Twin Paradox doppler resolution

Question

I've been doing some doppler thought experiment around the twin paradox,

(wikipedia) The paradox centers on the contention that, in relativity, either twin could regard the other as the traveler, in which case each should find the other younger—a logical contradiction.

Let's say Bob and Alice are twins who can measure each other's wavelength shift. As they move apart, they will see eachother red-shift. If you see a person red-shifted, you see them in slow motion, and if you see them blue-shifted you see them in fast forward. (Einstein thought about looking at the hands of the clock tower while moving away from them in a tram).

Scenario 1: Bob moves away and Alice stays put. Both instantaneously see eachother red-shift. At some point, Bob decides to stop and move back towards alice. Immediately Bob will see Alice blue shift. But Alice will have to wait until the light of the turnaround event has reached her, before she will see Bob blue-shift. Bob will see Alice red/blue 50% / 50% of his travel time, and Alice will see Bob more red than blue. So Bob will have aged less.

Scenario 2: Bob and Alice both move away from eachother at half the speed that Bob did in Scenario 1. Both turn around at the same time, and both will see eachother red/blue 50% of the time. They will have the same age. Both can clearly distinghuish this situation from the first, and also now they both have to agree on their turnaround timing, while in Scenario 1, Bob could decide by himself whenever he wanted to turn.

Scenario 3: Alice moves away, Bob stays put, the reverse of Scenario 1. Now Alice will instantly see Bob's color change as she reverses direction, thus making her the traveler.

It seems to me that after Bob and Alice reunite and compare measurements, it should be possible for them to figure out who was the traveler. There also seems to be no meaningful age comparison possible if the twins do not at some point reunite at the same location, because how would that work. Furthermore, can we conclude from these examples that changing course is what slows down the course-changer's clock? I would define changing course as being any change in velocity with respect to the other person. To move apart and rejoin, requires at least two course changes, so in the end: does this mean that, acceleration is the key to resolving the twin paradox? That the one who has a force acted upon her/him, will have his/her clock slowed down? If the answer is yes, it would be a relatively (no pun intended) logical direction of thought to reason that this could explain as to why gravity slows down clocks? But this raises another question: A course change requires a short time of force acted upon the traveler, if this slows down a clock, gravity seemingly exerts a continuous force on the traveler, but the clock does not keep on slowing down, or does it?

Answer 1

You describe the three scenarios correctly, and yes Alice and Bob can 'figure out who is the traveller' in the sense you mean- there is no problem with that.

Your suggestion that a course change is the key to resolving the paradox is correct, but not for the reason you suppose.

The twin paradox is a consequence of the geometry of spacetime. In spacetime, the elapsed time along a straight path between two events is always longer than the elapsed time along a kinked path between the same two events.

Acceleration is needed if you as an individual want to travel on a kinked path, as you have to change direction, but acceleration does not in itself cause the effect. You can consider a version of the twin paradox without acceleration, as follows...

Bob coasts past Alice for some distance until he meets Chloe who is coasting in the opposite direction, and Chloe later coasts past Alice. The elapsed time for Alice between meeting Bob and meeting Chloe will be longer than the sum of the elapsed time of Bob's leg of the experiment and Chloe's leg. No acceleration is involved.

Also, you shouldn't think of 'clocks slowing down' if you want to understand SR properly. The twin paradox arises because the elapsed time between two events depends on the path between them, and accurate clocks will correctly measure the different time periods- it is not because clocks' ability to measure time is hampered in some way. It is analogous to saying that 'odometers slow down' when cars taking different routes between two points report lower mileages for their journeys- the odometers are working fine; it is the actual distances that vary.