Multipole calculation of gravitational forces

In this paper we introduce a method to directly calculate the Newtonian gravitational forces using multipole moments. Gravitational torques for precision tests of Newtonian gravitation are regularly calculated with multipole expansions due to the elegance and efficiency of the calculations. Tests of Newtonian gravity which probe forces rather than torques often resort to less efficient numerical calculation of sextuple integrals. Unlike multipole expansions these cannot easily be adapted for numerous permutations of the system, and instead the calculation has to be repeated, often in full. The method derived in this paper calculates the forces from any 1/r potential given the outer multipoles of the system and the inner multipoles calculated at any arbitrary point. The result derived can be written as a simple recursion relation for efficient calculation.