My original explanation was wrong. The correct explanation is related to the curvature of space-time: there is no acceleration, your speed and trajectory throug

My original explanation was wrong. The correct explanation is related to the curvature of space-time: there is no acceleration, your speed and trajectory through space-time don't change. When the two people are far away from each other, they are both moving along through space time, having a speed that is, say, 1m/s through space, each towards the other, and (c-1) m/s in the time direction, both oriented towards the future. As they get close enough to each other, spacetime gets curved by their mass-energy, such that the direction of "the future" now points towards their shared center of mass. If we project this curvature back onto the flat plain in which they were originally moving, it will look as if their direction of movement has changed and their speed through space has increased, while their speed through time has decreased.