The geometric challenge of testing gravity with wide binaries
  Wide binaries provide promising laboratories for testing general relativity
(GR) in the low-acceleration regime. Recent observational studies have found
that the difference in the proper motions and/or radial velocities of the
components of nearby wide binaries appear larger than predicted by Kepler's
law's, indicating a potential breakdown of GR at low accelerations. These
studies have not accounted for projection effects owing to the different
position of the two stars on the celestial sphere. I show that two stars in a
wide binary with identical 3D space velocities often have significantly
different proper motions and radial velocities purely due to projection
effects. I construct a sample of simulated binaries that follow Kepler's laws
and have similar phase-space distributions to the observed samples of nearby
binaries. Beyond separations of $\sim$0.1 pc, direct comparison of the
components' proper motions would suggest strong tensions with GR, even though
the simulated binaries follow Kepler's laws by construction. The magnitude of
the apparent disagreement is similar to that found observationally, suggesting
that the apparent tension between observations and GR may largely be due to
projection effects. I discuss prospects for constraining gravity at low
accelerations with wide binaries. Robust tests of GR are possible with current
data but require measurements of 3D velocities. Further work is also needed to
model contamination from unbound moving groups and unrecognized hierarchical
triples.
