Why doesn't Galilean relativity lead to a contradiction in SR?

Question

Two identical spaceships commanded by Alice and Bob are at rest next to each other in outer space. The clocks of the spaceships are synchronised; and when they are close by Alice can see Bob's clock and vice versa. Both spaceships then accelerate away from each other in opposite directions. After a while both spaceships slow down, stop, turn round and return to their starting position passing each other at constant high velocity. The equations of motion of the two spaceships are identical, save for the fact that they went in opposite directions--in Cartesian coordinates you can replace x by -x.

As they pass each other at constant velocity Alice and Bob note the time on their own clock and the clock of the other spaceship. Alice looks at her clock; it says noon. She looks at Bob's clock; it says 11.55am. Alice knows about special relativity. In her inertial frame she is at rest. Bob is moving. She knows moving clocks run slow, and is not surprised that Bob's says 11.55am.

Bob looks at his own clock; it says 11.55am. He also knows about special relativity. In his inertial frame he is at rest. It is Alice who is moving. Alice's clock should run slow. It should show a time of about 11.50am. But Bob looks at Alice's clock and it reads noon.

Why is there not a contradiction here?

Answer 1

The elapsed property time over a worldline is:

$$ \Delta\tau = \int_W\frac{dt}{\gamma(t)} $$

where gamma is computed in the initial rest frame.

Since the two $W$ are related by reflection, they will measure the same $\Delta \tau$ at the return point.

They will each see the other clock running slower than their own, though.

Answer 2

Your question includes an incorrect assumption. If Alice and Bob start in the same place with synchronised clocks, and return to the same place having undergone symmetric journeys in opposite directions, then when they return their clocks will show the same reading.

Answer 3

As JEB said, you misanalyzed the problem: both clocks will read noon at the end. But more fundamentally, I don't see how this experiment is related to Galilean relativity in the first place.

Galilean relativity (as illustrated by Galileo's ship) says that you can't do any experiment in a closed lab that will determine its velocity relative to, say, the fixed stars. (You can't peek at the stars because the lab is closed.)

If you want to treat your two spaceships as two labs, the principle doesn't apply for two reasons: first, it only applies to velocity, not acceleration, and second, Alice and Bob can't look at each other's clocks because the labs are closed.

If you want to set the whole experiment with both ships in a single huge lab, then I don't see how the experiment relates to the motion of the lab.