In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature$\frac{{h\kappa }}{{2\pi k}} \approx 10^{ - 6} \left( {\frac{{M_ \odot }}{M}} \right){}^ \circ K$ where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases whereS is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.

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Particle Creation by Black Holes

We introduce the Virgo Consortium's EAGLE project, a suite of hydrodynamical simulations that follow the formation of galaxies and black holes in representative volumes. We discuss the limitations of such simulations in light of their finite resolution and poorly constrained subgrid physics, and how these affect their predictive power. One major improvement is our treatment of feedback from massive stars and AGN in which thermal energy is injected into the gas without the need to turn off cooling or hydrodynamical forces, allowing winds to develop without predetermined speed or mass loading factors. Because the feedback efficiencies cannot be predicted from first principles, we calibrate them to the z~0 galaxy stellar mass function and the amplitude of the galaxy-central black hole mass relation, also taking galaxy sizes into account. The observed galaxy mass function is reproduced to ≲0.2 dex over the full mass range, 108<M∗/M⊙≲1011, a level of agreement close to that attained by semi-analytic models, and unprecedented for hydrodynamical simulations. We compare our results to a representative set of low-redshift observables not considered in the calibration, and find good agreement with the observed galaxy specific star formation rates, passive fractions, Tully-Fisher relation, total stellar luminosities of galaxy clusters, and column density distributions of intergalactic CIV and OVI. While the mass-metallicity relations for gas and stars are consistent with observations for M∗≳109M⊙, they are insufficiently steep at lower masses. The gas fractions and temperatures are too high for clusters of galaxies, but for groups these discrepancies can be resolved by adopting a higher heating temperature in the subgrid prescription for AGN feedback. EAGLE constitutes a valuable new resource for studies of galaxy formation.

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The EAGLE project: Simulating the evolution and assembly of galaxies and their environments

Extended Theories of Gravity

This paper is divided into four parts. In part A, some general considerations about gravitational radiation are followed by a treatment of the scalar wave equation in the manner later to be applied to Einstein’s field equations. In part B, a co-ordinate system is specified which is suitable for investigation of outgoing gravitational waves from an isolated axi-symmetric reflexion-symmetric system . The metric is expanded in negative powers of a suitably defined radial co-ordinate r , and the vacuum field equations are investigated in detail. It is shown that the flow of information to infinity is controlled by a single function of two variables called the news function. Together with initial conditions specified on a light cone, this function fully defines the behaviour of the system . No constraints of any kind are encountered. In part C, the transformations leaving the metric in the chosen form are determined. An investigation of the corresponding transformations in Minkowski space suggests that no generality is lost by assuming that the transformations, like the metric, may be expanded in negative powers of r . In part D, the mass of the system is defined in a way which in static metrics agrees with the usual definition. The principal result of the paper is then deduced, namely, that the mass of a system is constant if and only if there is no news; if there is news, the mass decreases monotonically so long as it continues. The linear approximation is next discussed, chiefly for its heuristic value, and employed in the analysis of a receiver for gravitational waves. Sandwich waves are constructed, and certain non-radiative but non-static solutions are discussed. This part concludes with a tentative classification of time-dependent solutions of the types considered.

Gravitational Waves in General Relativity. VII. Waves from Axi-Symmetric Isolated Systems

The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the postNewtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many nonlinear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail eects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary’s orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.

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Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries.

Plane gravitational waves are here defined to be non-flat solutions of Einstein’s empty spacetime field equations which admit as much symmetry as do plane electromagnetic waves, namely, a 5-parameter group of motions. A general plane-wave metric is written down and the properties of plane wave space-times are studied in detail. In particular, their characterization as 4 plane ’ is justified further by the construction of 4 sandwich waves ’ bounded on both sides by (null) hyperplanes in flat space-time. It is shown that the passing of a sandwich wave produces a relative acceleration in free test particles, and inferred from this that such waves transport energy.

Gravitational waves in general relativity III. Exact plane waves

Truth and Beauty: Aesthetics and Motivations in Science. By S. Chandrasekhar University of Chicago Press: 1987. Pp. 170. $23.95, £19.25

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The good, the bad and the ugly

Earlier ideas associating an invariant integral of the energy invariant with the number of nucleons in a gravitating body are shown to be fallacious, and thus do not provide a means of following through the contraction of such a body. It is shown how the full field equations of general relativity give a feasible and rigorous method of examining contracting models. Schwarzschild-type co-ordinates are introduced and are used to examine the slow adiabatic contraction of a sphere of constant density. The particle paths are found and the pressure-density relation permitting such slow adiabatic contraction is examined. It is shown that the simple 4/3 power law of Newtonian theory has to be replaced by a steeper dependence of pressure on density for high gravitational potentials. Radiation co-ordinates are introduced to examine radiating contracting systems, and equations fully specifying such a system are obtained. A simple example is given in outline to illustrate the method.

The contraction of gravitating spheres

POLARIZED plane gravitational waves were first discovered by N. Rosen1, who, however, came to the conclusion that such waves could not exist because the metric would have to contain certain physical singularities. More recent work by Taub2 and McVittie3 showed that there were no unpolarized plane waves, and this result has tended to confirm the view that true plane gravitational waves do not exist in empty space in general relativity. Partly owing to this, Scheidegger4 and I5 have both expressed the opinion that there might be no energy-carrying gravitational waves at all in the theory. It is therefore of interest to point out, as was first shown by Robinson6 and has now been independently proved by me, that Rosen's argument is invalid and that true gravitational waves do in fact exist. Moreover, it is shown here that these waves carry energy, although it has not yet been possible to relate the intensity of the wave to the amount of energy carried.

Plane Gravitational Waves in General Relativity

The exact relativistic form of the equation of hydrostatic support by an isotropic pressure is found in an especially convenient form. A quantity u , the natural generalization of Schwarzschild’s m/r ratio at the surface, is used, and it is proved that the critical value u = ½ cannot be attained except perhaps under conditions of severe tension. It is shown that if u is everywhere less than different physical restrictions are significant in well-defined inner and outer zones as far as the attainment of high u values is concerned. A number of limiting configurations are derived for physically significant restrictions. In particular it is shown that if the density is nowhere negative then u u

Hermann Bondi

Papers

Gravitational waves in general relativity III. Exact plane waves

The good, the bad and the ugly

The contraction of gravitating spheres

Plane Gravitational Waves in General Relativity

Massive spheres in general relativity