  In this work we consider heterotic-gravity as the low-energy approximation to
heterotic string theory. We define a consistent non-relativistic limit of
heterotic gravity that includes the Yang-Mills Chern-Simons term. We perform
three tests on the heterotic limit: we use it to (i) derive the
non-relativistic transformation rules, (ii) show the existence of a finite
non-relativistic heterotic action and (iii) obtain the longitudinal
non-relativistic T-duality rules. We show that in all these cases the limit
procedure is well defined, free of divergences and leads to consistent
non-trivial results. We comment on the interpretation of the T-duality rules in
terms of a heterotic non-relativistic geometry underlying non-relativistic
heterotic string theory.
