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’.

The tidal Love numbers (TLNs) encode the deformability of a self-gravitating object immersed in a tidal environment and depend significantly both on the object's internal structure and on the dynamics of the gravitational field. An intriguing result in classical general relativity is the vanishing of the TLNs of black holes. We extend this result in three ways, aiming at testing the nature of compact objects: (i) we compute the TLNs of exotic compact objects, including different families of boson stars, gravastars, wormholes, and other toy models for quantum corrections at the horizon scale. In the black-hole limit, we find a universal logarithmic dependence of the TLNs on the location of the surface; (ii) we compute the TLNs of black holes beyond vacuum general relativity, including Einstein-Maxwell, Brans-Dicke and Chern-Simons gravity; (iii) We assess the ability of present and future gravitational-wave detectors to measure the TLNs of these objects, including the first analysis of TLNs with LISA. Both LIGO, ET and LISA can impose interesting constraints on boson stars, while LISA is able to probe even extremely compact objects. We argue that the TLNs provide a smoking gun of new physics at the horizon scale, and that future gravitational-wave measurements of the TLNs in a binary inspiral provide a novel way to test black holes and general relativity in the strong-field regime. CONTENTS

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Anisotropic stresses are ubiquitous in nature, but their modeling in General Relativity is poorly understood and frame dependent. We introduce the first study on the dynamical properties of anisotropic self-gravitating fluids in a covariant framework. Our description is particularly useful in the context of tests of the black hole paradigm, wherein ultracompact objects are used as black hole mimickers but otherwise lack a proper theoretical framework. We show that: (i) anisotropic stars can be as compact and as massive as black holes, even for very small anisotropy parameters; (ii) the nonlinear dynamics of the 1 + 1 system is in good agreement with linearized calculations, and shows that configurations below the maximum mass are nonlinearly stable; (iii) strongly anisotropic stars have vanishing tidal Love numbers in the black-hole limit; (iv) their formation will usually be accompanied by gravitational-wave echoes at late times.