Generalized Stefan-Boltzmann Law Research Articles

Quantum gravity phenomenology and the blackbody radiation

Abstract We analyze the blackbody radiation problem in the presence of quantum gravity effects encoded in modified dispersion relations. The spectral radiance and the generalized Stefan-Boltzmann law are studied in this context. Furthermore, the regime of low temperatures is also contemplated, where features related to the blackbody thermal laws and the thermodynamic quantities such as energy, pressure, entropy, and specific heat are obtained. Possible implications in compact objects such as neutron stars are also discussed.

Information paradox in a Kerr-Newman black hole under generalized Hawking radiation

Analogous to the calculation method of the interior volume inside a spherically symmetric black hole, we calculate the interior volume of a Kerr-Newman black hole. After that, we propose the generalized Stefan-Boltzmann law which can be used to investigate Hawking radiation with energy, charge and angular momentum. Based on it, the proportional relation between the entropy of the scalar field in the interior volume of the Kerr-Newman black hole and Bekenstein-Hawking entropy under generalized Hawking radiation has been investigated. Comparing to Hawking radiation without charge and angular momentum, we find that no matter the particles radiated from the black hole take charge and angular momentum or not, the two types of entropy remain the same proportional relation expression. Furthermore, the proportionality coefficient of the two types of entropy has been analyzed and discussed. It is found that the two types of entropy are approximately proportional to each other in an infinitesimal process except the late stage of Hawking radiation. Moreover, the proportional relation can degenerate to the Schwarzschild case when the charge and angular momentum of the black hole completely disappear. It illustrates that, for a Kerr-Newman black hole, different from Hawking radiation carrying only energy, the extremal black hole will technically not prevent and stop the evaporation anymore under generalized Hawking radiation, and the proportional relation obtained is applicable to investigate the evolution relation between the two types of entropy under an arbitrary Hawking radiation. Finally, based on the evolution of the proportional relation under generalized Hawking radiation, the black hole information paradox is brought up again and discussed more deeply.

Generalized Stefan–Boltzmann Law

We reconsider the thermodynamic derivation by L. Boltzmann of the Stefan law and we generalize it for various different physical systems {\it whose chemical potential vanishes}. Being only based on classical arguments, therefore independent of the quantum statistics, this derivation applies as well to the saturated Bose gas in various geometries as to "compensated" Fermi gas near a neutrality point, such as a gas of Weyl Fermions. It unifies in the same framework the thermodynamics of many different bosonic or fermionic non-interacting gases which were until now described in completely different contexts.

On the radiative and thermodynamic properties of the cosmic radiations using COBE FIRAS instrument data: III. Galactic far-infrared radiation

Using the three-component spectral model describing the FIRAS average continuum spectra, the exact analytical expressions for thermodynamic and radiative functions of Galactic far-infrared radiation are obtained. The COBE FIRAS instrument data in the 0.15–2.88 THz frequency interval at the mean temperatures of T1 = 17.72 K, T2 = 14 K and T3 = 6.73 K are used for calculating the radiative and thermodynamic functions, such as the total radiation power per unit area, total energy density, total emissivity, number density of photons, Helmholtz free energy density, entropy density, heat capacity at constant volume and pressure for the warm, intermediate-temperature and very cold components of the Galactic continuum spectra. The generalized Stefan-Boltzmann law for warm, intermediate-temperature and very cold components is constructed. The temperature dependence of each component is determined by the formula IS–B(T) = σ′T6. This result is important when we construct the cosmological models of radiative transfer that can be applied inside the Galaxy. Within the framework of the three-component spectral model, the total number of photons in our Galaxy and the total radiation power (total luminosity) emitted from a surface of the Galaxy are calculated. Their values are NGtotal = 1.3780 × 1068 and IGtotal(T) = 1.0482 × 1036 W. Other radiative and thermodynamic properties of the Galactic far-infrared radiation (photon gas) of the Galaxy are calculated. The expressions for astrophysical parameters, such as the entropy density/Boltzmann constant and number density of the Galactic far-infrared photons are obtained. We assume that the obtained analytical expressions for thermodynamic and radiative functions may be useful for describing the continuum spectra of the far-infrared radiation for other galaxies.

