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I remember to learn at school that our planet is approximately 4.543 billion years old.

But since the time is affected by mass of an object, this number is actually the age on the sea level, where the mass of our planet equals 1g.

But what about the center of the planet? In the very center there is no gravity (or better lets say the gravity is equally distributed to all directions) so in fact 0g. I guess there is still some momentum generated by Earth orbiting the Sun...but still - should not it be different than on the surface?

Does this mean that the cores of planets has in fact different age than their surface? And if so should we measure the age of cosmological objects by two numbers - by age on the surface and the age on the center of gravity?

While I understand how time dilation works when we travel away from the the surface, or when we are lets say on the surface of Mars...in case of going to center of Earth I actually do not know if the time "speed up" or "slow down".

Also while this idea...if correct...makes sense to me for planet Earth, for stars it seems to be a whole new level. I was trying to calculate the difference between the time on the surface of the Sun and the center of its gravity but to be honest I just have no idea how to begin. To me it seams that if the surface of the Sun is 4,6 billion years old and it has around 274 m*sec^-2 (or around 28g) on its surface and the gravity in the center is closer to 0g than in case of Earth...than the age of the center of gravity of the Sun is in negative numbers...? Is it by any chance possible that the age of the surface of the Sun was calculated like if it had 1g?...probably not... Can anyone help me to understand this and point out the errors in my logic?

I mean I kind of understand that object such as planet or start do not have two different ages and what is incorrect is the human definition of the force we call "time"...So let me little bit change the question - is it possible that during one periodic duration of certain physical process in the center of the planet the same process happens on the surface many more times? It seems even more confusing definition...sorry about that. But I hope you will understand what is the point of my question.

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Earth’s gravitational time dilation is completely negligible. The difference it makes in Earth’s age is much less than our uncertainty about that age.

Try calculating $\frac{2GM}{c^2R}$ to see how little the time dilation factor differs from $1$ at the surface. Hint: $\frac{2GM}{c^2}$ is about $9$ millimeters, while $R$ is about $6400$ kilometers! The time dilation at the center of the Earth is a bit different but equally negligible.

You can start worrying about gravitational time dilation when you travel to a neutron star or a black hole. Otherwise, you can forget that it exists unless you are doing ultra-precise experiments, navigating interplanetary spacecraft, or designing GPS systems.

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    $\begingroup$ FWIW, using that formula with the standard gravitational parameter of the Sun and radius 695,510 km gives $4.246\times10^{-6}$. Multiplying that by the age of the Sun comes to around 19,500 years. $\endgroup$
    – PM 2Ring
    Commented Dec 17, 2019 at 9:21
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    $\begingroup$ This reminds me about a caretaker at the Museum of Natural History in London (where they keep the dinosaurs). He used to tell visitors that the Diplodocus in the Hall was 70 million and ten years old. When asked why he was being so precise, he replied, "Well, they told me it was 70 million years old when I started work here - 10 years ago". $\endgroup$
    – Oscar Bravo
    Commented Dec 17, 2019 at 13:12
  • $\begingroup$ Newton's completely sufficient for interplanetary spacecraft navigation. GR might be necessary if you need your spacecraft to be at a meter-level specific position centuries in the future with no course corrections allowed, but noone builds to those mission parameters. $\endgroup$
    – notovny
    Commented Dec 17, 2019 at 16:09
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    $\begingroup$ @notovny According to The Planetary Society, “general relativity is routinely accounted for in spacecraft navigation” and “general relativity is definitely required for accurate navigation of MESSENGER”, a satellite mission to Mercury. $\endgroup$
    – G. Smith
    Commented Dec 17, 2019 at 18:14
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    $\begingroup$ @Miroslav Řešetka I saw a question you asked in a now-deleted answer. If there were two Earths separated by 100 meters, there would be time dilation at the midpoint between them, and it would be twice as much as for one Earth, even though there is no net force there. Gravitational time dilation depends on gravitational potential, not on gravitational force. $\endgroup$
    – G. Smith
    Commented Dec 17, 2019 at 18:24

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