  Fully relativistic and causal equations for the flow of charge in curved
spacetime are derived. It is believed that this is the first set of equations
to be published that correctly describes the flow of charge, and evolution of
the electromagnetic field, in highly dynamical relativistic environments on
time scales much shorter than the collapse time ($GM/c^3$). The equations will
be important for correctly investigating problems such as the dynamical
collapse of magnetized stellar cores to black holes and the production of jets
and gravitational waves. This system of equations, given the name of `charge
dynamics', is analogous to those of hydrodynamics (which describe the flow of
{\em mass} in spacetime rather than the flow of charge). The most important one
in the system is the relativistic generalized Ohm's law, which is used to
compute time-dependent four-current. Unlike previous equations for the current,
this one ensures that charge drift velocities remain less than the speed of
light, takes into account the finite current rise time, is expressed in a
covariant form, and is suitable for general relativistic computations in an
arbitrary metric. It includes the standard known effects (Lorentz force, Hall
effect, pressure effect, and resistivity) and reduces to known forms of Ohm's
law in the appropriate limits. In addition, the plasma particles are allowed to
have highly relativistic drift velocities, resulting in an implicit equation
for the `current beaming factor' $\gamma_q$.