A Cosmological Model Based on a Quadratic Equation of State Unifying Vacuum Energy, Radiation, and Dark Energy

We consider a cosmological model based on a quadratic equation of state (where is the Planck density and is the cosmological density) “unifying” vacuum energy, radiation, and dark energy. For , it reduces to leading to a phase of early accelerated expansion (early inflation) with a constant density equal to the Planck density g/m3 (vacuum energy). For , we recover the equation of state of radiation . For , we get leading to a phase of late accelerated expansion (late inflation) with a constant density equal to the cosmological density g/m3 (dark energy). The temperature is determined by a generalized Stefan-Boltzmann law. We show a nice “symmetry” between the early universe (vacuum energy + radiation) and the late universe (radiation + dark energy). In our model, they are described by two polytropic equations of state with index and respectively. Furthermore, the Planck density in the early universe plays a role similar to that of the cosmological density in the late universe. They represent fundamental upper and lower density bounds differing by 122 orders of magnitude. We add the contribution of baryonic matter and dark matter considered as independent species and obtain a simple cosmological model describing the whole evolution of the universe. We study the evolution of the scale factor, density, and temperature. This model gives the same results as the standard CDM model for , where is the Planck time and completes it by incorporating the phase of early inflation in a natural manner. Furthermore, this model does not present any singularity at and exists eternally in the past (although it may be incorrect to extrapolate the solution to the infinite past). Our study suggests that vacuum energy, radiation, and dark energy may be the manifestation of a unique form of “generalized radiation.” By contrast, the baryonic and dark matter components of the universe are treated as different species. This is at variance with usual models (quintessence, Chaplygin gas, ...) trying to unify dark matter and dark energy.

Generalized Wien’s Displacement Law in Determining the True Temperature of $$\text{ ZrB }_{2}$$ ZrB 2 –SiC-Based Ultrahigh-Temperature Ceramic: Thermodynamics of Thermal Radiation

The temperature dependence of the generalized Wien displacement law is investigated. For determining the true temperature of a \(\text{ ZrB }_{2}\)–SiC-based ultrahigh-temperature ceramic, the experimental values of the position of the maximum of the spectral density power are needed. Thermodynamics of the thermal radiation of \(\text{ ZrB }_{2}\)–SiC is constructed by using the temperature dependence of the generalized Stefan–Boltzmann law. The calculated values of the normal total emissivity for \(\text{ ZrB }_{2}\)–SiC at different temperatures are in good agreement with experimental data. The total radiation power emitted from a surface of \(\text{ ZrB }_{2}\)–SiC specimens at different temperatures is calculated. The temperature dependences of the Helmholtz free energy, entropy, heat capacity at constant volume, pressure, enthalpy, and internal energy of the thermal radiation of \(\text{ ZrB }_{2}\)–SiC are obtained. For determining the true temperature, experimental values of either the normal total emissivity or the normal total energy density are needed. The uncertainty in the determination of the true temperature is no greater than 1 %. A new universality class of bodies with a new relationship between the temperature \(T\) and the position of the spectral energy density maximum is established.

Instantaneous Radiation Energy Flux and Radiation Power Near the Event Horizon of Slowly Changing Vaidya Black Hole

By the thin film model of the black hole and the assumption of the local thermal equilibrium, the instantaneous radiation energy flux and radiation power of the slowly changing Vaidya black hole have been studied. The result has been obtained that the thermal radiation of the Vaidya black hole satisfies the generalized Stefan-Boltzmann law. When the cut-off distance and the thin film thickness are both fixed, the instantaneous radiation energy flux of the scalar field near the event horizon of the Vaidya black hole is not only related to the black hole mass, but also to the rate of the change of its event horizon and the average effusion velocity of the radiation particles in the thin film. While its instantaneous radiation power is related to the rate of the change of the event horizon and the average radial effusion velocity of the radiation particles in the thin film. These results indicate that the gravitational field around the black hole and the change of its event horizon will both affect the thermal radiation of the black hole.

The Generalized Stefan-Boltzmann Law of Slowly Changing Kerr-Newman Black Hole

Using the entropy density near the event horizon of Kerr-Newman black hole, the instantaneous radiation energy flux and the instantaneous radiation power of the slowly changing Kerr-Newman black hole have been studied. It is found that the thermal radiation of the Kerr-Newman black hole always satisfies the generalized Stefan-Boltzmann law and is affected by the gravitational field, the electromagnetic field around the black hole and the change of black hole event horizon. But the rate of the change of the event horizon usually makes very little affect on the instantaneous radiation energy flux and radiation power. Only when the rate of the change of the event horizon approaches to the light speed, it can make obviously affect on them.

