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 title immediately brings to mind a standard reference of almost the same title [1]. The authors are quick to point out the relationship between these two works: they are complementary. The purpose of this work is to explain what is known about a selection of exact solutions. As the authors state, it is often much easier to find a new solution of Einstein's equations than it is to understand it. Even at first glance it is very clear that great effort went into the production of this reference. The book is replete with beautifully detailed diagrams that reflect deep geometric intuition. In many parts of the text there are detailed calculations that are not readily available elsewhere. The book begins with a review of basic tools that allows the authors to set the notation. Then follows a discussion of Minkowski space with an emphasis on the conformal structure and applications such as simple cosmic strings. The next two chapters give an in-depth review of de Sitter space and then anti-de Sitter space. Both chapters contain a remarkable collection of useful diagrams. The standard model in cosmology these days is the ICDM model and whereas the chapter on the Friedmann-Lema?tre?Robertson?Walker space-times contains much useful information, I found the discussion of the currently popular a representation rather too brief. After a brief but interesting excursion into electrovacuum, the authors consider the Schwarzschild space-time. This chapter does mention the Swiss cheese model but the discussion is too brief and certainly dated. Space-times related to Schwarzschild are covered in some detail and include not only the addition of charge and the cosmological constant but also the addition of radiation (the Vaidya solution). Just prior to a discussion of the Kerr space-time, static axially symmetric space-times are reviewed. Here one can find a very interesting discussion of the Curzon?Chazy space-time. The chapter on rotating black holes is rather brief and, for example, does not contain reference to the insights found by Pretorius and Israel [2]. This is perhaps justifiable in view of the many specialized texts devoted to the Kerr space-time (e.g. [3]). The large clear diagrams that one becomes accustomed to in this book show off the Taub-NUT (and related) space-times in the next chapter. After perhaps a somewhat standard discussion of stationary axially symmetric space-times, there is a very informative discussion of accelerating black holes. For example, the global structure of the C-metric is considered in detail. This is followed by a brief discussion of solutions for uniformly accelerating particles. The discussion of the Pleba?ski-Demia?ski solutions contains two very useful flow charts that help to systematize two rather complex families of solutions. After a somewhat brief discussion of plane and pp-waves, the authors give an extensive discussion of the Kunt solutions. I note here that after this text was in production the importance of the Kunt space-times as regards the characterization of space-times by scalar curvature invariants was made clear [4]. The discussion of the Robinson-Trautman solutions that follows is extensive, containing, for example, details of the singularity structure and of the global structure. The final formal chapter in this text covers colliding plane waves. This contains, for example, discussions of the Khan?Penrose, Ferrari?Iba?ez and Chandrasekhar?Xanthopoulos solutions. The text ends with a `final miscellany'. This covers a number of interesting topics, but I found the discussion of the Lema?tre?Tolman solutions rather weak (compare e.g. [5]). The book has two quite useful appendices covering 2-spaces and 3-spaces of constant curvature. To conclude, I will quote from the dust jacket: `The book is an invaluable resource for both graduate students and academic researchers working in gravitational physics'. I highly recommend it. References [1] Stephani H, Kramer D, MacCallum M, Hoenselaers C and Herlt E 2003 Exact Solutions of Einstein's Field Equations (Second Edition) (Cambridge: Cambridge University Press) [2] Pretorius F and Israel W 1998 Class. Quantum Grav.15 2289 [3] Wiltshire D, Visser M and Scott S (ed) 2008 The Kerr Spacetime: Rotating Black Holes in General Relativity (Cambridge: Cambridge University Press) [4] Coley A, Hervik S and Pelavas N 2009 Class. Quantum Grav. 26 025013 [5] Pleba?ski J and Krasi?ski A 2006 An Introduction to General Relativity and Cosmology (Cambridge: Cambridge University Press)

Exact Space-Times in Einstein's General Relativity

Constraints on dark matter density and axion mass from the large-scale structure of spacetime

Dynamical horizons are considered in full, nonlinear general relativity. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local, the energy flux is positive, and change in the horizon area is related to these fluxes. The flux formulas also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black-hole mechanics.

Dynamical Horizons: Energy, Angular Momentum, Fluxes and Balance Laws

An emergent gravity metric incorporating $k-$essence scalar fields $\phi$ having a Born-Infeld type lagrangian is mapped into a metric whose structure is similar to that of a blackhole of large mass $M$ that has swallowed a global monopole. However, here the field is not that of a monopole but rather that of a $k-$essence scalar field. If $\phi_{emergent}$ be solutions of the emergent gravity equations of motion under cosmological boundary conditions at $\infty$, then for $r\rightarrow\infty$ the rescaled field $\frac {\phi_{emergent}}{2GM-1}$ has exact correspondence with $\phi$ with $\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)$. The Hawking temperature of this metric is $T_{\mathrm emergent}= \frac{\hbar c^{3}}{8\pi GM k_{\mathrm B}}(1-K)^{2}\equiv \frac{\hbar}{8\pi GM k_{\mathrm B}}(1-K)^{2}$, taking the speed of light $c=1$. Here $K=\dot\phi_{2}^{2}$ is the kinetic energy of the $k-$essence field $\phi$ and $K$ is always less than unity, $k_{\mathrm B}$ is the Boltzmann constant. This is phenomenologically interesting in the context of Belgiorno {\it et al's} gravitational analogue experiment.

