The grand challenges of contemporary fundamental physics—dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem—all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on ‘Black holes, Gravitational waves and Fundamental Physics’.

Transitional millisecond pulsars (tMSPs) switch, on roughly multi-year timescales, between rotation-powered radio millisecond pulsar (RMSP) and accretion-powered low-mass X-ray binary (LMXB) states. The tMSPs have raised several questions related to the nature of accretion flow in their LMXB state and the mechanism that causes the state switch. The discovery of coherent X-ray pulsations from PSRJ1023+0038 (while in the LMXB state) provides us withthe first opportunity to perform timing observationsand to compare the neutron star's spin variation during this state to the measured spin-down in the RMSP state. Whereas the X-ray pulsations in the LMXB state likely indicate that some material is accreting onto the neutron star's magnetic polar caps, radio continuum observations indicate thepresence of an outflow. The fraction of the inflowing material being ejected is not clear, but it may be much larger than that reaching the neutron star's surface. Timing observations can measure the total torque on the neutron star. We have phase-connected nine XMM-Newton observations of PSRJ1023 +0038over the last 2.5 years of the LMXB stateto establish a precise measurement of spin evolution. We find that the average spin-down rate as an LMXB is 26.8±0.4% faster than the rate (−2.39×10 −15 Hz s −1 ) determined during the RMSP state. This shows that negative angular momentum contributions (dipolar magnetic braking, andoutflow) exceed positive ones (accreted material), and suggests that the pulsar wind continues to operate at a largely unmodified level. We discuss implications of this tight observational constraint in the context of possible accretion models.