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This chapter looks at the detection of black holes through gravitational waves. While further improvements can be expected in the ability to detect and measure electromagnetic radiation, it is possible that the next great advances in observational astrophysics will come from the detection of other kinds of information altogether. Currently, there is a great excitement about the possibility of directly detecting an entirely new “celestial messenger,” namely, gravitational radiation. The existence of gravitational waves is a prediction of general relativity, and current technology is very close to being able to detect them directly. The strongest sources of gravitational radiation are expected to be merging black holes. Since such mergers are expected to occur, both between stellar-mass and supermassive black holes, the detection of gravitational radiation would provide a new way not only to explore gravitational physics but also to look for and to study celestial black holes.

This chapter begins with a discussion of Newton's gravity law. It then covers general relativity, observations and experiments, Einstein's equations, field sources, Lagrangians, fluid sources, Newtonian approximation, Minkowskian approximation, high-frequency gravitational waves, and coupled electromagnetic and gravitational waves.

In the weak-field limit Einstein's equation can be linearized and it takes on form of the familiar wave equation. Gravitational waves may be viewed as ripples of curvature propagating in a background of flat spacetime. The strategy of detecting such tidal forces by a gravitational wave interferometer is outlined. The rate of energy loss due to the quadrupole radiation by a circulating binary system is calculated, and found in excellent agreement with the observed orbit decay rate of the Hulse-Taylor binary pulsar.

This chapter discusses how gravitational waves emerge from general relativity, and what their properties are. The most straightforward approach is ‘linearized theory’, where the Einstein equations are expanded around the flat Minkowski metric. It is shown how a wave equation emerges and how the solutions can be put in an especially simple form by an appropriate gauge choice. Using standard tools of general relativity such as the geodesic equation and the equation of the geodesic deviation, how these waves interact with a set of test masses is detailed. The energy and momentum carried by GWs are then computed and discussed. This chapter approaches the problem from a geometric point of view, identifying the energy-momentum tensor of GWs from their effect on the background curvature. Finally, GW propagation in curved space is discussed.

Gravitational waves are one of the most important physical phenomena associated with the presence of strong and dynamic gravitational fields, and as such they are of great interest in numerical relativity. There are two main approaches to the extraction of gravitational wave information from a numerical simulation. Traditional approach has been based on the theory of perturbations of a Schwarzschild spacetime developed originally by Regge and Wheeler, Zerilli, and a number of other authors, and later recast as a gauge invariant framework by Moncrief. In recent years, however, it has become increasingly common in numerical relativity to extract gravitational wave information in terms of the components of the Weyl curvature tensor with respect to a frame of null vectors, using what is known as the Newman-Penrose formalism. This chapter presents a brief introduction to both these approaches, and describes how to calculate the energy and momentum radiated by gravitational waves in each case.

This chapter draws on the treatment of progressive waves for non-linear equations used in Sections III.12 and III.13 to construct weak gravitational and electromagnetic waves on a given electrovac Einsteinian spacetime. Topics covered include quasilinear systems, quasilinear first-order systems, progressive waves in relativistic fluids, quasilinear quasidiagonal second-order systems, non quasidiagonal second-order systems, fields and equations, and strong gravitational waves.

This Chapter contains a review of many important aspects of modern black hole theory and its applications. It begins with a general definition of a (not‐necessary stationary) black hole and formulation of the most important results on generic properties of black holes, including the Penrose theorem on the structure of the event horizon, the Hawking theorems on the topology and area of the event horizon and black hole uniqueness theorems. Gravitational radiation from black holes in a binary system and modern status and perspectives of the gravitation waves search from black holes and other compact sources are discussed. We also describe black hole models proposed for the explanation of the gamma‐ray bursts. Modeling of black hole properties, in particular their Hawking radiation, in the laboratory experiments is reviewed. We also discuss recent models with large extra dimensions and possibility of micro black hole creation in the collider experiments. This subject is directly connected with the problem of the higher dimensional black holes. Higher dimensional generalization of the Kerr metric, and a variety of new exact solutions for higher dimensional black objects with the non‐spherical topology of the horizon are discussed. The Chapter ends with remarks on two closely related problems on the wormhole and ‘time machine’ existence. It is shown that in order to create and support macroscopic objects of this type a new exotic form of the matter is requires. It seams that this and possible instabilities make the existence of such objects questionable at least at the present state of our knowledge. These and other fascinating open problems are still wait for their solution.

This chapter focuses on how gamma-ray bursts (GRBs) are emerging as unique tools in the study of broad areas of astronomy and physics by virtue of their special properties. The unassailable fact about GRBs that makes them such great probes is that they are fantastically bright and so can be seen to the farthest reaches of the observable Universe. In parallel with the ongoing study of GRB events and progenitors, new lines of inquiry have burgeoned: using GRBs as unique probes of the Universe in ways that are almost completely divorced from the nature of GRBs themselves. Topics discussed include studies of gas, dust, and galaxies; the history of star formation; measuring reionization and the first objects in the universe; neutrinos, gravitational waves, and cosmic rays; quantum gravity and the expansion of the universe; and the future of GRBs.

