General Relativity Paradox?

I have had a very tough question about relativity. Consider two immortal observers ; Adam lives on the Earth, Eve lives in a space station 100 000 km away from the Earth. Now, since Eve is further from the center of the Earth's gravity well than Adam, her flow of time will run faster according to the flow of time of Adam. However, both will percieve their own flow of time as flowing normally. Now suppose Eve stares at the Earth through a telescope and counts the number of days that passes on Earth. Each time North America appears to her makes 1 more day. Adam does exactly the same thing ; each time the Sun rises makes 1 more day. They both started counting at the same time. They carry this little experiment for a billion years. After that time, Eve comes back to Earth and compares the number of days she counted with the number of days counted by Adam. Since time went on faster for Eve according to Adam, Eve will have counted more days than Adam. However, Eve would have witnessed through her telescope every full rotation that the Earth made around its axis during the whole time. So she would have witnessed directly the day-night cycles that went on on Earth. She will have witnessed more day-night cycles that went on on Earth than Adam. How's that possible? Where did the extra days went? That's the question I've had asked to me. Is the only way out of this is to conclude that they both counted the same number of days but the day-night cycles Eve witnessed took more than 24h of Adam's time flow but lasted 24h of Eve's time flow? Someone brought me forth that problem as an attempt to discredit relativity.
- Anonymous

Close. Of course Eve and Adam agree on the number of Earth day-nights that have happened between Eve's departure and return. Eve sees all of the Earth-clocks as running slow. That includes the Earth's spin. She thinks it takes more than 24 hours, as defined by her wristwatch. So she thinks the spinning Earth and Adam and his clocks are all slow by some shared factor, about (1/(1+gR/c²)), where g is the gravitational acceleration at the earth's surface, R is the radius, and c is the speed of light. That's (1- 6.7*10^-10).  There's no real problem or even complication here.

Mike W.

(published on 02/13/2011)