General relativity tells us that as you get closer to a mass clocks run a bit more slowly. The effect can be observed on Earth with precise clocks at different distances from the center of the Earth and is pronounced enough that you have to correct for it to get the precise timing signals from GPS satellites right.
It follows that time runs more slowly at the center of a mass like the Earth. Apparently Feynman mentioned that, over the life of the Earth, the center of the Earth would be a day or two younger than its surface. The calculation is quite easy and one might expect it assigned as an undergrad physics problem.
It turns out that when you do a back of the envelop calculation you get about a year and a half. Feynman may have made a simple mistake or perhaps he was incorrectly quoted - apparently he never wrote the result down. I was unaware of his result, but I would have probably accepted it even though I had worked it out for distances from the earth (tall buildings, mountains, orbiting satellites).
The paper paper works out the result at the level of an undergrad physics class.
The young center of the Earth
U.I. Uggerhøj,1 R.E. Mikkelsen,1 and J. Faye2
1Department of Physics and Astronomy, Aarhus University, Denmark 2Department of Media, Cognition and Communication, University of Copenhagen, Denmark (Dated: April 20, 2016)
We treat, as an illustrative example of gravitational time dilation in relativity, the observation that the center of the Earth is younger than the surface by an appreciable amount. Richard Feynman first made this insightful point and presented an estimate of the size of the effect in a talk; a transcription was later published in which the time difference is quoted as ’one or two days’. However, a back- of-the-envelope calculation shows that the result is in fact a few years. In this paper we present this estimate alongside a more elaborate analysis yielding a difference of two and a half years. The aim is to provide a fairly complete solution to the relativity of the ’aging’ of an object due to differences in the gravitational potential. This solution - accessible at the undergraduate level - can be used for educational purposes, as an example in the classroom. Finally, we also briefly discuss why exchanging ’years’ for ’days’ - which in retrospect is a quite simple, but significant, mistake - has been repeated seemingly uncritically, albeit in a few cases only. The pedagogical value of this discussion is to show students that any number or observation, no matter who brought it forward, must be critically examined.