Black Hole Radiation and Volume Statistical Entropy

The simplest possible equation for Hawking radiation and other black hole radiated power is derived in terms of black hole density, ρ . Black hole density also leads to the simplest possible model of a gas of elementary constituents confined inside a gravitational bottle of Schwarzchild radius at tremendous pressure, which yields identically the same functional dependence as the traditional black hole entropy S _bh∝ (kAc ³)/ℏ G. Variations of S _bh can be obtained which depend on the occupancy of phase space cells. A relation is derived between the constituent momenta and the black hole radius R _H, p = which is similar tothe Compton wavelength relation.     

Periodicity and area spectrum of the three dimensional BTZ black hole

Very recently, a new scheme to quantize the horizon area of a black hole has been proposed by Zeng and Liu et?al. In this paper, we further apply the analysis to investigate area spectrum of three dimensional BTZ black hole with the cosmological constant ${\Lambda=-1/l^{2}}$ . The results show that the area spectrum and entropy spectrum are independent of the cosmological constant. The area spectrum of the black hole is ${\Delta A=8\pi l_{P}^{2}}$ , which confirms the initial proposal of Bekenstein that the area spectrum is independent of the black hole parameters and the spacing is ${8\pi l_{P}^{2}}$ . This result also confirms the speculation of Maggiore that the periodicity of a black hole may be the origin of the area quantization. In addition, for the rotating and non-rotating BTZ black holes, we obtain the same entropy spectrum ${\triangle S=2\pi}$ , which is consistent with the result for other black holes. This implies that the entropy spectrum is more fundamental than the area spectrum.     

On the parametrisation of unitary matrices by the moduli of their elements

The parametrisation of ann×n unitary matrix by the moduli of its elements is not a well posed problem, i.e. there are continuous and discrete ambiguities which naturally appear. We show that the continuous ambiguity is (n–1)(n–3)-dimensional in the general case and in the symmetric caseS _ij=S_ij. We give also lower bounds on the number of discrete ambiguities, the number of solutions being at least in the first case and for the symmetric one, where [r] denotes the integral part ofr.     

Four Quantum Conservation Laws for Black Hole Stationary Equilibrium Radiation Processes

The classical first law of thermodynamics for a Kerr–Newman black hole (KNBH)is generalized to a law in quantum form on the event horizon. Then four quantumconservation laws on the KNBH equilibrium radiation process are derived. TheBekenstein–Hawking relation S = A/4 is exactly established. It can be inferredthat the classical entropy of black hole arises from the quantum entropy of fieldquanta or quasiparticles inside the hole.     

An analytical expression for the specific capacity of lithium ion cells

The concentration of lithium ions in the cathode of lithium ion cells has been obtained by solving the materials balance equation $$\frac{{\partial c}}{{\partial t}} = \varepsilon ^{1/2} D\frac{{\partial ^2 c}}{{\partial x^2 }} + \frac{{aj_n (1--t_ + )}}{\varepsilon }$$ by Laplace transform. On the assumption that the cell is fully discharged when there are zero lithium ions at the current collector of the cathode, the discharge timet _d is obtained as $$\tau = \frac{{r^2 }}{{\pi ^2 \varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{{r^2 }}\left( {\frac{{\varepsilon ^{1/2} }}{J} + \frac{{r^2 }}{6}} \right)} \right]$$ which, when substituted into the equationC=It _d /M, whereI is the discharge current andM is the mass of the separator and positive electrode, an analytical expression for the specific capacity of the lithium cell is given as $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

More corrections to the sixth-order anomalous magnetic moment of the muon

The contribution to the sixth-order muon anomaly from second-order electron vacuum polarization is determined analytically to orderm _e/m _μ. The result, including the contributions from graphs containing proper and improper fourth-order electron vacuum polarization subgraphs, is $$\begin{gathered} \left( {\frac{\alpha }{\pi }} \right)^3 \left\{ {\frac{2}{9}\log ^2 } \right.\frac{{m_\mu }}{{m_e }} + \left[ {\frac{{31}}{{27}}} \right. + \frac{{\pi ^2 }}{9} - \frac{2}{3}\pi ^2 \log 2 \hfill \\ \left. { + \zeta \left( 3 \right)} \right]\log \frac{{m_\mu }}{{m_e }} + \left[ {\frac{{1075}}{{216}}} \right. - \frac{{25}}{{18}}\pi ^2 + \frac{{5\pi ^2 }}{3}\log 2 \hfill \\ \left. { - 3\zeta \left( 3 \right) + \frac{{11}}{{216}}\pi ^4 - \frac{2}{9}\pi ^2 \log ^2 2 - \frac{1}{9}log^4 2 - \frac{8}{3}a_4 } \right] \hfill \\ + \left[ {\frac{{3199}}{{1080}}\pi ^2 - \frac{{16}}{9}\pi ^2 \log 2 - \frac{{13}}{8}\pi ^3 } \right]\left. {\frac{{m_e }}{{m_\mu }}} \right\} \hfill \\ \end{gathered} $$ . To obtain the total sixth-order contribution toa _μ?a _e, one must add the light-by-light contribution to the above expression.     

