  A pair plasma in hidrostatic equilibrium near the horizon of a Schwarzschild
black hole is an exact state. In this work, we show that an exact
non--hidrostatic equilibrium state can be sustained for a pair plasma, in which
its constituents are moving near the horizon of the black hole. This
equilibrium state solution is not due to a vortical formalism, but by the
balance of forces in the plasma. This new equilibrium state has a pressure for
the hydrostatic equilibrium, although the plasma moves. The equilibrium is
achieved by the generation of a magnetic field that emerges by the interplay
between the plasma dynamics and the gravitational field, having no flat
spacetime analogue. No electric fields are needed to sustain the equilibria.
The magnetic field shows an exponential decaying behavior, governed by a
function that depends on the properties of the pair plasma and the black, and
that plays the role of a characteristc length of the dynamics of the magnetic
field.
