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This book overviews the extensive literature on apparent cosmological and black hole horizons.

In theoretical gravity, dynamical situations such as gravitational collapse, black hole evaporation, and black holes interacting with non-trivial environments, as well as the attempts to model gravitational waves occurring in highly dynamical astrophysical processes, require that the concept of event horizon be generalized. Inequivalent notions of horizon abound in the technical literature and are discussed in this manuscript.

The book begins with a quick review of basic material in the first one and a half chapters, establishing a unified notation. Chapter 2 reminds the reader of the basic tools used in the analysis of horizons and reviews the various definitions of horizons appearing in the literature. Cosmological horizons are the playground in which one should take baby steps in understanding horizon physics. Chapter 3 analyzes cosmological horizons, their proposed thermodynamics, and several coordinate systems. The remaining chapters discuss analytical solutions of the field equations of General Relativity, scalar-tensor, and f(R) gravity which exhibit time-varying apparent horizons and horizons which appear and/or disappear in pairs. An extensive bibliography enriches the volume.

The intended audience is master and PhD level students and researchers in theoretical physics with knowledge of standard gravity.

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Stationary black holes in General Relativity.- Introduction.- Stationary black holes of General Relativity.- Schwarzschild spacetime.- Reissner-Nordström metric.- Kerr spacetime.- Kerr-Newman metric.- Energy conditions.- Conclusions.- Problems.- Horizons.- Introduction.- Null geodesic congruences and trapped surfaces.- Rindler horizons for accelerated observers in Minkowski spacetime.- Event horizons.- Killing horizons.- Apparent horizons.- Trapping horizons.- Isolated and dynamical horizons.- Kodama vector and surface gravity.- Surface gravities.- Spherical symmetry.- Rindler horizons revisited.- Conclusions.- Problems.- Cosmological horizons.- Introduction.- 3.2 Hyperspherical coordinates for FLRW space.- Kruskal-Szekeres coordinates for de Sitter space.- Painlev e-Gullstrand and Schwarzschild-like coordinates for k = 0 FLRW space.- Schwarzschild-like coordinates for general FLRW spaces.- Painlevé-Gullstrand coordinates for general FLRW spaces.- Congruences of radial null geodesics in FLRW space.- Horizons in FLRW space.- Dynamics of cosmological horizons.- Another notation.- de Sitter space.- Thermodynamics of cosmological horizons in General Relativity.- Thermodynamics of de Sitter space.- Thermodynamics of apparent/trapping horizons in FLRW space.- Conclusions.- Problems.- Inhomogeneities in cosmological "backgrounds" in Einstein theory.- Introduction.- Schwarzschild-de Sitter-Kottler spacetime.- McVittie solution.- Charged McVittie spacetime.- An application to the quantization of black hole areas.- Generalized McVittie spacetimes.- Sultana-Dyer spacetime.- Husain-Martinez-Nu¤ez spacetime.- Fonarev solutions.- Other analytic cosmological black hole solutions of the Einstein Equations.- Conclusions.- Cosmological inhomogeneities in alternative gravity.- Introduction.- Brans-Dicke cosmological black holes.- f (R) cosmological black holes.- Conclusions.- A Appendix.- A.1 Painlevé-Gullstrand coordinates for general spherically symmetric metrics.- A.2 Kodama vector in FLRW space.- A2.1 Pseudo-Painlev e-Gullstrand coordinates.- A.2.2 Comoving coordinates.- References.- Index.

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