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Will a person on top of hill will age faster than one at sea level due to Time Dilation?

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  • $\begingroup$ That is correct. You just want a confirmation, or is there anything in gravitational time dilation that you find puzzling? $\endgroup$ – Johannes Aug 4 '13 at 5:18
  • $\begingroup$ @Johannes But the differnce would be too minute...right? $\endgroup$ – Sid Aug 4 '13 at 5:23
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    $\begingroup$ Faster is the crux here and how to measure it. worldlifeexpectancy.com/longevity-hot-spots . Two of these hot spots are on the sea level. $\endgroup$ – anna v Aug 4 '13 at 5:24
  • $\begingroup$ Similar to question physics.stackexchange.com/q/53889/353 $\endgroup$ – DarenW Aug 4 '13 at 5:49
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It is indeed the case that due to gravitational time dilation a person on top of a hill would age slightly faster than a person at sea level. You don't need to climb hills to measure this effect. Gravitational time dilations due to height differences of a few feet have been measured in the laboratory.

The effect is tiny though. In the neighborhood of a large spherically-symmetric massive object such as earth, and compared to being infinitely far away from the object, your aging slows down by a factor

$$\sqrt{1\ -\ \frac{v_{esc}^2}{c^2}}$$

here $v_{esc}$ is the velocity needed at your particular position to escape from the gravitational pull of the object, and $c$ the speed of light.

When comparing two positions near to each other and close to the object, we can derive a simple equation that describes the relative time dilation. For two positions close to earth with height differences much smaller than the radius of earth, the fractional time dilation is given by $g h / c^2$, where $g$ denotes the local gravitational acceleration and $h$ the height difference.

For a hill on earth that peaks 90 m (300 ft) high, and using $g=10\ m/s^2$ and $c=3\ 10^8 m/s$, we find $g h / c^2 \approx 10^{-14}$. In other words, over a period of 3 years, a person on top of the hill would age one millionth of a second (one microsecond) more compared to a person at the foot of the hill.

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