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This work devises a formalism to obtain the equations of motion for a blackhole-fluid configuration. Our approach is based on a Post-Newtonian expansionand adapted to scenarios where obtaining the relevant dynamics requires longtime-scale evolutions. These systems are typically studied with Newtonianapproaches, which have the advantage that larger time-steps can be employedthan in full general-relativistic simulations, but have the downside thatimportant physical effects are not accounted for. The formalism presented hereprovides a relatively straightforward way to incorporate those effects inexisting implementations, up to 2.5PN order, with lower computational coststhan fully relativistic simulations.

A Post-Newtonian approach to black hole-fluid systems