical Cesium clocks in each of two airliners which flew around the entire Earth in opposite directions!) NASA had simply not tried to do an accurate enough experiment! You will see the math below that their Lorentz equation calculation of the Time Dilation they had expected to detect could only have been one part in a trillion at best. Even if their airliners had continued to repeatedly fly around the Earth for an entire year, that could still not have resulted in more than 1/30,000 of a second difference, even if they had correctly also factored in the apparent time-rate effect of the acceleration of General Relativity (as calculated below).
Around 1880, math was developed by Lorentz and FitzGerald which later became the math basis for the Special Relativity effect of Time Dilation. Unfortunately, little in our Universe is experimentallly known really precisely. The size (radius) (r) of any planet or satellite is generally not known very well, so our knowledge of the circumference (2 π*r) is usually not precise, although we often know the rotation rate (t) of many objects quite accurately. Therefore, our knowledge of the rotational velocity (v) is often not known very precisely, with a single exception. That Exception is the Earth. We know the Equatorial Radius of the Earth extremely accurately, to around one meter precision, of the 6,378.137▒ kilometer radius (r) (or one part in around 6,000,000). We also know the length of our Sidereal Day (t) to impressively great precision, 86,164.0905 seconds, around one-ten-thousandth of a second accuracy. Therefore, we know the velocity of the rotation of the Earth's equator (v) (circumference divided by Sidereal day length) at precisely 1,674.366 102 km/hr, which is 0.465 101 695 km/sec, far more accurately than any other measurement in outer space. We see that the Lorentz factor (▀) or the apparent Time Dilation factor of Special Relativity only depends on that velocity (v) and the speed of light (c), 299,792.458 km/sec, which was defined as exactly this number. Therefore, we can use the Lorentz formula to calculate the apparent Time Dilation Factor (of Special Relativity) for someone standing at the surface of the Equator of the Earth to impressive 18-digit precision. It dawned on me that the Equator of the Earth is the only location in the Universe where we can calculate that Time Dilation Factor so accurately that gives us ▀ = 0.999 999 999 998 796 560 .
I was impressed by the amazing precision which we can be assured of for the apparent Time Dilation factor, for that single location of the Equator of the Earth. At first, I did not see any reason for wanting to know this number so precisely, except for being a Physicist who obsesses on such things. All I saw was that the apparent Time Dilation effect of Special Relativity is amazingly tiny, only a difference of about one part in a trillion in the 31 million seconds of a year, or about 1/30,000 second per year difference for a guy standing at the Equator of the Earth.
You can duplicate these calculations. The speed of light (c), is 299,792.458 km/sec. The Equatorial radius of the Earth is 6,378.137▒ 0.001 km (d). The downward Centripetal Acceleration at the Equator of the Earth (a) given in the Handbook of Chemistry and Physics is given as 0.033 915 785 m/sec2The resulting Equivalency formula regarding the man at the Equator circling the Earth every day gives us an Apparent Relativistic Equivalency Factor of 1.000 000 000 001 203 440 .
At first, I did not realize the importance of the relationship between those two amazingly precise numbers. I knew that they both referred to an apparent tiny difference in apparent time-rate. I did notice that they each referred to an apparent time-rate difference of about one part in a trillion per year. I was first mystified that they seemed to be in opposite directions! One seemed to be an apparent Time Dilation effect (SR) and the other seemed to be an apparent opposite effect of some sort of "time speeding" (GR).
I eventually realized that we are forever circling the Earth in our daily path, which means that we are forever subject to the SR effect of our velocity, and we are also forever subject to the GR effect of our downward (centripetal) acceleration. So I realized that I could not avoid having to consider both of the Relativistic time-rate effects, so I had to multiply the two time-rate numbers mentioned above, to know the total apparent Relativistic time-rate effect on that man standing at the Equator.