The Hawking temperature in the context of dark energy

We show that the Hawking temperature is modified in the presence of dark energy in an emergent gravity scenario for Kerr–Newman(KN) and Kerr–Newman–AdS(KNAdS) background metrics. The emergent gravity metric is not conformally equivalent to the gravitational metric. We calculate the Hawking temperatures for these emergent gravity metrics along $$\theta =0$$
 . Also we show that the emergent black hole metrics are satisfying Einstein’s equations for large r and $$\theta =0$$
 . Our analysis is done in the context of dark energy in an emergent gravity scenario having $$k-$$
 essence scalar fields $$\phi $$
 with a Dirac–Born–Infeld type Lagrangian. In KN and KNAdS background, the scalar field $$\phi (r,t)=\phi _{1}(r)+\phi _{2}(t)$$
 satisfies the emergent gravity equations of motion at $$r\rightarrow \infty $$
 for $$\theta =0$$
 .

/pdf/the-hawking-temperature-in-the-context-of-dark-energy-for-2ocovq9e0o.pdf

The Hawking temperature in the context of dark energy for Kerr-Newman and Kerr-Newman-AdS backgrounds

We establish a formal connection between the $k$-essence emergent gravity scenario and generalizations of Vaidya spacetime. Choosing the $k$-essence action to be of the Dirac-Born-Infeld variety, the physical spacetime to be a general static spherically symmetric black hole, and restricting the $k$-essence scalar field to be a function solely of the advanced or the retarded time, we show that the emergent gravity metric resembles closely the generalized Vaidya metrics for null fluid collapse proposed by Husain. Imposing null energy conditions on the emergent energy-momentum tensor derived from the emergent Einstein equation, restrictions are obtained on the functions characterizing the emergent metric for consistent identification with generalized Vaidya spacetimes. We discuss the possibility of dynamical horizons in the $k$-essence emergent Vaidya spacetime. Admissible explicit black hole background metrics are discussed as examples.

k -essence emergent spacetime as a generalized Vaidya geometry

The k-essence emergent Reissner–Nordstrom–de Sitter spacetime has exactly mapped on to the Robinson–Trautman (RT)-type spacetime with cosmological constant $$\varLambda $$ for certain configuration of k-essence scalar field. Theoretically, we evaluated that the thermodynamical quantities for the RT-type emergent black hole is different from the usual one in the presence of kinetic energy of the k-essence scalar field i.e., the dark energy density. We restrict ourselves into the fact that the dark energy density (K) is to be unity, then the effective temperature and pressure both are negative for the RT-type emergent black hole which implies that the system is thermodynamically unstable when the charge $$Q\ne 0$$, and the emergent spacetime is only dark energy dominated, and it does not radiate when $$Q=0$$. The thermodynamically unstable situation is physically plausible only when we consider spin degrees of freedom of a system. We have made this analysis in the context of dark energy in an emergent gravity scenario having k-essence scalar fields $$\phi $$ with a Dirac–Born–Infeld-type Lagrangian. The scalar field also satisfies the emergent equation of motion at $$r\rightarrow \infty $$.

Thermodynamics for the k-essence emergent Reissner-Nordstrom-de Sitter spacetime

We show that the Hawking temperature is modified in the presence of dark energy in an emergent gravity scenario for Kerr-Newman(KN) and Kerr-Newman-AdS(KNAdS) background metrics. The emergent gravity metric is not conformally equivalent to the gravitational metric. We calculate the Hawking temperatures for these emergent gravity metrics along $\theta=0$. Also we show that the emergent black hole metrics are satisfying Einstein's equations for large $r$ and $\theta=0$. Our analysis is done in the context of dark energy in an emergent gravity scenario having $k-$essence scalar fields $\phi$ with a Dirac-Born-Infeld type lagrangian. In KN and KNAdS background, the scalar field $\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)$ satisfies the emergent gravity equations of motion at $r\rightarrow\infty$ for $\theta=0$.

/pdf/the-hawking-temperature-in-the-context-of-dark-energy-for-3dzuri0w4b.pdf

We study the Hawking radiation with the dynamical horizon in the {\bf K-}essence Vaidya geometry. By considering the {\bf K-}essence action to be of the Dirac-Born-Infeld variety, the physical spacetime to be a general static spherically symmetric black hole, and by restricting the {\bf K-}essence scalar field to be a function solely of the advanced or the retarded time, Manna et. al. have established the connection between the {\bf K-}essence emergent gravity scenario and generalizations of Vaidya spacetime. Based on modified definition of the dynamical horizon by Sawayama, we investigate the Hawking effect in the {\bf K-}essence Vaidya Schwarzschild spacetime. Especially, we investigate the Hawking Radiation in the two ways, by using the dynamical horizon equation and using the tunneling formalism. The results are different from the usual Vaidya spacetime geometry.

Bivash Majumder

Papers

The Hawking temperature in the context of dark energy for Kerr-Newman and Kerr-Newman-AdS backgrounds

k -essence emergent spacetime as a generalized Vaidya geometry

Thermodynamics for the k-essence emergent Reissner-Nordstrom-de Sitter spacetime

The Hawking temperature in the context of dark energy for Kerr-Newman and Kerr-Newman-AdS backgrounds

Hawking radiation with the dynamical horizon in the K-essence emergent Vaidya spacetime