  We present a new, semi-analytic formalism to model the acceleration and
collimation of relativistic jets in a gravitational potential. The
gravitational energy density includes the kinetic, thermal, and electromagnetic
mass contributions. The solutions are close to self-similar throughout the
integration, from very close to the black hole to the region where gravity is
unimportant. The field lines are tied to the conditions very close to the
central object and eventually overcollimate, possibly leading to a collimation
shock. This collimation shock could provide the conditions for diffusive shock
acceleration, leading to the observed electron populations with a power-law
energy distribution in jets.
  We provide the derivation, a detailed analysis of a solution, and describe
the effects the parameters have on the properties of the solutions, such as the
Lorentz factor and location of the collimation shock. We also discuss the
deviations from self-similarity.
  By comparing the new gravity term with the gravity term obtained from a
non-relativistic formalism in a previous work, we show they are equivalent in
the non-relativistic limit. This equivalence shows the approach taken in that
work is valid and allows us to comment on its limitations.
