If you read the whole article, you get the point: this proves more than Einstein's theory - it has implications throughout science.

Most Precise Test Yet Of Einstein's Gravitational Redshift

Airplane and rocket experiments have shown that gravity at altitude makes clocks tick slowler, a central prediction of Albert Einstein's general theory of relativity. A new atom interferometer experiment now measures this slowdown 10,000 times more accurately than before, exactly what Einstein predicted. The result shows once again how well Einstein's theory describes the real world.

That gravity changes the flow of time is fundamental to the theory of general relativity. We often call the effect gravitational redshift because light wave oscillations slow down, become redder, when tugged by gravity.

Matter is both a particle and a wave. The cesium atom can be represented by matter waves oscillating 3 x 10^25 times per second, or 30 million billion billion times a second. When the cesium atom matter wave encounters a carefully tuned laser light flash, quantum mechanics steps in. Each cesium atom enters two alternate realities. In one, the laser has pushed the atom up one-tenth of a millimeter, 4/1000 of an inch, giving it a tiny lift to where Earth's gravitational field is slightly weaker. In the other, the atom remains unmoved inside Earth's gravitational well, where time flies slower.

The cesium matter wave frequency, too high to measure directly, uses interference between alternate reality cesium matter waves to measure the resulting difference between their oscillations, and thus the redshift. General relativity precisely predicted time’s slowing to one part in 100 million, 10,000 times more accurately than measurements made 30 years ago using two hydrogen maser clocks, one on Earth and the other at a height of 10,000 kilometers.

The results have practical implications for Earth's global positioning satellite system, precision timekeeping, and gravitational wave detectors. Our best clocks have 17-digit precision. In global positioning satellites, we could determine location to the millimeter. But lifting a clock by 10,000 times higher, 1 meter, creates a change in the 16th digit. As we make better clocks, we need to understand how gravity influences time better.

Gravity being a manifestation of curved space and time is among the greatest discoveries of humankind. It means that what we think of as the influence of gravity is really following the shortest time-space path. In flat geometry, the quickest route is a straight line. In General Relativity, time flow becomes a function of location, so the shortest path is an elliptical orbit or a plumb line to the ground. Experiments have tested the theory with higher and higher precision, but direct gravitational redshift measurements have struggled with Earth’s tiny gravitational field. Hydrogen maser clock measurements in a 1976 NASA experiment reached a 7 x 10^-5 precision.

Just as an optical interferometer uses interfering light waves to measure time or distance to within to a fraction of a wavelength, an atom interferometer uses interfering matter waves. Because matter waves oscillate at a much higher frequency than light waves, they can measure correspondingly smaller times and distances.

We have cooled and trapped atoms with lasers to build the most precise atom interferometers. An experiment on cesium atoms in free fall precisely measured the acceleration of gravity. Scientists captured a million cesium atoms, trapped them with a laser, chilled them to a few millionths of a degree above absolute zero, and zapped them with a vertical laser beam tuned to give them a kick upwards with 50% probability. An instant later later a second laser pulse sent the high-flying matter waves downward to merge with the stationary ones. A third laser pulse then recombined the two. Measuring the recombined matter wave amplitude revealed the phase difference between the two.

The rest mass contribution to the matter wave oscillation “Compton frequency” is normally ignored because the resulting frequencies are too fast to measure. But in this experiment, it allowed an extremely precise measurement of the different clock rates.

Relativity theory demands that the energy E also includes atom’s rest mass energy given by E = mc^2. An atom’s rest mass is enormous, resulting in an atomic clock that ticks at 3 x 10^25 Hertz. During the 0.3 second freefall, the matter waves on the higher route experience 2 x 10^-20 seconds more time than the lower route. Because of the sheer magnitude of the Compton frequency, they oscillated a million times more often. Since the atom interferometer measures the difference to within a thousandth of an oscillation, the experiment produced a 9-digit accuracy. This corresponds to measuring the time difference to 10^-28 seconds.

In perspective, if freefall time was extended to the age of the universe, 14 billion years, the upper and lower route time difference would be 1/100th of a second. The accuracy of the measurement would be 60 picoseconds, the time it takes light to travel 1/2 inch.

Some day an atom interferometer could measure gravitational redshift with millimeter separation. One future milestone will be a separation of a meter. By separating the atoms by a meter, we could observe the gravity waves generated by interactions between massive stars or black holes. To filter out graitational noise, like a passing truck, such an experiment would involve at least two atom interferometers separated by a large distance. An ideal spot for the experiment would be the Deep Underground Science and Engineering Laboratory at the former Homestake mine in South Dakota.