What is the age of planet Earth, if we are not looking for the age at sea level?

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.

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.
• 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. Commented Dec 17, 2019 at 9:21