Relativistic effects of an orbit around sun in a different direction than Earth

Question

If I manage to orbit around the sun at the same speed but in a different direction than the earth, will I perceive the time on earth going faster or slower than my own time? And what about people on earth, will they perceive my time faster or slower than theirs?

Answer 1

The metric for the rest frame of a rotating observer is complicated so calculating how your clock looks to the Earth or how the Earth's clock looks to you is hard. However there is a simple argument we can use to show that overall your and the Earth's clocks will tick at the same rate and both will tick slower than a clock on the Sun.

Consider this diagram showing you and the Earth orbiting the Sun at the same distance but in different planes:

Your orbit and the orbit of the Earth will intersect twice in every orbit at the points I've marked $A$ and $B$. I am sitting with a stopwatch at point $A$ so once in every orbit you, the Earth and I am at the same point. When the three of us are at point $A$ we all reset our clocks to zero. Observers at the same spacetime point can compare their clocks directly so we all agree our clocks show zero. You and the Earth then head away on your orbits while I stay motionless at point $A$. One orbit later the three of us meet again and we compare our clocks. Your question now reduces to asking how our three clocks compare.

In my frame both your and the Earth's clocks are running slow, and the time dilation is simply $1/\gamma$ (if you're interested I prove this in my answer to Can a ultracentrifuge be used to test general relativity?). Now my argument is:

1. in my frame the time dilation is $1/\gamma$ for both you and the Earth because it makes no difference what angle you orbit at. Therefore when we all meet again at point $A$ I will observe your and the Earth's clocks to read the same.

2. because we are all at the same spacetime point $A$ we must all agree on what the clocks read. Therefore if I observe your and the Earth's clocks to read the same both you and the Earth must also observe your and the Earth's clocks to read the same.

3. therefore when averaged over the course of one orbit your and the Earth's clocks must tick at the same rate.

This shows that when averaged over any integral number of orbits your and the Earth's cloks tick at the same rate. During an orbit the sync between your and the Earth's clocks will drift relative to each other in a complicated way, but in the long term they will stay the same.