I was rather shocked to see the product of those two apparent time-rate factors to be 1.000 000 000 000 000 000 000 000 . In other words, the two Relativistic Apparent effects amazingly exactly cancel each other out. Special Relativity would cause the Appearance of a tiny difference of time-rate which we would call Time Dilation, but we can never see it because it is invariably always exactly canceled out by an exact opposite apparent time-rate effect of General Relativity.
The Equivalency Factor used above (and shown here again) is commonly used in Relativistic Physics but it is very, very slightly incorrect. The a * d factors used in the Equivalency factor can be exactly replaced by v2, by a standard Newtonian motion formula v2 = a * d. There is still a factor of 1/2 that then exists, but when a number is very close to one, the square root of that number is then very close to 1 + 1/2 of that number. This following formula is actually slightly more precise, but for our man at the Equator of the Earth, the improvement in accuracy is only in about the twentieth digit, and everyone has ignored it. Please note that this (precise) equation for the Equivalency Factor looks exactly like the Lorentz factor used above in the Special Relativity calculation except that a + sign exists where a - sign does there. This comment actually clears up why the product of the two given above is only precise to about 20 digits, because the (correct) square root is used in one and the 1/2 approximation is used in the other.
As a result of this, using the Lorentz Equation and Equivalence Principle formula used above for the man circling the Earth at the Equator every day, the numbers which apply for our orbiting the Sun are far less precise. The apparent time-rate effect given by the Lorentz equation is 0.999 999 995 . And the apparent time-rate effect given by the Equivalency formula is 1.000 000 004 95. Both of these effects are much greater than the effects of the Earth daily spinning, since we travel around the Sun much faster than we spin on our axis. They each would result in an effect of about 1/6 of a second per year. However, like for our spinning on our axis every day, both the orbital SR and GR time-rate effects constantly and continuously affect us, so we have to multiply the two values just given, which again results in a net apparent Relativistic time-rate effect of 1.000 000 000 000 000 0, as they also cancel each other out for us.
We do not have as accurate data as we have for the radius of the Earth, but the same results apply, that, for us on Earth, the two Relativistic time-rate effects exactly cancel each other out for us.
The first experimental proof of time dilation was that laboratories on the surface of the Earth, many miles below, were detecting those Muons! In 1941, the Rossi-Hall experiment first demonstrated the situation, followed by many more. That should have been impossible!
A Muon which was created maybe 50 miles high in the atmosphere, was known to not be able to even go half a mile before decaying. So there was no chance whatever that any Muon could possibly get down to Earth-based labs to be detected. Time dilation was the only possible explanation! Here is the scientific explanation of that experiment. From OUR human point-of-view, the Muon's velocity was sooo fast (0.9954c) that its apparent Time Dilation rate of time passage was far slower than ours, around 1/100 as fast, where it was able to make that far longer distance trip before decaying. From the Muon's point-of-view, the scientific explanation was different, but again due to the extremely fast differential velocity, the apparent thickness of the Earth's atmosphere was less than half a mile, so there is no problem of getting all the way through it before decaying (even though time seems to pass at normal rate for the Muon!) We know that the 50-mile-thick atmosphere did not shrink to half a mile thick, but the appearance was due to the apparent Time Dilation as seen by the Muon, it appears to. Time Dilation is a consequence of great constant velocity.
An incredible important "detail" has always been absolutely
overlooked by scientists! The science should be corrected now.
Slightly before that trip by the Muon,
a Cosmic Ray accelerated the relatively stationary Muon particle
in our upper atmosphere up to very near the speed of light (measured
to be 0.995c to 0.9954c). That means that,
for an extremely brief moment, the Muon experienced fantastic
General Relativity, which has an opposing apparent time-rate effect on the Muon.
In fact, when the Muon crashed to Earth in the laboratory, it again
has to experience fantastic acceleration (actually, extreme deceleration)
which causes bremsstrahlung radiation in
stopping. Yes, in our reference frame, we see the Muon appear to live
a hundred times longer during the constant-velocity trip down to Earth,
but we do not have any equipment which is capable of detecting the
apparent time-rate effect of the intense acceleration, that is due to General Relativity.
