  Particle-in-cell codes usually represent large groups of particles as a
single macroparticle. These codes are computationally efficient but lose
information about the internal structure of the macroparticle. To improve the
accuracy of these codes, this work presents a method in which, as well as
tracking the macroparticle, the moments of the macroparticle are also tracked.
Although the equations needed to track these moments are known, the coordinate
transformations for moments where the space and time coordinates are mixed
cannot be calculated using the standard method for representing moments. These
coordinate transformations are important in astrophysical plasma, where there
is no preferred coordinate system. This work uses the language of Schwartz
distributions to calculate the coordinate transformations of moments. Both the
moment tracking and coordinate transformation equations are tested by modelling
the motion of uncharged particles in a circular orbit around a black hole in
both Schwarzschild and Kruskal-Szekeres coordinates. Numerical testing shows
that the error in tracking moments is small, and scales quadratically. This
error can be improved by including higher order moments. By choosing an
appropriate method for using these moments to deposit the charge back onto the
grid, a full particle-in-cell code can be developed.
