In the hundred years since Einstein proposed General Relativity, no one has yet actually strictly experimentally proven that he was right (or wrong)! We can do that now!
The universally assumed idea that time is passing at exactly the same rate for everywhere in the Universe is certainly totally wrong! The nearby presence of any large mass, such as the Earth, causes a gravitational field to exist where time passes slightly more quickly! Albert Einstein taught us that, because of his General Relativity, but no one has yet experimentally confirmed that he was right.
Einstein's General Relativity and the Equivalency Principle Factor (of ) tells us the exact rate at which an atomic clock would run in any gravitational field. For example, for an atomic clock at the Earth's North Pole, (c) represents the speed of light, 299,792.458 km/s2; (a) represents the local polar gravitational acceleration of 9.83224 meters/sec2 , and (d) represents the Earth's Polar radius of 6356.7523 km. This simple calculation results in the Equivalency Factor being 1.000 000 000 347 563, which tells us that the clock at the North Pole runs slightly FASTER than it would in deep space (by that factor). (Do the math!) This causes time to pass at a slightly different rate on and near each planet or satellite. It is at an easily measurable fraction of a second per year faster in each case!
Einstein described in his General Relativity Theory that there is a clear and simple relationship between (inertial) mechanical acceleration or (weight) a gravitational field and the rate of passage of time. Unfortunately, he expressed it in an incredibly esoteric set of ten curved-space Tensor Riemannian Calculus simultaneous equations, which no one has yet even completely solved a hundred years later! However, the relationship between Acceleration and Gravitation is considered to be quite simple, given by a single Equivalency Principle Factor Equation (presented above).
There is a new approach which seems to offer a truly strict and accurate basis for the possible confirmation of General Relativity. According to Einstein's General Relativity, the rate at which time passes on the surface of any massive object should be dependent on the size and the mass of that body, per that Equivalency Factor. We should have a rather simple and precise experiment available. Specifically, the rate at which time passes on the surface of the Moon should be clearly different than the rate that we experience here on the surface of the Earth. The proposed experiment is rather simple, of placing a Cesium atomic clock on the surface of the Moon and comparing the time it establishes with an identical Cesium clock here on the surface of the Earth. The Equivalency math indicates that due to the greater mass of the Earth, the Earth clock should "tick" around 10,976 times more every hour than the identical Cesium clock on the Moon. (This would display as an output difference of about 1,194 nano-seconds every hour on the two identical clocks.) If this turns out to be experimentally true, then we will finally have an actual and precise proof that General Relativity is valid!
Various scientists have made weakly supported speculations and then claimed that they had provided some proof regarding General Relativity, but such alleged proof has always been totally dependent on the often questionable validity of the assumptions.
Only three possible methods have traditionally been suggested for confirming Einstein's General Relativity, and experiments on all three include aspects of crudeness where valid questions have been discussed. The most famous is regarding a proposed General Relativistic effect on the perihelion of the orbit of the planet Mercury, where the orbit very gradually changes, by approximately 43 arc-seconds per century of the position of the perihelion of Mercury's orbit. That observation had been detected (by Fourier Analysis) prior to Einstein but Classical Physics had not been adequately able to explain the existence of this (known) effect, and Einstein's General Relativity and his complex Tensor Calculus equations seemed to mathematically explain it. However, modern Theoretical Physicists argue over the fact that an exact match does not seem to exist and that General Relativity seems to only provide an approximate mathematical answer (within about 1.5%).
A second proof of the General Theory of Relativity was established in 1919 and 1922 (and repeatedly afterward) at solar eclipses. Accurate photographs of stars whose light had passed near the Sun were slightly gravitationally affected by the Sun, so they appeared in very slightly different positions in the sky during the Eclipse. Again, there are many experimental complications of such research, and true precision has never yet been achieved. Modern experiments seem to confirm Einstein's prediction to be accurate within about 1% accuracy, but that is still not good enough for Theoretical Physicists!
A third proof of General Relativity involves extremely minimal propagation delays in signals sent between spacecraft on the opposite side of the Sun and us, but again, a variety of experimental complications affect the real precision that Theoretical Physicists look for.
Even after a hundred years of countless Research Grants paying for attempts at proof, the best we have seen so far is around 1% accuracy, which is good, but not sufficient for Theoretical Physicists.
The assorted assumptions which have been made in all three cases even seem to cause an assumed situation where General Relativity and Special Relativity both cause Time Dilation. A careful examination of the precise 18-digit accurate Time Dilation calculations regarding a man standing at the Equator of the Earth and flaws in the popular Twins Paradox story show that is not the case, that General Relativity causes an opposite effect from Time Dilation, which I call Time-Rate Speeding. The proposed experiment here would immediately experimentally establish whichever actually the truth is.
Since 2006, I have tried to get NASA to do a fairly simple and cheap experiment, which would finally accurately prove whether Einstein was right about General Relativity. So far, NASA has shown no interest. Neither has ESA (the European Space Agency). The Chinese Space Agency has not shown any interest yet either. I have not tried to interest the India Space Agency yet. I am hoping that JAXA (Japanese Aerospace) might include such a clock in their planned 2018 launch for a soft-landing on the Moon.