This chapter addresses one of the key themes of modern general relativity: gravitational waves. In 1916, Einstein performed the first relevant calculations, including a first derivation of the celebrated quadrupole formula, which contained a mistake that he corrected in 1918. The corrected formula became the basis for the long-term search and the first observational verification of these waves. These calculations provoked a discussion about the reality of these waves, which continued well into the 1950s and beyond. What is worth emphasizing is how Einstein's predominant interest in this phenomenon, which developed immediately after the completion of his general theory, had faded away completely by the time he delivered the Princeton lectures.

In this Chapter we ‘derive’ the Einstein equations, study their weak field approximation, and briefly discuss gravitational waves and their generation. We finish the Chapter by remarks on the higher dimensional gravity, spacetimes with large extra dimensions, and their properties.

This chapter reviews the 40-year history that led to the first detection of gravitational waves, and goes on to outline techniques which will allow the detectors to be substantially improved. Following a review of the gravitational wave spectrum and the early attempts at detection, it emphasizes the theme of optomechanics, and the underlying physics of parametric transducers, which creates a connection between early resonant bar detectors and modern interferometers and techniques for enhancing their sensitivity. Developments are presented in an historical context, while themes and connections between earlier and later work are emphasized.

Civita criticized Einstein’s papers on gravitational waves: their energy momentum is frame dependent and therefore does not fit the covariance of Einstein’s gravity theory. Infeld and Rosen did not believe gravitational waves existed, and Einstein changed his mind on their existence repeatedly. Others did believe in them, such as Fock and Feynman. Weber constructed his “Weber bar” to detect gravitational waves, but when he claimed success, he was criticized. He then proposed using a Michelson-Morley type of interferometer with lasers to detect gravitational waves, as did Weiss. Merging black holes and neutron stars were proposed as detectable sources of gravitational waves. Taylor and Hulse, using the large Arecibo radio telescope, indirectly detected gravitational waves from inspiraling neutron stars. Primordial gravitational waves, still emanating from the Big Bang, were claimed to have been detected by BICEP2, but the waves were eventually shown to be a result of foreground dust.

A subject closely related to equations of motion is the gravitational physics of few body systems. First, the chapter reviews choreographic solutions for the N‐body problem in Newton's theory of gravity. A solution is called choreographic if each massive particle moves periodically in a single closed orbit. One such choreographic solution is a stable figure‐eight orbit for a three‐body system. The chapter describes connections between the choreographic solutions and the gravitational waves that are predicted by Einstein's theory of general relativity and thus general relativistic choreography is discussed. General relativistic effects cause, for example to binary orbits, a complicated flower‐like pattern by the periastron advance. It is not trivial whether general relativistic effects admit a choreographic solution such as the figure eight. At least at the first and second post Newtonian orders (neglecting dissipative effects starting at the second and half post Newtonian order) the tricky figure eight remarkably remains true.

At a press conference on February 11, 2016, David Reitz, LIGO Executive Director, announced, “We did it!” They detected gravitational waves for the first time. Both LIGO sites, in Washington state and Louisiana, registered the incoming gravitational waves from two black holes colliding and merging far away. Over the following months, more mergers were detected. Gravitational waves are caused by the acceleration of a massive object, which stretches and compresses spacetime in a wave-like motion that is incredibly small and difficult to detect. Numerical relativity research over decades has produced over a quarter of a million template solutions of Einstein’s equations. The best template fit to the wave form data identifies the masses and spins of the two merging black holes. Much of this chapter describes the technology of the LIGO apparatus. On October 3, 2017, Barish, Thorne, and Weiss, the founders of LIGO, received the Nobel Prize for Physics.

This chapter presents the physical motivation for general relativity, derives the Einstein field equation and gives concise derivations of the main results of the theory. It begins with the equivalence principle, tidal forces in Newtonian gravity and their connection to curved spacetime geometry. This leads to a derivation of the field equation. Tests of general relativity are considered: Mercury’s perihelion advance, gravitational redshift, the deflection of starlight and gravitational lenses. The exterior and interior Schwarzschild solutions are discussed. Eddington–Finkelstein coordinates are used to describe objects falling into non-rotating black holes. The Kerr metric is used to describe rotating black holes and their astrophysical consequences. Gravitational waves are described and used to explain the orbital decay of binary neutron stars. Their recent detection by LIGO and the beginning of a new era of gravitational wave astronomy is discussed. Finally, the gravitational field equations are derived from the Einstein–Hilbert action.

This chapter provides a brief survey of gravitational-wave astronomy, including the recent recent breakthrough detection. It sets the stage for the rest of the book via simple back-of-the-envelope estimates for different sets of sources. The chapter also describes the first detection of a black hole merger (GW150914) as well as the first observed neutron star binary event (GW170817) and introduces some of the ideas required to understand these breakthroughs.

The last chapter of the book deals with physical systems whose conditions require the solution both of the Einstein equations and of the hydrodynamics equations. The first examples considered are those of stationary isolated stars, including gravastars and rotating stars, followed by the analysis of compact stars collapsing to a black hole, which are treated both through the dust solution of Oppenheimer–Snyder and through fluid solutions. Since the nonlinearity and complexity of the equations that need to be solved make it increasingly difficult to obtain analytic solutions, the role of numerical simulations becomes increasingly important. Numerical simulations are indeed crucial for the investigation of complex systems such as neutron-star binaries and black-hole–neutron-star binaries, which are treated with an eye on their possible detection through the emission of gravitational waves.