Which planet will have a faster clock?

Question

This is a complication to the twin paradox.

Eddy stays on Earth. George goes flying quickly through space then he turns around and heads back toward Earth. As George is coasting back toward Earth he sees a rogue planet that happens to be traveling with him. George lands on the planet and meets Peggy. Everyone involved knows about Relativity. George and Peggy riding on this rogue planet go speeding past Earth.

George knows he should be aging slower than Eddy who is still on Earth since he left Earth, traveled quickly and is now flying past Earth at speed. I.E. Eddy is the inertial observer.

Peggy thinks she is the inertial observer because she has not been accelerated and she knows the Earth zipping past must have a slower clock because the Earth is moving at speed relative to her.

So Peggy thinks Eddy's clock on Earth should run slower than hers, but George thinks Eddy's clock should run faster than his.

Who will have the faster clock? Who will have the slower clock? Why?

Answer 1

Time dilation is not clocks running faster or slower. It occurs when two observers measure a different elapsed time between the same two spacetime points. Time is what clocks measure i.e. a clock is a measuring device that measures elapsed time between two points in the same way that a milometer measures the distance in space travelled between two points.

This may seem a fine distinction, but it is a very important point not usually appreciated in popular science descriptions, and it is key to understanding this situation.

If you consider just the twins then we can define the spacetime point $A$ to be when they part and the spacetime point $B$ to be the time when they meet again. Both twins measure the elapsed time between these two points using the clocks they are carrying, so they can compare their measurements of the elapsed time, and as usual in the twin paradox the accelerating twin George will measure the smaller elapsed time. You can describe this as George's clock running slow, but it really just means George's clock measured less elapsed time.

However Peggy only passes through spacetime point $B$ i.e. the point in which all three people meet back on Earth. Peggy was never at point $A$ so she cannot use her clock to measure the elapsed time between the two points. Hence it is meaningless to ask whether Peggy's clock is running slow, fast or whatever relative to the other two clocks.

Peggy can use the Lorentz transformations to calculate how her clock compares to the other two clocks, but this is just a calculation and not a direct comparison. Doing the calculation would lead you to the apparent paradox that Peggy thinks Eddy's clock runs slow while Eddy thinks Peggy's clock runs slow. It isn't a paradox because the two cannot make a proper comparison of their clocks.

For more on the twin paradox see What is the proper way to explain the twin paradox?. For some background on what we mean by time in relativity see What is time, does it flow, and if so what defines its direction? and What is time dilation really?.