[chris (OP)]

One of Einstein's predictions from general relativity is that time ticks more slowly owing to the effects of gravity. So, close to a black hole, for example, time would pass more slowly than elsewhere in space.

This being the case, does that mean that for large bodies like the Earth, because gravity acts through the centre of mass, that the core is "older" than the rest of the planet?

[jeffreyH]

I have thought about this one quite a bit but I am still not sure about it. I would also be interested in comments from other members.

[Blame]

This being a thought experiment lets make the simplifying, if unlikely, assumption that the earth came into existence all at the same time.

The core would be more time dilated. Therefor less local time would have passed. It would be younger from that point of view. But from where I am standing it has been around just as long as the rest of the dirt and has no right to call me grandad. That's relativity for you.

[evan_au]

the simplifying, if unlikely, assumption that the earth came into existence all at the same time.

This assumption provides the answer according to General Relativity.

Relaxing this assumption, we can get an answer from astronomy and geology...
It is thought that the Earth formed by accretion of interstellar dust that formed a protoplanetary disk.

Initially, its composition would have been fairly uniform.
But the heat of collisions, the energy released by gravitational collapse and heat released by decaying radioactive elements would have melted the interior, causing it to differentiate with the denser nickel/iron in the center (along with elements that dissolve in these metals), with the less dense silicates around them, and less dense water and gases on the outside.

So by this measure, the core separated out about the same time as the other layers of the Earth.
It has been slowly cooling since then, eventually solidifying perhaps 2-4 billion years ago. The solid core has been slowly growing ever since.

Recent laboratory experiments simulating conditions in the Earth's core have given quite different answers for the thermal conductivity of the core material, and hence different estimates of its age.

Today's continental crust and ocean floor is younger than the other layers, because it is being continually being eroded, subducted and reformed via plate tectonics. 

See: https://en.wikipedia.org/wiki/Inner_core#History

[jerrygg38]

Einstein's relativity states that the time clock moves more slowly due to strong gravitational fields and high speeds as compared to the surface of the earth. But when we look at the age of an object we use our clock and ruler. Thus the center of the earth is basically the same age as the surface except for the time to form the earth.
  Relativity then means nothing in this regard. The biological functions of a person traveling around the earth in orbit may slow slightly but the effects of weightlessness will do more harm to the astronaut  than the slight time improvement due to relativity.

[chris (OP)]

Thus the center of the earth is basically the same age as the surface except for the time to form the earth.

Richard Feynman was of the opinion that the core was "older" than the surface owing to time dilation...

[jeffreyH]

At the centre of the earth you can imagine all the mass to be composed of successive shells of infinitesimal thickness all the way to the surface. The radius to each shell and the area of its surface will be proportional to the force exerted be the mass of the shell. Summing all the contributions will give a quantity for the total force concentrated at the centre. This cannot be done any other way since the forces 'cancel' at the centre. So time dilation is not straightforward to calculate here.