Alternatives to Schwarzschild in the weak field limit of General
  Relativity
  The metric outside an isolated object made up of ordinary matter is bound to
be the classical Schwarzschild vacuum solution of General Relativity.
Nevertheless, some solutions are known (e.g. Morris-Thorne wormholes) that do
not match Schwarzschild asymptotically. On a phenomenological point of view,
gravitational lensing in metrics falling as $1/r^q$ has recently attracted
great interest. In this work, we explore the conditions on the source matter
for constructing static spherically symmetric metrics exhibiting an arbitrary
power-law as Newtonian limit. For such space-times we also derive the
expressions of gravitational redshift and force on probe masses, which,
together with light deflection, can be used in astrophysical searches of
non-Schwarzschild objects made up of exotic matter. Interestingly, we prove
that even a minimally coupled scalar field with a power-law potential can
support non-Schwarzschild metrics with arbitrary asymptotic behaviour.
