  A geometric theory for spacetimes whose world lines associated with physical
particles have an upper bound for the proper acceleration is developed. After
some fundamental remarks on the requirements that the classical dynamics for
point particles should hold, the notion of generalized metric and a theory of
maximal proper acceleration are introduced. A perturbative approach to metrics
of maximal proper acceleration is discussed and we show how it provides a
consistent theory where the associated Lorentzian metric corresponds to the
limit when the maximal proper acceleration goes to infinity. Then several of
the physical and kinematical properties of the maximal acceleration metric are
investigated, including a discussion of the rudiments of the causal theory and
the introduction of the notions of radar distance and celerity function. We
discuss the corresponding modification of the Einstein mass-energy relation
when the associated Lorentzian geometry is flat. In such context it is also
proved that the physical dispersion relation is relativistic. Two possible
physical scenarios where the modified mass-energy relation could be confronted
against experiment are briefly discussed.
