When we're near the earth, we instinctively use an earth-based reference frame. So when we say we accelerate, we're referring to the change our velocity with respect to the earth.
Let's try to visualize a spacetime picture. That's not too hard here because we only need one space coordinate, the one along the line between the center of the earth and the center of you. Let's make the other, vertical, coordinate in a 2-D plot be time.
Say you start at rest with respect to the earth. In our plot, that means that you and the earth start out with parallel world lines. For convenience, we pick the earth frame so these are just vertical lines. The earth's warping of spacetime causes your world line to bend toward it, until they intersect.
Here's a start on how to think about this effect. If you make two identical clocks, and put one near the earth and one farther away, the near one will run slower. This effect can be seen and agreed upon by observers at either clock, although they may choose to call their own clock "correct". This effect applies to all clocks. Now the rate at which things tick, the frequency, is identical to the energy, except for a factor of Planck's constant. So the position-dependent frequency is really the same effect as the classical position-dependent energy. Take any quantum wave packet representing a particle. Let the quantum phase advance faster (higher frequency) on one side than the other. That shifts the interference pattens, causing the average velocity of the particle to change. That gives the acceleration.
Here we've just ignored the spatial distortions, which become equally important for figuring out the trajectories of particles moving at the speed of light with respect to the earth. You can search around here for "light bending gravity" to find some discussions of that.
Mike W.
(published on 10/17/2012)