QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced. The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.

"relative State" Formulation of Quantum Mechanics

These lectures are based on a book “THE RELATIVISTIC NUCLEAR MANY-BODY PROBLEM” written with Brian Serot which will appear as volume 16 of the series Advances in Nuclear Physics edited by J. W. Negele and E. Vogt [R1]. I am distributing copies of the table of contents.

The Relativistic Nuclear Many-Body Problem

Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

Part 1 Foundations: introduction the era of decentralization. Part 2 Constructions: constructionism LEGO/logo StarLogo objects and parallelism. Part 3 Explorations: simulations and stimulations slime mould artificial ants traffic jams termites turtles and frogs turtle ecology new turtle geometry forest fire recursive trees. Part 4 Reflections: the centralized mindset beyond the centralized mindset. Part 5 Projections: growing up. Appendices: student participants StarLogo overview.

Turtles, Termites, and Traffic Jams: Explorations in Massively Parallel Microworlds

On the possible role of gravity in the reduction of the wave function

In addition to the Riemannian metricization of the thermodynamic state space, local relaxation times offer a natural time scale, too. Generalizing existing proposals, we relate a thermodynamic time scale to the standard kinetic coefficients of irreversible thermodynamics. The notion of thermodynamic speed is generalized to higher dimensions. Criteria for minimum entropy production in slow, slightly irreversible processes are discussed. Euler–Lagrange equations are derived for optimum thermodynamic control for fixed clock time period as well as for fixed thermodynamic time period. It is emphasized that the correct derivation of the principle of constant thermodynamic speed, proposed earlier by others, requires the entropy minimization at fixed thermodynamic time instead of clock‐time. Most remarkably, optimum paths are Riemannian geodesics which would not be the case had we used ordinary time. To interpret thermodynamic time, an easy‐to‐implement stepwise algorithm is constructed to realize control at constant thermodynamic speed. Thermodynamic time is shown to correspond to the number of steps, and the sophisticated task of determining thermodynamic time in real control problems is achieved by measuring ordinary intensive variables.

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Thermodynamic length, time, speed, and optimum path to minimize entropy production

On the minimum uncertainty of space-time geodesics

Single-particle spectra and two-particle Bose-Einstein correlation functions are determined analytically utilizing a self-similar solution of nonrelativistic hydrodynamics for ellipsoidally symmetric, expanding fireballs, by assuming that the symmetry axes of the ellipsoids are tilted in the frame of the observation. The directed, elliptic, and third flows are calculated analytically. The mass dependences of the slope parameters in the principal directions of the expansion, together with the mass and angular dependences of the HBT radius parameters, reflect directly the ellipsoidal properties of the flow.

B. Lukács

Papers

On the possible role of gravity in the reduction of the wave function

Thermodynamic length, time, speed, and optimum path to minimize entropy production

On the minimum uncertainty of space-time geodesics

Observables and initial conditions for self-similar ellipsoidal flows

The quark-nucleon phase diagram and quantum chromodynamics