Radiative Properties of Stoichiometric Hafnium, Titanium, and Zirconium Carbides: Thermodynamics of Thermal Radiation

The true temperatures of the thermal radiation of stoichiometric hafnium, titanium, and zirconium carbides are defined from the generalized Wien displacement law. It is shown that Wien’s displacement law for the investigated stoichiometric carbides decreases linearly with increasing temperature. The uncertainties in the determination of the true temperature are no greater than 1 %. For determining the true temperature of stoichiometric carbides, the experimental values of the position of the maximum of the spectral density power are needed. By extrapolating the generalized Wien displacement laws in the ultra-high-temperature region, the positions of the maximum of the normal energy density of hafnium, titanium, and zirconium carbides at melting temperatures are obtained. Thermodynamics of thermal radiation of stoichiometric carbides is constructed by using the temperature dependences of the generalized Stefan–Boltzmann law. The calculated values of the normal total emissivity for the investigated carbides at different temperatures are in good agreement with experimental data. For determining the true temperatures of the thermal radiation of stoichiometric carbides, experimental values of either the normal total emissivity or the normal total energy density are needed. The temperature dependences of the Helmholtz free energy, entropy, heat capacity at constant volume, pressure, enthalpy, and internal energy of the thermal radiation of stoichiometric carbides at high temperature are obtained. It is shown that thermodynamic function values increase with increasing temperature as a power law.

Radiation Energy Flux and Radiation Power of Schwarzschild Black Hole

Applying the entropy density near the event horizon, we obtained the result that the radiation energy flux of the black hole is always proportional to the quartic of the temperature of its event horizon. That is to say, the thermal radiation of the black hole always satisfies the generalized Stefan–Boltzmann law. The derived generalized Stefan–Boltzmann coefficient is no longer a constant. When the cut-off distance and the thin film thickness are both fixed, it is a proportional coefficient which is related to the black hole mass, the kinds of radiation particles and space–time metric near the event horizon. In this paper, we have put forward a thermal particle model in curved space–time. By this model, the result has been obtained that when the thin film thickness and the cut-off distance are both fixed, the radiation energy flux received by observer far away from the Schwarzschild black hole is proportional to the average radial effusion velocity of the radiation particles in the thin film, and inversely proportional to the square of the distance between the observer and the black hole.

The generalized Stefan–Boltzmann law of a rectilinear non-uniformly accelerating Kinnersley black hole

Using entropy density of Dirac field near the event horizon of a rectilinear non-uniformly accelerating Kinnersley black hole, the law for the thermal radiation of black hole is studied and the instantaneous radiation energy density is obtained. It is found that the instantaneous radiation energy density of a black hole is always proportional to the quartic of the temperature on event horizon in the same direction. That is to say, the thermal radiation of a black hole always satisfies the generalized Stefan–Boltzmann law. In addition, the derived generalized Stefan–Boltzmann coefficient is no longer a constant, but a dynamic coefficient related to the space–time metric near the event horizon and the changing rate of the event horizon in black holes.

Generalized area law under multiparameter rotating black hole spacetime

We study the statistical mechanics of quantum scalar fields under multiparameter rotating black hole spacetime in arbitrary D dimensions. The method of analysis is general because the metric does not depend on explicit black hole solutions. A generalized Stefan–Boltzmann law for scalar fields is derived by properly considering the allowed energy region. Then a generalized area law for scalar field entropy is derived by introducing an invariant regularization parameter in Rindler spacetime. The derived area law is applied to Kerr–AdS black holes in four and five dimensions. The thermodynamic implications are also discussed.

On the generalized blackbody distribution

We consider an application of the generalized statistical mechanics due to Tsallis to systems of quantum particles (fermions or bosons). Tsallis' approximation to finding the analytic quantum distribution function and its application to blackbody radiation are discussed. A generalized Stefan-Boltzmann law is proposed for the q < 1 case and the q dependence of the Stefan constant is derived.

Calculational Rules for Evaluating Transition Rates of Reactions Taking Place in a Closed System

Systematic calculational rules for evaluating the reaction rate of a generic dynamical process taking place in a closed system are presented. As applications, various reactions in the early universe and in the quark-gluon plasma are kept in mind. In the systems of practical interest, VE3 ⪢ 1 holds, with E the energy and V the volume of the system. In such cases, the reaction rate is found to be expanded in powers of l/(VE3)1/4, and each term of the series is written in terms of the corresponding finite-temperature reaction rate and the Helmholtz free energy at the temperature T, both of which have been extensively studied so far. The temperature T is related to E through the generalized Stefan-Boltzmann law. The rules are applied to the toy example, calculation of the production rate of a heavy scalar from a closed system of light scalars.

Stefan-Boltzmann law and correct statistics for the neutrino sea

It is argued that the Fermi-Dirac statistics is the correct statistics for the neutrino sea and yields the generalized StefanBoltzmann law for the neutrino power flow. (LBS)