In "The Elegant Universe," two individuals, George and Gracie, are noted to be moving relative to one another in space and wearing clocks. They sync their clocks upon passing each other. Each observes the other's clock to be ticking more slowly. Gracie communicates with George via cell.
Gracie communicates her time to George as he recedes into the distance. Initially, Gracie thinks that George will hear her reported time before his clock reaches the same time. But then, she takes into account travel time and realizes the travel time will more than compensate, and George will actually hear her time after his clock has passed that time.
Then, the author notes that
Gracie realizes that even if George takes the travel time into account, he will conclude from Gracie's communication that her clock is running slower than his.
By this last statement, does it mean that the time difference will be greater than what can be accounted for by mere communication travel time? I would think that some of the time difference is due to Gracie (from George's perspective) transmitting the time after that time has already elapsed on his clock. After all, Gracie's clock is moving more slowly (due to time dilation) relative to his clock and that accounts for the extra time difference (beyond the communication travel time). Do I have that correct?