A central basis of General Relativity is the Equivalency Principle. This is a mathematical statement that gravitational mass and force is identical to inertial mass and force. Einstein called that Equivalency. There is a relatively simple equation (above) that calculates a time effect due to either mechanical (inertial) acceleration or acceleration due to a gravitational field.
What I have proposed is to soft-land a Cesium (atomic) clock on the surface of the Moon.
Then a reliable radio communication would be established between such a clock and an identical Cesium clock in a laboratory on the Earth. If Einstein was right about General Relativity then every hour, the Earth Cesium clock should record about 10,976 more "ticks" per hour than the Moon clock should! The common time display on such clocks is in nano-seconds, and this experiment should show a major difference of about 1,194 nano-seconds difference every hour (with the Earth clock always showing the faster time reading. This difference would accumulate every hour and would result in about 1/100 second [actually 0.0105 second] difference per year.)
Here is the factor called the Equivalency Principle Factor in physics.
(c) represents the speed of light, 299,792.458 km/s2; (a) represents either a mechanical (inertial) acceleration or the local acceleration due to a gravitational field at that location; (d) represents a distance, which is necessarily assumed to be the distance from the exact center of the massive body, in that gravitational field (which is the planetary radius at that location.)
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.
In a laboratory at sea level at the North Pole on the surface of the Earth the radius is 6356.7523 km from the Earth's center and the gravitational acceleration there is 9.83224 meters/sec2. This results in the Equivalency Factor being 1.000 000 000 347 709
In a laboratory at sea level at the Equator on the surface of the Earth the radius is 6378.1370 km from the Earth's center and the gravitational acceleration there is 9.780327 meters/sec2. This results in the Equivalency Factor being 1.000 000 000 347 037
This then results in the (average) General Relativity gravitational time effect on the Earth being 1.000 000 000 347 373 (as compared to deep space).
On the surface of the Moon the radius is 1738.78 km from the Moon's center and the gravitational acceleration there is around 1.6231 meters/sec2. This results in the Equivalency Factor on the Moon's surface being 1.000 000 000 015 701
This then results in the General Relativity time effect on the Moon being 1.000 000 000 015 701 (again, as compared to deep space)
The relative General Relativity difference in the rate of time passage on the Earth and Moon is therefore 1.000 000 000 331 672
Since Cesium clocks count about 9,192,631,770 ticks per second, this implies that the two clocks should have clearly different counts, of about 3.0489 ticks every second! In the first hour that the two clocks were communicating, they should have a difference of about 10,976 ticks, a very easy and obvious experimental difference! Every following hour should show the same easily experimentally measured difference.
These two clocks should have a total time difference of about 0.0105-second every year.
The clock on Earth should get ahead of the clock on the Moon by that difference. You should be older than you would have been had you lived your life on the Moon, by part of a second! As an old man now, I am certainly about 0.73 second older having lived my life on Earth than if I had lived in the lesser gravitational field on the Moon.
The exact measured different rate on the Earth and Moon would be different from the 10,976 clicks per hour, depending on the Latitude of the location of the clock on Earth. Our calculation uses an AVERAGE Latitude on Earth where most people live, but if the Earth clock was placed at the North Pole, the experimentally measured difference in clock rates might be slightly greater, at about 11,003 clicks per hour.
The suggested experiment could technically also be performed on Earth, with one of the clocks being at the North Pole and the other at the Equator. This effect is due to the oblate shape of the Earth, where the radius and the gravitational field strength at the Pole and the Equator are different. Per Einstein's General Relativity and the Equivalency Factor calculations above, (and not counting on the significant Special Relativity factor differences at the Pole and Equator) the time-rate difference effect of GR should be
1.000 000 000 347 709 - 1.000 000 000 347 037 or 0.000 000 000 000 672 or about one part in one and a half trillion, a very small effect. These two clocks might click around 0.00618 clicks per second different or about 22.2 clicks per hour different (with the Pole clock being slightly faster). These two clocks might then show a difference of about 2.420 nano-seconds after an hour or 58 nano-seconds per full day.
Unfortunately, there is a significant Special Relativity Time Dilation effect at the Equator due to the 1,674.366 102 km/hr Equatorial rotational velocity (as calculated in the related http://mb-soft.com/public9/dilatio4.doc web-page) and the Lorentz Time Dilation factor of which is then 0.999 999 999 998 796 560 . As the North Pole has no rotational velocity, the SR Time Dilation factor there is exactly 1.000, and the difference in the SR time-rate effects of the two clocks is 0.999 999 999 998 796 560, or about one part in 0.83 trillion, also a very small effect. Due to the SR time rate effect of Lorentz, these two clocks might tick at around 0.01106 ticks per second different or about 39.8 clicks per hour different (with the Pole clock again being slightly faster). The SR time rate difference would then show about 6.750 nano-seconds different after an hour or 162 nano-seconds difference after a full day. So such an experiment, on Earth, would likely show a total clock rate difference of about 62 clicks different every hour. After an hour, the two clocks might show a difference of about 6.75 nano-seconds or 162 nano-seconds different after a full day experiment. That amount would be easily measurable, but as it is a combination of the SR and GR effects, the results would not prove anything. These calculations show that the two clocks might show a difference of 162 nano-seconds after a 24-hour experiment. A far more valuable experiment would be to let these clocks run for 30 days or 365 days, where then the difference shown between the two clocks would be 4,860 nano-seconds or 59,130 nano-seconds. Such longer experiments would be conclusive in identifying if ONLY GR was acting (1,120 nano-seconds or 13,627 nano-seconds) or if ONLY SR was acting (3,117 nano-seconds or 37,927 nano-seconds) difference. In fact, if the general Physics community happened to be right that GR causes Time Dilation, we could see that as well, where the clocks would show a difference after a month or a year would be (1,997 nano-seconds or 24,300 nano-seconds) difference. But people seem to be in too much of a hurry to get quick results to actually leave such clocks running for a month or a year!
