I just read this answer that explains that the time dilation due to the gravity and the time dilation due to velocity are the same thing.
Does that mean that, if I fall from the top of a skyscraper on Earth, my clock ticks faster than the clock of an observer that remains at the top of the skyscraper because even though I'm moving faster than him, my body is motionless in the space component of space-time?
In other words, does this gravitational/velocity time dilation equivalence implies that any observers subject to gravity (not falling) is actually moving in the space component of space-time?