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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

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

The observable universe could be a 1 + 3-surface (the “brane”) embedded in a 1 + 3 + d-dimensional spacetime (the “bulk”), with Standard Model particles and fields trapped on the brane while gravity is free to access the bulk. At least one of the d extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (∼ TeV) level. This revolutionary picture arises in the framework of recent developments in M theory. The 1 + 10-dimensional M theory encompasses the known 1 + 9-dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity “leaks” into the bulk, behaving in a truly higher-dimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting and potentially testable implications for high-energy astrophysics, black holes, and cosmology. Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. This review discusses the geometry, dynamics and perturbations of simple brane-world models for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall-Sundrum models.

Brane-World Gravity

Treating the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the critical behaviour of charged AdS black holes. We complete the analogy of this system with the liquid-gas system and study its critical point, which occurs at the point of divergence of specific heat at constant pressure. We calculate the critical exponents and show that they coincide with those of the Van der Waals system.

P − V criticality of charged AdS black holes

In this work we probe the Einstein-Born-Infeld (EBI) black hole in the isolated horizon framework. It turns out that the EBI black hole is consistent with the heuristic model for colored black holes proposed by Ashtekar et al. [Class. Quantum Grav. 18, 919 (2001)] when one considers the Born-Infeld (BI) parameter playing the role of the color parameter. This remarkable coincidence points to a wide range of validity of the Ashtekar model proved, in this case, for black holes that go from Schwarzschild to Reissner-Nordstr\"om black holes, passing through a continuous set of nonlinear electromagnetic black holes with fixed mass and charge but with distinct BI parameter.

Born-Infeld black hole in the isolated horizon framework

We present a model for an inhomogeneous and anisotropic early universe filled with a nonlinear electromagnetic field of Born–Infeld (BI) type. The effects of the BI field are compared with the linear case (Maxwell). Since the curvature invariants are well behaved then we conjecture that our model does not present an initial big bang singularity. The existence of the BI field modifies the curvature invariants at t=0 as well as sets bounds on the amplitude of the conformal metric function.

Born-infeld cosmologies

It is known that for nonlinear electrodynamics the First Law of Black Hole Mechanics holds, however the Smarr’s formula for the total mass does not. In this contribution we discuss the point and determine the corresponding expressions for the Bardeen black hole solution that represents a nonlinear magnetic monopole. The same is done for the regular black hole solution derived by Ayon–Beato and Garcia [1], showing that in the case that variations of the electric charge are involved, the Smarr’s formula is no longer valid.

Smarr's formula for black holes with non-linear electrodynamics

In this paper we study the trajectories of test particles in a geometry that is the nonlinear electromagnetic generalization of the Reissner–Nordstrom black hole. The solution was originally derived by Garcia, Salazar and Plebanski (1984 Nuovo Cimento 84 65–90). The studied spacetime is an Einstein–Born–Infeld solution, nonsingular outside a regular event horizon and characterized by three parameters: mass M, charge Q and the Born–Infeld parameter b related to the magnitude of the electric field at the origin. Asymptotically it is a Reissner–Nordstrom solution.

Geodesic structure of the Born-Infeld black hole

It is known that for nonlinear electrodynamics the First Law of Black Hole Mechanics holds, however the Smarr's formula for the total mass does not. In this contribution we discuss the point and determine the corresponding expressions for the Bardeen black hole solution that represents a nonlinear magnetic monopole. The same is done for the regular black hole solution derived by Ayon-Beato and Garcia, showing that in the case that variations of the electric charge are involved, the Smarr's formula does not longer is valid.

Nora Bretón

Papers

Born-Infeld black hole in the isolated horizon framework

Born-infeld cosmologies

Smarr's formula for black holes with non-linear electrodynamics

Geodesic structure of the Born-Infeld black hole

Smarr's formula for black holes with non-linear electrodynamics