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To be more specific the depth GMm/r of a stationary-object's potential-energy well on earth is (dt/dτ-1)mc2, in much the same way that a moving object's kinetic-energy K far away from earth is (dt/dτ-1)mc2. In both cases here dt/dτ is the rate of far-time t's passage (well away from our planet) per unit proper-time τ on the clock of the object. To see where this connection between energy and time comes from, take a look at the first figure here on the traveler-kinematic in the curved spacetime of a Schwarzschild metric.
The practical everyday fact that time passes more quickly in global positioning system (GPS) satellite orbits, due to their added height, than time passes here on earth is more than a curiosity. That's because this differential-aging effect measures the very feature of spacetime-curvature which is responsible for the potential-energy differences that Newton ascribed to a gravitational force. In other words, the potential energy it takes to lift a massive object here on earth is linked directly to the increase in time's rate of passage with height.
As shown in the figure, the stationary (i.e. zero proper-velocity) Lorentz-factor in the far-time Schwarzschild metric, namely dt/dτ = 1/Sqrt[1-2GM/(c2r)] ≈ 1+(1/c2)GM/r, is directly linked to the gravitational potential-energy per unit mass (GM/r) that we need to escape the planet every day. In particular, near a planet's surface the differences in time's passage at two different heights is directly linked to the gravitational potential energy difference mgh between those heights, i.e. dtlo/dthi ≈ 1-mgh/(mc2) = 1-gh/c2.
As discussed later, such differential-aging effects also occur for accelerated-frames in flat space-time.
Thus when a dropped object accelerates downward it is confirmation that objects closer to the earth's center are aging more slowly than objects further away. This differential-aging must be taken directly into account when estimating your location with a global positioning system, because GPS satellites operate at altitudes where the effect of gravitational time-dilation is larger than the time-dilation associated with satellite motion.
Example Problem 4.1.2a: If you spend one quarter of your first decade standing up as a 1.8 meter tall adult, and very little of that time standing on your head to cancel the effect, about how much older is your head than your feet?
Example Problem 4.1.2b: If the official formation-date for the earth was 4.5 billion years ago, how much older is the earth's surface, than its center, today?
Note that these problems involve small differences between relatively large numbers, so they might have to be done with very high precision calculations. For example, clocks 1.8 meters above the earth's surface only pick up about 1.96 additional attoseconds per second of elapsed time at the earth's surface.
The moving Schwarzschild Lorentz-factor dt/dτ = Sqrt[1+(w/c)2]/Sqrt[1-2GM/(c2r)], which is a product of stationary and motion-related terms, can of course also be used by students to examine time-dilation effects associated with satellite orbits as illustrated in the 2nd figure of this section. One must of course consider time at two radii (e.g. on earth and in orbit) instead of Schwarzschild far or bookkeeper-time t per se, in which case one finds that low-earth orbit effects are dominated by satellite motion while high earth-orbit effects are dominated by gravitational time-dilation, as shown in the 2nd figure on this page.
Related Reference/Links
Another way to see gravity's link to time is in a youtube video inspired by this spacetime stretcher.
Epstein, Lewis Carroll (1895/1994). Relativity Visualized, (Insight Press, San Fransisco), ch. 10,11,12.
Rickard M. Jonsson (2001) Visualizing curved spacetime (Licentiate Thesis, Chalmers University of Technology and Goteborg University, Sweden) pdf, contains derviations behind the x-cτ embedments used in the visualizations above.
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