  We investigate a foliation of Schwarzschild spacetime determined by observers
freely falling in the radial direction. This is described using a
generalisation of Gullstrand-Painlev\'e coordinates which allows for any
possible radial velocity. This foliation provides a contrast with the usual
static foliation implied by Schwarzschild coordinates. The $3$-dimensional
spaces are distinct for the static and falling observers, so the embedding
diagrams, spatial measurement, simultaneity, and time at infinity are also
distinct, though the $4$-dimensional spacetime is unchanged. Our motivation is
conceptual understanding, to counter Newton-like viewpoints. In future work,
this alternate foliation may shed light on open questions regarding quantum
fields, analogue gravity, entropy, energy, and other quantities. This article
is aimed at experienced relativists, whereas a forthcoming series is intended
for a general audience of physicists, mathematicians, and philosophers.
