@dalehileman,
Let's assume there's a train, parked at the station, and it suddenly begins to move (it is now accelerating).
Let's say it continues to accelerate at a very slow rate, such as an increase of 1 mph. So, after 60 hours it is going 60 mph.
Now, according to what the physicist said, all during that entire 60 hours, a guy on the train will "see" an earth clock running faster, not slower.
Now let's assume that for one minute, the train quits accelerating, and maintains a uniform speed of 60 mph. Now, according to SR, the train passenger must assume that he is not moving, but that the earth is, even though, for the past 60 hours, the opposite was the case.
Now let's say the train starts slowing down at a the same slow rate of decrease (1 mph) until it comes to a complete rest with respect to the earth. During these 60 hours, the passenger also "sees" himself as "moving" and therefore also assumes that the earth clock is running faster than his, not slower.
What "magic" thing happened during that brief one minute (out of a total of 7,201 minutes) that would lead him to "suddenly" conclude that his clock was running faster?
My answer: Nothing. He wouldn't see it that way. Unless he is a fool.
The mere fact that he quit accelerating for one minute would NOT tell him that he quit "moving." The laws of physics (e.g. inertia) would in fact tell him just the opposite (i.e., that he is STILL moving, even though he is no longer accelerating).