  The X-ray emission of neutron stars and black holes presents a rich
phenomenology that can lead us to a better understanding of their nature and to
address more general physics questions: Does general relativity apply in the
strong gravity regime? Is spacetime around black holes described by the Kerr
metric? This white paper considers how we can investigate these questions by
studying reverberation mapping and quasi-periodic oscillations in accreting
systems with a combination of high-spectral and high-timing resolution. In the
near future, we will be able to study compact objects in the X-rays in a new
way: advancements in transition-edge sensors (TES) technology will allow for
electron-volt-resolution spectroscopy combined with nanoseconds-precision
timing.
