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According to General relativity the clocks in our satellites in the atmosphere tick faster than those on Earth, as they are farther from the gravitational well of Earth and are free falling.

According to Newtonian mechanics and Gauss theorem, a hollow massive spherical shell has no net gravitational field at any point inside the shell as the geometry cancels out the pull from each extrema.

I would imagine that this does not change in GR (am I wrong?), an object inside should be free falling. But does time behave different inside the shell and outside very far from the shell?

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Yes, time will go differently.

Imagine a single photon, emitted from inside the shell and absorbed far outside. The energy outside should be lower than the energy inside because some energy was 'spent' to get out from gravitational well.

Now imagine two persons, one inside, the other outside. The 'inside' one sends radio waves at some frequency. Let's say the frequency of radio waves he transmits is $N$, the transmission time is $t$ so that the transmission has $N*t$ full periods.

His colleague will receive a transmission having $N*t$ full periods as well.

But the wave he receives is red-shifted (see the experiment with photons). So, the frequency of the wave is lower and the time of transmission is longer for him. Time goes slower for the person inside the shell, even though the space-time is flat around him.

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  • $\begingroup$ I was expecting more maths but your argument was very clear and intuitive. Thanks ! $\endgroup$ – Mauricio Nov 27 '17 at 14:51

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