Paradoxes in special relativity almost always stem from forgetting about the relativity of simultaneity.
No, the clocks of Fred and Steve are progressing at different rates no matter what. Fred sees* Steve’s clock as slow. Steve sees Fred’s clock as slow. This is true on both the outbound and the return journey. Thus… a paradox is born.
While outbound Steve and returning Steve agree about Fred’s clock speed, these two Steves (or two different reference frames for Steve) disagree about which events in the story are simultaneous. That’s the critical piece.
Example: Steve leaves Fred at speed 0.6c, travels 3 light-years (as measured by Fred), flips around nearly instantly 5 years after departure (as measured by Fred), and then travels back at the same speed. During the whole trip, Fred says, “Steve’s clock is slower than mine. He will age 8 years before returning here rather than my 10 years.”
Right before the turnaround, Steve says:
- “Fred’s clock is slower than mine.”
- “My clock shows 4 years since departure since I only traveled 2.4 light-years (as measured by me) to this spot, but Fred’s clock shows 3.2 years since departure.”
- “That matches with the fact that his clock is going slower.”
Right after the turnaround, Steve says:
- “Fred’s clock remains slower than mine.”
- “My clock shows 4 years since departure still, since turnaround was instant and my clock stayed with me, but Fred’s clock shows 6.8 years since departure.”
- “Given the 3.2 Fred-years it will take for my return trip, Fred will show 10 years of aging (6.8 yr + 3.2 yr) when I get there.”
That apparent time jump is because simultaneity assessments differ in the old and new reference frames. In the old frame, the following are seen as simultaneous: Steve is at the turnaround point; Fred sees “3.2 years” on his clock. But those events happen far apart from one another, so a different reference frame will judge them as not simultaneous. In the new frame, the following two events are simultaneous: Steve is at the turnaround point; Fred sees “6.8 years” on his clock.
*“See” is a hazardous verb to use here since there is also a mundane Doppler effect on anything literally “seen”, as through a telescope. So, read “see” here as “would see if he had co-moving clones of himself at all points in the universe so that he can read all clocks locally next to the clock in question” or more simply “would later ascertain through a comparing of notes and calculations”. Or even better: just don’t worry about this detail and happily ignore what “see” means in practice. It doesn’t actually matter for understanding this paradox.
More details
The bouncing photon “light clock” you described can be used to work out various consequences of the constancy of c. Time dilation is one, of course. Length contraction is another. But relativity of simultaneity is a critical third, yet it is often skipped over in discussions. You need all three. These can all be encapsulated in (or derived from) a single mathematical framework for spacetime, so you don’t have to think about them piecemeal in practice. But if you do think about them piecemeal, and you don’t consider all three carefully, you stumble into apparent paradoxes.
Quantitatively, the relative timing shift T between two events separated by a distance d along a direction of relative motion of speed v is given by T=vd/c^2. In this case, we switch from +v to -v, so there is a double-sized shift, leaving T_{\mathrm{shift}}=2(0.6c)(3\; c\mathord{\cdot}\mathrm{yr})/c^2=3.6\;\mathrm{yr}, as above.
Ultimately this seemingly out-of-the-blue time shift goes back to a deeper question: how are clocks in different reference frames synchronized with each other in the first place? You can also eliminate the “instant turnaround” conceit by having Steve’s brother Schmeve handle the return trip, where Schmeve was traveling toward Fred from far away this whole time. It’s a different analysis of the same situation, but it shows that the details of the turnaround aren’t actually important.