There was a bowl of M&Ms for after the oral exams Our grades were based on homework assignments and three oral exams. Four of us would take our turn at the blackboard. The professor had this natural ability to ask question you hadn't thought about, but should be able to solve.

*Mr Crandall: Does a clock at the center of the Earth run faster or slower than one on the surface? If so how big is the difference since the Earth was formed?*

(It wasn't as bad as that might sound - he was nothing like John Houseman.)

First a little diversion is in order. You've probably heard the Global Positioning System needs to account for relativity. Two effects are going on. In one the satellites are moving with respect to GPS receiver on the ground. Each satellite has a very stable atomic clock that is synchronized with the other GPS satellites. Your receiver look at position and time information from at least four satellites to determine your position. To get the necessary level of accuracy the system needs to account for the effects of both special and general relativity.

Special relativity tells us moving clock appears to run a bit slower than one at rest. Almost everything around us moves too slowly for this to be important , but a clock on a GPS satellite would lose about seven microseconds a day. That may not seem like much, but that's enough to cause a position error that increases by about two kilometers a day.

The curvature of spacetime is the bigger problem. A result of general relativity is the direction where time runs more slowly.^{1} The further a clock is from a mass, the faster it runs. Clocks on orbiting GPS satellites gain about forty six microseconds a day. Add the special and general relativistic effects and the satellite clocks gain a bit under thirty nine microseconds a day. Left uncorrected the system would be useless in a few minutes and the error would increase by over seven miles in a day. The satellites correct for both of these effects before the signal is transmitted.

All of this is key to the original question. Time at the center of the Earth will pass more slowly. The question is how much?

The purpose of these oral exams was not so much to test if you had learned the course material, but to encourage you to quickly frame a problem and come up with a ballpark estimate. In the spirt of the type of education pioneered by Comenius it was supposed to be playful and even fun. This type of thinking with a piece of chalk is core to thinking through new ideas and working with others on certain stages of problems. A few marks of chalk might represent dozens of pages of calculations if done in detail. Unfortunately its protrayal on television and in movies can make the field seem artificially difficult to outsiders.

A quick calculation showed the center of the Earth is about 1.4 years "younger" than the surface.^{2} Long lived radioactive elements at the core decay slightly less than their counterparts at the surface. I can still remember the "wow" in my mind as took a handful of M&Ms. A not very great fountain of youth is under your feet. Since then I've learned exercise and eating right are better approaches to aging.

Since then I've done a more accurate calculation. The inner Earth isn't homogeneous. I found an estimate of how density changes and came up with about 2.5 years. The initial estimate was ten or fifteen minutes with a piece of chalk vs at least a day's worth of work. In the process I found time dilation due to special relativity was a thousand times smaller and reasonable to ignore. Moving to the Sun I got about 30,000 years with the simple estimate for the Sun's density.

The more important point is many fields develop mechanisms for dealing with either too much or too little data by coming up with simple ways to reframe and think about the task. I've seen it in math, art and sports and assume it's a common approach to playful problem solving - probably so common you don't think about it. The playful and fun part is key. That's a space where you can be creative.

__________

^{1} You can learn special relativity with high school math and a bit of logic. It's not at all difficult. General relativity, on the other hand, requires much deeper math. And a tip. "Down is the direction where time runs more slowly" on a t-shirt is a good way to find other physicists in a crowd.

The effect is known as gravitational redshift as you usually measure this by looking at frequency shifts. Atomic clocks are so good these days that you can measure the effect in a lab by raising one a few meters.

2 If you want the details send me a note. First derive an equation for the gravitational potential inside a homogeneous sphere. Next think about the frequency of a photon in a gravitation well to get change as a function of change in gravitational potential. Then it's just remembering some constants to get the number.

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