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’.

We show that gravity theories involving disformally transformed metrics in their matter coupling lead to spontaneous growth of various fields in a similar fashion to the spontaneous scalarization scenario in scalar-tensor theories. Scalar-dependent disformal transformations have been investigated in this context, and our focus is understanding the transformations that depend on more general fields. We show that vector-dependent disformal couplings can be obtained in various different ways, each leading to spontaneous vectorization as indicated by the instabilities in linearized equations of motion. However, we also show that spontaneous growth is not evident beyond vectors. For example, we could not identify a spontaneous growth mechanism for a spinor field through disformal transformations, even though there is a known example for conformal transformations. This invites further work on the fundamental differences between the two types of metric transformations. We argue that our results are relevant for observations in strong gravity such as gravitational wave detections due to their promise of large deviations from general relativity in this regime.

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A Hamilton-Jacobi formalism for thermodynamics was formulated by Rajeev [Ann. Phys. 323, 2265] based on the contact structure of the odd dimensional thermodynamic phase space. This allows one to derive the equations of state of a family of substances by solving a Hamilton-Jacobi equation (HJE). In the same work it was applied to chargeless non-rotating black holes, and the use of Born-Infeld electromagnetism was proposed to apply it to charged black holes as well. This paper fulfills this suggestion by deriving the HJE for charged non-rotating black holes using Born-Infeld theory and a negative cosmological constant. The most general solution of this HJE is found. It is shown that there exists solutions which are distinct from the equations of state of the Einstein-Born-Infeld-AdS black hole. The meaning of these solutions is discussed.

Following earlier works of Dereli and collaborators, we study a three dimensional toy model where we extend the topologically massive gravity with electrodynamics by the most general RF 2-type non-minimal coupling terms. Here R denotes the possible curvature terms and F denotes the electromagnetic 2-form. We derive the variational feld equations and look for exact solutions on constant negative curvature space-times with a constant, self-dual electromagnetic feld. The notion of self-dual electromagnetic felds in three dimensions is introduced by Dereli and collaborators in the study of exact solutions of models with gravity-electromagnetism couplings. We note the conditions that the parameters of the model have to satisfy for these self-dual solutions to exist.