Gravitational waves from extreme mass ratio inspirals: Challenges in mapping the spacetime of massive, compact objects

In its final year of inspiral, a stellar mass ($1 - 10 M_\odot$) body orbits a massive ($10^5 - 10^7 M_\odot$) compact object about $10^5$ times, spiralling from several Schwarzschild radii to the last stable orbit. These orbits are deep in the massive object's strong field, so the gravitational waves that they produce probe the strong field nature of the object's spacetime. Measuring these waves can, in principle, be used to ``map'' this spacetime, allowing observers to test whether the object is a black hole or something more exotic. Such measurements will require a good theoretical understanding of wave generation during inspiral. In this article, I discuss the major theoretical challenges standing in the way of building such maps from gravitational-wave observations, as well as recent progress in producing extreme mass ratio inspirals and waveforms.