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.

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Energy conversion by gravitational waves

THE fundamental postulate of the special theory of relativity, and a vital principle of the general theory of relativity, is that there exists a maximum speed of propagation, confining the link between cause and effect to the surface or interior of the light cone. It is important to realize that, particularly in the special theory, this postulate is used essentially not in the mathematical but in the logical sense. That is to say, it is asserted that causal links and links of information outside the light cone do not exist, rather than that such links are more difficult to describe mathematically than links on or within the light cone. This logical assertion is used to the full in the denial of the possibility of clock synchronization.

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Relativity and Indeterminacy

This paper studies time sequences of axially symmetric static configurations which can be continuously deformed into each other. We find the restriction which characterizes those time sequences which are physically allowed. This restriction can be interpreted as the law governing the non-radiative, or near field, transfer of gravitational energy, - m = ∯ P . d S and helps to clarify the concept of mass-energy in general relativity. We consider only the regions of space surrounding a source where, for quasi-static systems, the exact, static, empty space solutions of Weyl and Levi-Civita are good approximations at all times, and exact whenever the motion stops. Our results are then valid for arbitrarily strong fields. The restriction on the time sequences can be expressed as a restriction on the time dependence of the multipole moments A l , B l (which are defined by the Weyl and Levi Civita solutions), becoming then -d/d t [ A 0 + ½ G ∑ l = 0 ∞ (2 l + 1) A l B l ] = ½ G ∑ l = 0 ∞ ( A l B l - A l B l ). The right-hand side of this expression corresponds exactly to the flux of energy across a surface, as given by Bondi’s Newtonian Poynting vector P = (1/8 π G ) ( ϕ ∇ ϕ - ϕ ∇ ϕ ). It is natural to identify A 0 + ½ G ∑ l = 0 ∞ (2 l + 1) A l B l with the total mass-energy m enclosed by the surface. m can also be expressed as a surface intergal. Our restriction was derived from the vanishing of a surface integral in much the same manner as the equations of motion are derived in post-Newtonain approximations. However, it holds in arbitrarily strong fields, whereas the usual post-Newtonian methods use in lowest order a trivial solution (flat space) and can only be used in weak fields.

Hermann Bondi

Papers

Energy conversion by gravitational waves

Relativity and Indeterminacy

Transfer of energy in general relativity.

Keeping a balance

Rival Theories of Cosmology