The Great Atomic Clock of Kansas

Question

It is said that astronauts on the ISS experience time slightly slower than the rest of us. (I believe that the lesser gravitational field would increase their rate of time, so that affect must be more than canceled by the dilation affect of their velocity)

One could imagine setting up an atomic clock in a flat sparsely populated area such as Kansas, with a huge display that can be seen from space. We could than synchronize this clock with an atomic clock on the ISS.

The above wikipedia article mentions that the ISS looses about 0.007 seconds every 6 months.

When the astronauts of the ISS pass over Kansas, will they see their clock losing time or gaining time versus the Great Atomic Clock of Kansas?

Answer 1

Let's ignore gravitational time time dilation for now to keep things simple.

Suppose we place me and a second clock 300km (the altitude of the ISS orbit) above the Kansas clock, and by some means we hover there so we are stationary with respect to the Kansas clock. That means both clocks are in the same inertial frame so they run at the same rate and will show the same time. So when I report times they will be the same times that ground control report.

You're on the ISS moving at 7.65 kilometres per second and you have your own clock. From my perspective your clock will be running slower than mine by the 7 ms/month that you mention.

The point of placing me and the second clock at the ISS orbit is that the ISS can pass right by us. So once an orbit you and I and both clocks are at the same spacetime point, and all observers at the same spacetime point will agree on what they read from the clocks. That means you and I will agree what your clock shows and you and I will also agree on what my clock shows.

And with this setup we can now answer your question. Every time you pass me I can read both my clock and yours, and I can see that your clock reads slightly less than mine. But our readings must agree because we're at the same spacetime point, so you will also see that your clock reads less than mine. We will both agree that your clock is running slow.

At first glance this may seem to violate the principle of relativity because we expect that two observers in relative motion will both see the other person's clock running slower than theirs. It does not do so because the statement that both observers see the other clock running slow applies only to unaccelerated motion, and if you're travelling in a circle you are accelerating.

If you're interested in pursuing this further I analyse the time dilation in circular motion in my answer to Is gravitational time dilation different from other forms of time dilation?. If you can work through that you've taken the first steps towards understanding General Relativity!