  The characteristic physical timescales near stellar-mass compact objects are
measured in milliseconds. These timescales -- the free-fall time, the fastest
stable orbital period, and stellar spin periods -- encode the fundamental
physical properties of compact objects: mass, radius, and angular momentum. The
characteristic temperature of matter in the vicinity of neutron stars is such
that the principal electromagnetic window into their realms is the X-ray band.
Because of these connections to the fundamental properties of neutron stars,
X-ray timing studies remain today the most direct means of probing their
structure and dynamics. While current X-ray observatories have revealed many
relevant and fascinating phenomena, they lack the sensitivity to fully exploit
them to uncover the fundamental properties of compact objects and their extreme
physics. With this white paper, we summarize and highlight the science
opportunities that will accompany an order-of-magnitude improvement in X-ray
timing sensitivity, a goal attainable in the coming decade.
