Recently, the old notion of causal boundary for a spacetime \(V\) has been redefined consistently. The computation of this boundary \(\partial V\) on any standard conformally stationary spacetime \(V=\mathbb{R}\times M\), suggests a natural compactification \(M_B\) associated to any Riemannian metric on \(M\) or, more generally, to any Finslerian one. The corresponding boundary \(\partial_BM\) is constructed in terms of Busemann-type functions. Roughly, \(\partial_BM\) represents the set of all the directions in \(M\) including both, asymptotic and “finite” (or “incomplete”) directions.

This Busemann boundary \(\partial_BM\) is related to two classical boundaries: the Cauchy boundary \(\partial_{C}M\) and the Gromov boundary \(\partial_GM\).

The authors' aims are: (1) to study the subtleties of both, the Cauchy boundary for any generalized (possibly non-symmetric) distance and the Gromov compactification for any (possibly incomplete) Finsler manifold, (2) to introduce the new Busemann compactification \(M_B\), relating it with the previous two completions, and (3) to give a full description of the causal boundary \(\partial V\) of any standard conformally stationary spacetime.