Secular-free solution up to third order of a model nonlinear equation for dispersive waves

The perturbation method of Lindstedt is applied to study the non linear effect of a nonlinear equation $$\nabla ^2 {\rm E} - \frac{1}{{c^2 }}\frac{{\partial ^2 {\rm E}}}{{\partial t^2 }} - \frac{{\omega _0^2 }}{{c^2 }}{\rm E} + \frac{{2v}}{{c^2 }}\frac{{\partial {\rm E}}}{{\partial t}} + E^2 \left[ {\frac{{\partial {\rm E}}}{{\partial t}} \times A} \right] = 0,$$ where (A. E)=0 andA,c, ω ₀ andν are constants in space and time. Amplitude dependent frequency shifts and the solution up to third order are derived.     

Three-loop on-shell charge renormalization without integration:\Lambda _{QED}^{\overline {MS} } to four loops

Using algebraic methods, we find the three-loop relation between the bare and physical couplings of one-flavourD-dimensional QED, in terms of Γ functions and a singleF ₃₂ series, whose expansion nearD=4 is obtained, by wreath-product transformations, to the order required for five-loop calculations. Taking the limitD→4, we find that the \(\overline {MS} \) coupling \(\bar \alpha (\mu )\) satisfies the boundary condition $$\begin{gathered} \frac{{\bar \alpha (m)}}{\pi } = \frac{\alpha }{\pi } + \frac{{15}}{{16}}\frac{{\alpha ^3 }}{{\pi ^3 }} + \left\{ {\frac{{11}}{{96}}\zeta (3) - \frac{1}{3}\pi ^2 \log 2} \right. \hfill \\ \left. { + \frac{{23}}{{72}}\pi ^2 - \frac{{4867}}{{5184}}} \right\}\frac{{\alpha ^4 }}{{\pi ^4 }} + \mathcal{O}(\alpha ^5 ), \hfill \\ \end{gathered} $$ wherem is the physical lepton mass and α is the physical fine structure constant. Combining this new result for the finite part of three-loop on-shell charge renormalization with the recently revised four-loop term in the \(\overline {MS} \) β-function, we obtain $$\begin{gathered} \Lambda _{QED}^{\overline {MS} } \approx \frac{{me^{3\pi /2\alpha } }}{{(3\pi /\alpha )^{9/8} }}\left( {1 - \frac{{175}}{{64}}\frac{\alpha }{\pi } + \left\{ { - \frac{{63}}{{64}}\zeta (3)} \right.} \right. \hfill \\ \left. { + \frac{1}{2}\pi ^2 \log 2 - \frac{{23}}{{48}}\pi ^2 + \frac{{492473}}{{73728}}} \right\}\left. {\frac{{\alpha ^2 }}{{\pi ^2 }}} \right), \hfill \\ \end{gathered} $$ at the four-loop level of one-flavour QED.     

R- and T-Regions in Space-Time with Spherically Symmetric Space

The properties of solutions of Einstein's equations for spherically symmetric gravitational fields are studied. The notions of R- and T-universe regions are introduced. T-regions are shown to be regions of essential instability in the distinguished direction of the flow of time. The geometrical properties and those of the deformation of reference frames in these regions are discussed. Birkhoff's theorem for the Schwarzschild field is stated more precisely. In a T-region of this field there exists a frame of reference with the interval in the form

Path integral treatment of a noncentral electric potential

We present a rigorous path integral treatment of a dynamical system in the axially symmetric potential $V(r,\theta ) = V(r) + \tfrac{1} {{r^2 }}V(\theta ) $ . It is shown that the Green’s function can be calculated in spherical coordinate system for $V(\theta ) = \frac{{\hbar ^2 }} {{2\mu }}\frac{{\gamma + \beta \sin ^2 \theta + \alpha \sin ^4 \theta }} {{\sin ^2 \theta \cos ^2 \theta }} $ . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number l ≥ 0, are recovered.