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Geometric inequalities for axially symmetric black holes
 Classical and Quantum Gravity
"... A geometric inequality in General Relativity relates quantities that have both a physical interpretation and a geometrical definition. It is well known that the parameters that characterize the KerrNewman black hole satisfy several important geometric inequalities. Remarkably enough, some of these ..."
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A geometric inequality in General Relativity relates quantities that have both a physical interpretation and a geometrical definition. It is well known that the parameters that characterize the KerrNewman black hole satisfy several important geometric inequalities. Remarkably enough, some of these inequalities also hold for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse, they are closed related with the cosmic censorship conjecture. Axially symmetric black holes are the natural candidates to study these inequalities because the quasilocal angular momentum is well defined for them. We review recent results in this subject and we also describe the main ideas behind the proofs. Finally, a list of relevant open problem is presented. 1
Inital data for perturbed Kerr black holes on hyperboloidal slices
, 2013
"... Abstract. We construct initial data corresponding to a single perturbed Kerr black hole in vacuum. These data are defined on specific hyperboloidal (“ACMC”) slices on which the mean extrinsic curvature K asymptotically approaches a constant at future null infinity I +. More precisely, we require th ..."
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Abstract. We construct initial data corresponding to a single perturbed Kerr black hole in vacuum. These data are defined on specific hyperboloidal (“ACMC”) slices on which the mean extrinsic curvature K asymptotically approaches a constant at future null infinity I +. More precisely, we require that K obeys the Taylor expansion K = K0 + O(σ4) where K0 is a constant and σ describes a compactified spatial coordinate such that I + is represented by σ = 0. We excise the singular interior of the black hole and assume a marginally outer trapped surface as inner boundary of the computational domain. The momentum and Hamiltonian constraints are solved by means of pseudospectral methods and we find exponential rates of convergence of our numerical solutions. Some physical properties of the initial data are studied with the calculation of the Bondi Mass, together with a multipole decomposition of the horizon. We probe the standard picture of gravitational collapse by assessing a family of Penroselike inequalities and discuss in particular their rigidity aspects. Dynamical evolutions are planned in a future project.