No scientist in a hundred years has recognized this fact! Every one of
those Muons experienced an entire trip which included General
Relativity, then Special Relativity, and then General Relativity.
The total time involved for the entire Muon trip, acceleration due to the Cosmic Ray creating the Muon, then the trip down through the atmosphere to the Earth-based laboratory and then sudden deceleration and the bremsstrahlung radiation, is the exact same total time, whether seen by the Muon or by an Earth scientist. As we showed in our calculations for the Earth Equator man, the acceleration and deceleration portions of the trip cause opposite apparent time-rate effects from the much more easily observed apparent (time dilation) time-rate changes in the Muon trip down through our atmosphere.
Only a single location in the Universe exists where we have truly precise data where we can calculate Einstein's Special Relativity and its apparent Time Dilation factor to eighteen-digit accuracy. It is for a man standing at the Equator of the Earth, who is rotating with our planet.
For true precision, we need to observe this from an Inertial Rest Frame of Reference in order to be able to use Euclidean Geometry (also called Plane Geometry). Our observer cannot be accelerating, such as when standing at the North Pole. Due to the apparent Time Dilation Effect of Special Relativity, this observer would see such a rapidly moving Equator man appear to have time pass slightly slower. 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. (That is less than 1.000 so it is an observed time-slowing effect) as the apparent Time Dilation time-rate factor which you (at the North Pole) see occurring for that person standing at the Equator due to Special Relativity due to the rotational speed. (It is not "experienced" by the guy at the Equator but it is only as seen by you, a motionless observer at the North Pole). We know this so precisely because we accurately know the size of the Earth and its rotational speed (the sidereal length of day).
For that same man at the Equator, 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 Apparent Relativistic Equivalency Factor ( ) being 1.000 000 000 001 203 440. That is then the apparent General Relativity time-rate factor as observed from the North Pole of the person standing at the Equator being 1.000 000 000 001 203 440. (That is more than 1.000 so it is an observed time-speeding effect) (this is due to the Equator man's centripetal radially downward acceleration).
The Special and General Relativistic time-rate factors always both constantly apply and so we must multiply these two time-rate factors to get the net time-rate effect of Relativity on that Equator man. The result is 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 at the North Pole viewing the person at the Equator). (The guy at the Equator does not actually experience or detect either of the two Relativity effects on him).
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. (Due to the different strengths of the gravitational fields of the Earth and Moon, 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 this 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. He lives a mundane life. Only the (usually distant) observer sees any appearance where the rate of time passage seems to have changed. (He also observes some other apparent effects, such as an apparent change in inertial mass and in radial distances, also by the same ▀ factor. None of these phenomena is actually sensed by the moving person).
We place you to be standing at the North Pole, where
you are neither moving nor accelerating due to the daily spinning of the
Earth. You will be our "stationary, Inertial 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 length of 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 apparent 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 an observed time-slowing effect ) as the apparent Time Dilation time-rate (slowing) factor which you (at the North Pole) observe 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 apparent 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 . That is then the apparent General Relativity time-rate factor (as observed from the North Pole) on the person standing at the Equator being 1.000 000 000 001 203 440 . (That is more than 1.000 so it is an apparent time-speeding effect) (due to acceleration) (due to the radially downward centripetal acceleration of the Equator man).
Since both Relativistic effects constantly exist, we must multiply these two factors to determine the net apparent Relativistic time effect. That is: SR apparent Time Dilation factor of 0.999 999 999 998 796 560 times the GR apparent time-rate factor of 1.000 000 000 001 203 440. That results in the net apparent 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, and ignored all effects of General Relativity. 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 totally 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 any 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 radial 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 an apparent Time Speeding General Relativity effect which opposes the known apparent 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, 2018, 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 to the surface of the Moon. 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 his Equivalency Principle and therefore General Relativity. My calculations, based on Einstein, indicate that the Earth clock should count about 10,976 extra 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 his Equivalency Principle of gravitation and acceleration. 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, the Hafele-Keating experiment, the Twins Paradox, alleged Time Travel, wormholes, 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