Privatization of irrigation schemes in New Zealand

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Ionospheric E-region Irregularities Produced by Non-linear Coupling of Unstable Plasma Waves

Ionospheric E region irregularities produced by nonlinear coupling of unstable plasma wave

Spin-2 Amplitudes in Black-Hole Evaporation

Quantum amplitudes for $s=2$ gravitational-wave perturbations of Einstein/scalar collapse to a black hole are treated by analogy with $s=1$ Maxwell perturbations. The spin-2 perturbations split into parts with odd and even parity. We use the Regge-Wheeler gauge; at a certain point we make a gauge transformation to an asymptotically-flat gauge, such that the metric perturbations have the expected falloff behaviour at large radii. By analogy with $s=1$, for $s=2$ natural 'coordinate' variables are given by the magnetic part $H_{ij} (i,j=1,2,3)$ of the Weyl tensor, which can be taken as boundary data on a final space-like hypersurface $\Sigma_F$. For simplicity, we take the data on the initial surface $\Sigma_I$ to be exactly spherically-symmetric. The (large) Lorentzian proper-time interval between $\Sigma_I$ and $\Sigma_F$, measured at spatial infinity, is denoted by $T$. We follow Feynman's $+i\epsilon$ prescription and rotate $T$ into the complex: $T\to{\mid}T{\mid} \exp(-i\theta)$, for $0<\theta\leq\pi/2$. The corresponding complexified {\it classical} boundary-value problem is expected to be well-posed. The Lorentzian quantum amplitude is recovered by taking the limit as $\theta\to 0_+$. For boundary data well below the Planck scale, and for a locally supersymmetric theory, this involves only the semi-classical amplitude $\exp(iS^{(2)}_{\rm class}$, where $S^{(2)}_{\rm class}$ denotes the second-variation classical action. The relations between the $s=1$ and $s=2$ natural boundary data, involving supersymmetry, are investigated using 2-component spinor language in terms of the Maxwell field strength $\phi_{AB}=\phi_{(AB)}$ and the Weyl spinor $\Psi_{ABCD}=\Psi_{(ABCD)}$

Relic Radiation from an Evaporating Black Hole

We present a non-string-theoretic calculation of the microcanonical entropy of relic integer-spin Hawking radiation -- at fixed total energy $E$. The only conserved macroscopic quantity is the total energy $E$ (the total energy of the relic radiation). Data for a boundary-value approach, with massless, integer-spin perturbations, are set on initial and final space-like hypersurfaces. In the resulting 1-dimensional statistical-mechanics problem, the real part of the (complex) time separation at spatial infinity, $T = {\mid}T{\mid}\exp(-i\delta), \delta >0$, is the variable conjugate to the total energy. We count the number of weak-field configurations on the final space-like hypersurface with energy $E$. One recovers the Cardy formula and the Bekenstein-Hawking entropy, if Re(T) is of the order of the black-hole life- time, leading to a statistical interpretation of black-hole entropy. The microcanonical entropy includes a logarithmic correction to the black-hole area law, which is {\it universal} (independent of black-hole parameters). Here, the discreteness of the energy levels is crucial. This approach is compared with that of string theory for the transition to the fundamental-string r\'egime in the final stages of evaporation. The squared coupling, $g^2$, regulating the transition to a highly-excited string state and {\it vice versa}, can be related to the angle, $\delta$, of complex-time rotation above. The strong-coupling r\'egime corresponds to a Euclidean black hole, while the physical limit of a Lorentzian space-time (as $\delta \to 0_+$) corresponds to the weak-coupling r\'egime. This resembles the transition to a highly-excited string-like state which subsequently decays into massless particles, thereby avoiding the naked singularity.Comment: To appear in International Journal of Modern Physics

Black hole evaporation in a spherically symmetric non-commutative space-time

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in noncommutative geometry have shown that, in general relativity, the effects of non-commutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy-momentum tensor as a source on the right-hand side. Relying on the recently obtained non-commutativity effect on a static, spherically symmetric metric, we have considered from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes has been shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F have been derived which are compatible with the adiabatic approximation.Comment: 8 pages, Latex file with IOP macros, prepared for the QFEXT07 Conference, Leipzig, September 200