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We construct multipole moments for stationary, asymptotically flat, spacetimesolutions to higher-order curvature theories of gravity. The moments aredefined using $3+1$ techniques involving timelike Killing vector constructionsas in the classic papers by Geroch and Hansen. Using the fact that theKerr-Newman metric is a vacuum solution to a particular class of $f(R)$theories of gravity, we compute all its moments, and find that they admitrecurrence relations similar to those for the Kerr solution in generalrelativity. It has been proposed previously that modelling the measuredfrequencies of quasi-periodic oscillations from galactic microquasars enablesexperimental tests of the no-hair theorem. We explore the possibility that,even if the no-hair relation is found to break down in the context of generalrelativity, there may be an $f(R)$ counterpart that is preserved. We apply theresults to the microquasars GRS $1915$+$105$ and GRO J$1655$-$40$ using thediskoseismology and kinematic resonance models, and constrain the spins and`charges' [which are not really electric charges in the $f(R)$ context] oftheir black holes.

Testing modified gravity and no-hair relations for the Kerr-Newman metric through quasi-periodic oscillations of galactic microquasars