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

Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)

Convection Heat Transfer

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Gravitation And Cosmology

Wind energy is the fastest growing alternate source of energy in the world since its purely economic potential is complemented by its great positive environmental impact. The wind turbine, whether it may be a Horizontal-Axis Wind Turbine (HAWT) or a Vertical-Axis Wind Turbine (VAWT), offers a practical way to convert the wind energy into electrical or mechanical energy. Although this book focuses on the aerodynamic design and performance of VAWTs based on the Darrieus concept, it also discusses the comparison between HAWTs and VAWTs, future trends in design and the inherent socio-economic and environmental friendly aspects of wind energy as an alternate source of energy.

Wind turbine design with emphasis on Darrieus concept Ion Paraschivoiu

Motivated by recent attempts to solve the cosmological constant problem, we examine the observational consequences of a vacuum energy which decays in time. In both radiation and matter dominated eras, the ratio of the vacuum to the total energy density of the universe must be small. Although the vacuum cannot provide the “missing mass” required to close the universe today, its presence earlier in the history of the universe could have important consequences. Element abundances from primordial nucleosynthesis require the ratio x = ϱ vac /( ϱ vac + ϱ rad ) ⩽ 0.1 of neutrino (or equivalent light) species to exceed N ν > 4, a case ruled out in the standard cosmological model. If the vacuum decays into low energy photons, the lack of observed spectral distortions in the microwave background gives tighter bounds, x −4 . In the matter-dominated era, the presence of a vacuum term may allow more time for growth of protogalactic perturbations.

Cosmology with Decaying Vacuum Energy

An entropic-force scenario, i.e., entropic cosmology, assumes that the horizon of the Universe has an entropy and a temperature. In the present paper, in order to examine entropic cosmology, we derive entropic-force terms not only from the Bekenstein entropy but also from a generalized black-hole entropy proposed by Tsallis and Cirto [Eur. Phys. J. C 73, 2487 (2013)]. Unlike the Bekenstein entropy, which is proportional to area, the generalized entropy is proportional to volume because of appropriate nonadditive generalizations. The entropic-force term derived from the generalized entropy is found to behave as if it were an extra driving term for bulk viscous cosmology, in which a bulk viscosity of cosmological fluids is assumed. Using an effective description similar to bulk viscous cosmology, we formulate the modified Friedmann, acceleration, and continuity equations for entropic cosmology. Based on this formulation, we propose two entropic-force models derived from the Bekenstein and generalized entropies. In order to examine the properties of the two models, we consider a homogeneous, isotropic, and spatially flat universe, focusing on a single-fluid-dominated universe. The two entropic-force models agree well with the observed supernova data. Interestingly, the entropic-force model derived from the generalized entropy

Entropic cosmology for a generalized black-hole entropy

Cosmological equations were recently derived by Padmanabhan from the expansion of cosmic space due to the difference between the degrees of freedom on the surface and in the bulk in a region of space. In this study, a modified Renyi entropy is applied to Padmanabhan’s ‘holographic equipartition law’, by regarding the Bekenstein–Hawking entropy as a nonextensive Tsallis entropy and using a logarithmic formula of the original Renyi entropy. Consequently, the acceleration equation including an extra driving term (such as a time-varying cosmological term) can be derived in a homogeneous, isotropic, and spatially flat universe. When a specific condition is mathematically satisfied, the extra driving term is found to be constant-like as if it is a cosmological constant. Interestingly, the order of the constant-like term is naturally consistent with the order of the cosmological constant measured by observations, because the specific condition constrains the value of the constant-like term.

