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I've spent half an hour reading QAs and wiki. I wanted to know what is the time dilation in the center of the Earth. There is that question here already: Gravitational time dilation at the earth's center. But the answer to this: Calculation of gravitational time dilation? suggests differently than the accepted answer to the former (in my understanding) "If you are 1 km from the center of the Sun you are not in a vacuum, so you need a different mathematical description of what's going on." (but does not give the answer to that case).

What is the formula to calculate time dilation in the very center of a massive body? (non-rotating, uniform density if needed).

Martian2020
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  • The two answers aren't in contradiction. The formula quoted by the OP in the second link only applies in the vacuum region outside a body. That is why the accepted answer in the first link doesn't use it and instead derives a different expression that correctly gives the time dilation inside the body (with some simplifying assumptions). That is the answer you're looking for. – Kris Walker Feb 04 '24 at 07:58
  • @Kris, does formula in the first link applies to both inside and outside the radius? Because it is said "The rule I mentioned in another question, " and that question has similar formula for the Voyager. – Martian2020 Feb 04 '24 at 13:49
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    The (approximate) time dilation factor $1+\Delta\Phi/c^2$ holds everywhere in a weak gravitational field. However, the form of $\Delta\Phi$ in the vacuum region is different to its form inside the body. In his answer, Ted derives the form of $\Delta\Phi$ inside the Earth, where $g(r)=GMr/R^3$. In the vacuum region, we instead have $g(r)=GM/r^2$, and you can follow the same steps to determine $\Delta\Phi$ and the dilation factor for the vacuum region. (And if you further set the radius of the distant observer to infinity, you end up with an approximation of the formula in the second link.) – Kris Walker Feb 04 '24 at 14:59
  • Kris is correct. This is a duplicate of the indicated question and the accepted answer is correct – Dale Feb 05 '24 at 00:54

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