  To perform realistic tests of theories of gravity, we need to be able to look
beyond general relativity and evaluate the consistency of alternative theories
with observational data from, especially, gravitational wave detections using,
for example, an agnostic Bayesian approach. In this paper we further examine
properties of one class of such viable, alternative theories, based on metrics
arising from ungauged supergravity. In particular, we examine the massless,
neutral, minimally coupled scalar wave equation in a general stationary,
axisymmetric background metric such as that of a charged rotating black hole,
when the scalar field is either time independent or in the low-frequency,
near-zone limit, with a view to calculating the Love numbers of tidal
perturbations, and of obtaining harmonic coordinates for the background metric.
For a four-parameter family of charged asymptotically flat rotating black hole
solutions of ungauged supergravity theory known as STU black holes, which
includes Kaluza-Klein black holes and the Kerr-Sen black hole as special cases,
we find that all time-independent solutions, and hence the harmonic coordinates
of the metrics, are identical to those of the Kerr solution. In the
low-frequency limit we find the scalar fields exhibit the same $SL(2,R)$
symmetry as holds in the case of the Kerr solution. We point out extensions of
our results to a wider class of metrics, which includes solutions of
Einstein-Maxwell-Dilaton theory.