/pdf/cosmological-model-from-the-holographic-equipartition-law-1xbq4u40nj.pdf

Cosmological model from the holographic equipartition law with a modified Rényi entropy

We study two types of entropic-force models in a homogeneous, isotropic, spatially flat, matter-dominated universe. The first type is a $\mathrm{\ensuremath{\Lambda}}(t)$ type similar to $\mathrm{\ensuremath{\Lambda}}(t)\mathrm{CDM}$ (varying-lambda cold dark matter) models in which both the Friedmann equation and the acceleration equation include an extra driving term. The second type is a BV type similar to bulk viscous models in which the acceleration equation includes an extra driving term but the Friedmann equation does not. In order to examine the two types systematically, we consider an extended entropic-force model that includes a Hubble parameter ($H$) term and a constant term in entropic-force terms. The $H$ term is derived from a volume entropy, whereas the constant term is derived from an entropy proportional to the square of an area entropy. Based on the extended entropic-force model, we examine four models obtained from combining the $H$ and constant terms with the $\mathrm{\ensuremath{\Lambda}}(t)$ and BV types. The four models agree well with the observed supernova data and describe the background evolution of the late universe properly. However, the evolution of first-order density perturbations is different in each of the four models, especially for low redshift, assuming that an effective sound speed is negligible. The $\mathrm{\ensuremath{\Lambda}}(t)$ type is found to be consistent with the observed growth rate of clustering, in contrast with the BV type examined in this study. A unified formulation is proposed as well, in order to systematically examine density perturbations of the two types.

Evolution of the universe in entropic cosmologies via different formulations

Entropic cosmology assumes several forms of entropy on the horizon of the universe, where the entropy can be considered to behave as if it were related to the exchange (the transfer) of energy. To discuss this exchangeability, the consistency of the two continuity equations obtained from two different methods is examined, focusing on a homogeneous, isotropic, spatially flat, and matter-dominated universe. The first continuity equation is derived from the first law of thermodynamics, whereas the second equation is from the Friedmann and acceleration equations. To study the influence of forms of entropy on the consistency, a phenomenological entropic-force model is examined, using a general form of entropy proportional to the $n$th power of the Hubble horizon. In this formulation, the Bekenstein entropy (an area entropy), the Tsallis-Cirto black-hole entropy (a volume entropy), and a quartic entropy are represented by $n=2$, 3, and 4, respectively. The two continuity equations for the present model are found to be consistent with each other, especially when $n=2$, i.e., the Bekenstein entropy. The exchange of energy between the bulk (the universe) and the boundary (the horizon of the universe) should be a viable scenario consistent with the holographic principle.

/pdf/general-form-of-entropy-on-the-horizon-of-the-universe-in-37555xriqx.pdf

General form of entropy on the horizon of the universe in entropic cosmology

In ``entropic cosmology,'' instead of a cosmological constant $\ensuremath{\Lambda}$, an extra driving term is added to the Friedmann equation and the acceleration equation, taking into account the entropy and the temperature on the horizon of the universe. By means of the modified Friedmann and acceleration equations, we examine a non-adiabatic-like accelerated expansion of the universe in entropic cosmology. In this study, we consider a homogeneous, isotropic, and spatially flat universe, focusing on the single-fluid- (single-component-) dominated universe at late times. To examine the properties of the late universe, we solve the modified Friedmann and acceleration equations, neglecting high-order corrections for the early universe. We derive the continuity (conservation) equation from the first law of thermodynamics, assuming nonadiabatic expansion caused by the entropy and temperature on the horizon. Using the continuity equation, we formulate the generalized Friedmann and acceleration equations, and propose a simple model. Through the luminosity distance, it is demonstrated that the simple model agrees well with both the observed accelerated expansion of the Universe and a fine-tuned standard $\ensuremath{\Lambda}\mathrm{CDM}$ (lambda cold dark matter) model. However, we find that the increase of the entropy for the simple model is likely uniform, while the increase of the entropy for the standard $\ensuremath{\Lambda}\mathrm{CDM}$ model tends to become gradually slower, especially after the present time. In other words, the simple model predicts that the present time is not a special time, unlike for the prediction of the standard $\ensuremath{\Lambda}\mathrm{CDM}$ model.

Nobuyoshi Komatsu

Papers

Entropic cosmology for a generalized black-hole entropy

Cosmological model from the holographic equipartition law with a modified Rényi entropy

Evolution of the universe in entropic cosmologies via different formulations

General form of entropy on the horizon of the universe in entropic cosmology

Non-adiabatic-like accelerated expansion of the late universe in entropic cosmology