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 spectrum of known black-hole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all black-hole-equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the Einstein-Maxwell system.

The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature. The existence of smooth hyperboloidal initial data for Einstein's field equations is demonstrated.

Abstract. We present a set of global invariants, called "mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the "boundary at infinity" has spherical topology one single invariant is obtained, called the mass; we show positivity thereof. We apply the definition to conformally compactifiable manifolds, and show that the mass is completion-independent. We also prove the result, closely related to the problem at hand, that conformal completions of conformally compactifiable manifolds are unique.Introduction.

Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularityfree, vacuum space-times which are stationary in a neighborhood of i 0 ; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global I ; we prove existence of initial data for many black holes which are exactly Kerr -or exactly Schwarzschild -both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to r −m terms, for any fixed m, and with multipole moments freely prescribable within certain ranges. * Partially supported by a Polish Research Committee grant 2 P03B 073 24; email piotr@ gargan.math.univ-tours.fr † Visiting Scientist. Permanent address: Département de Mathématiques, Faculté des Sciences, Parc de Grandmont, F37200 Tours, France. ‡ Partially supported by the ACI program of the French Ministry of Research; email delay@ gargan.math.univ-tours.fr Lemma 2.3 Suppose that dim M ≥ 2, then P * ′ (x, ξ) is injective for ξ = 0.Proof: We define a linear map α from the space S 2 of two-covariant symmetric tensors into itself by the formula α(S) = S − (tr S)g .(2.6)Let ξ = 0, if (Y, N ) is in the kernel of P * ′ (x, ξ) then α(ξ (i Y j) ) = 0 , so that ξ (i Y j) = 0, and Y = 0. It follows that α(ξ i ξ j )N = 0 , which implies N = 0. ✷ The lemma implies:2 See Appendix A for the definitions of the function spaces we use.one is led toWe have thus showed that for C 2 compactly supported vector fields we have 16) and it should be clear that this remains true for vector fields which are only differentiable once. To continue, we use (2.16) with Y replaced with φψY ; the hypothesis that Ric (g) ∈ φ −2 L ∞ φ allows us to write |b(φψY )| + 2α(∇ (i (φψY j) )) L 2 + ψY L 2 ≥ c ∇(φψY ) L 2 .We have 2α(∇ (i (φY j) )) L 2 ψ = 2α(ψ∇ (i (φY j) )) L 2

Abstract. We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of time functions remains true on domains of dependence with continuous metrics, and that C 0,1 differentiability of the metric suffices for many key results of the smooth causality theory.PACS numbers: 04.20 Gz arXiv:1111.0400v3 [gr-qc] May 2012On Lorentzian causality with continuous metrics 2 Causality for continuous metricsOne of the factors that constrains the differentiability requirements of the proof of the celebrated Choquet-Bruhat-Geroch theorem [1], of existence and uniqueness of maximal globally hyperbolic vacuum developments of general relativistic initial data, is the degree of differentiability needed to carry out the Lorentzian causality arguments that arise. Here one should keep in mind that classical local existence and uniqueness of solutions of vacuum Einstein equations in dimension 3 + 1 applies to initial data (g, K) in the product of Sobolev spaces H s × H s−1 for s > 5/2, and that the recent studies of the Einstein equations [2][3][4][5][6][7][8] assume even less differentiability. On the other hand, the standard references on causality seem to assume smoothness of the metric [9][10][11][12][13][14], while the presentation in [15,16] requires C 2 metrics. ‡ Hence the need to revisit the causality theory for Lorentzian metrics which are merely assumed to be continuous. Surprisingly enough, some standard facts of the C 2 theory fail to hold for metrics with lower differentiability. For example, we will show that the following statements are wrong:(i) light-cones are topological hypersurfaces of codimension one;(ii) a piecewise differentiable causal curve which is not null everywhere can be deformed, with end points fixed, to a timelike curve.Concerning point (i) above, we exhibit metrics with light-cones which have nonempty interior.Researchers familiar with Lorentzian geometry will recognize point (ii) as an essential tool in many arguments. One needs therefore to reexamine the corresponding results, documenting their failure or finding a replacement for the proof.In the course of the analysis, one is naturally led to the notion of a causal bubble which, roughly speaking, is defined to be an open set which can be reached from, say a point p, by causal curves but not by timelike ones.The object of this paper is to present the above, reassessing that part of causality theory which has been presented in [16] from the perspective of continuous metrics. One of our main results is the proof that domains of dependence equipped with continuous metrics continue to admit Cauchy time functions. § Another key result is the observation that the causality theory developed in [16] for C 2 metrics remains true for C 0,1 metrics. An application of our work to the general relativistic...

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We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary dimensions. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.