  Flatly Foliated Relativity (FFR) is a new theory which conceptually lies
between Special Relativity (SR) and General Relativity (GR), in which spacetime
is foliated by flat Euclidean spaces. While GR is based on the idea that
"matter curves spacetime", FFR is based on the idea that "matter curves
spacetime, but not space". This idea, inspired by the observed spatial flatness
of our local universe, is realized by considering the same action as used in
GR, but restricting it only to metrics which are foliated by flat spatial
slices. FFR can be thought of as describing gravity without gravitational
waves.
  In FFR, a positive cosmological constant implies several interesting
properties which do not follow in GR: the metric equations are elliptic on each
euclidean slice, there exists a unique vacuum solution among those spherically
symmetric at infinity, and there exists a geometric way to define the arrow of
time. Furthermore, as gravitational waves do not exist in FFR, there are simple
analogs to the positive mass theorem and Penrose-type inequalities.
  Importantly, given that gravitational waves have a negligible effect on the
curvature of spacetime, and that the universe appears to be locally flat, FFR
may be a good approximation of GR. Moreover, FFR still admits many notable
features of GR including the big bang, an accelerating expansion of the
universe, and the Schwarzschild spacetime. Lastly, FFR is already known to have
an existence theory for some simplified cases, which provokes an interesting
discussion regarding the possibility of a more general existence theory, which
may be relevant to understanding existence of solutions to GR.