You may have noted that if anyone would do ALL the necessary math for both SR and GR, even the CORRECT results of the famous Hafele-Keating experiment might be calculated.
A bunch of (Physicists) (including Stephen Hawking) have made far more simple (and flashy) public demonstrations of such comparison of clocks, where one of them was driven up to be on a mountain for a day or two. Their experiments are never at the Equator or the Pole, so EACH of the effects of SR and the (gravitational field) GR and the (mechanical centripetal acceleration) GR needed to be calculated for useful results. They SHOULD HAVE DONE ALL THREE of the calculations shown above (SR and GR and GR) for both the lower and upper locations of the clocks during their experiments. The higher altitude clock has a longer circumference to travel each day, so (v) is greater and so is the SR time effect. However, the higher altitude also has a reduced local gravitational field strength (a) (per Newton's Gravitational Vector formula) and an increased distance from the center of the Earth (d). These changed values therefore change both the Equivalency factors so the GR time effects must also be calculated. Note that each of the two locations (altitudes) on the mountain also have yet another mathematical effect, that of a separate General Relativity effect due to the MECHANICAL ACCELERATION of the centripetal acceleration being different at the different altitudes. Their experimental results actually also are slightly affected by where the Moon is in the sky! The precise local gravitational field strength is a Vector quantity, and so it is slightly affected by whether the Moon and its gravitational effect is on the horizon or at the zenith. No one seems willing to do all that math. They each then proudly claim that the experimental difference of the two clocks "confirms what they believe" (but in reality, as in our example above, it never actually does, since they never did all the necessary math).
Any reader of this could replicate the math shown above for two different altitudes on a given mountain, possibly also for any difference in all those effects if their two clocks were at different Latitudes during the experiment, as (r) and (a) and (d) might have changed enough to affect the results (due to the oblate shape of the Earth). YOU could supply the correction for any of those Physicists whose mountain results never quite confirmed what they insisted they were confirming! Yes, their two (identical) atomic clocks had slightly different readings after their day or two experiments (they usually show that the clocks are different by around 20 nano-seconds) but they don't seem to realize that their (allegedly careful) experiment had not involved all the math that was necessary. They all seem to downplay the fact that they never seem to claim any specific predicted clock readings, but they merrily claim that ANY difference actually proved something! Even if they claim (incorrectly) that General Relativity causes a Time Dilation effect, they (or YOU) could do that math as well. All I demand is that they do ALL the appropriate math for their mountain experiments, which clearly requires at least SIX calculations (for both gravitational and inertial GR at the top, SR at the top, both gravitational and inertial GR at the bottom, and SR at the bottom). Why don't they follow correct scientific methods? Even someone as respected as Hawking does such sloppy math??? Amazing.
If it were possible for us to have lived on the surface of the Sun, the Time Speeding effect would be around 3,000 times as fast as here, and you would now have become nearly a hour older than you are now on Earth. (and you would have burned feet!)
For the clocks comparison experiment involving the Moon, the communicating of the two clocks (on the Moon and on the Earth) is critical, as the orbit of the Moon is elliptic in a continuously changing manner, and the propagation distance between the two clocks is therefore also constant changing. However, those distance changes are very accurately known. It is also realistic to record each hour of count difference for many hours, and after the orbital radius correction, an extremely accurate value for the value of the Equivalency Principle could be statistically calculated.
The experiment involving the Moon will give far larger clock time reading differences, around 1,194 nano-seconds difference every hour. Very importantly, that experiment exclusively involves only GR effects so the accuracy of the results can be excellent. The Earth Pole-Equator experiment would have less difference, of around 162 nano-seconds every hour, but that experiment necessarily involves effects of both SR and GR, so the precision of the results is not as certain. The (popular) experiments at different heights on a mountain have even far less observed difference of around 1 nano-second per hour (or about the 20 nano-seconds per day that are flaunted to the public, but again, those mountain experiments involve both GR and SR effects and so they have minimal scientific value (unless they were extended to a month or a year duration.)
A more comprehensive presentation on this subject is at http://mb-soft.com/public3/gravit33.html
Carl W. Johnson, Theoretical Physicist, Physics Degree from University of Chicago