Quantum amplitudes for $s=1$ at Maxwell fields and for $s=2$ linearised gravitational wave perturbations of a spherically symmetric Einstein/massless scalar background, describing gravitational collapse to a black hole, are treated by analogy with a previous treatment of $s=0$ scalar-field perturbations of gravitational collapse at late times. In both the $s=1$ and $s=2$ cases, we isolate suitable 'co-ordinate' variables which can be taken as boundary data on a final space-like hypersurface $\Sigma_F$. For simplicity, we take the data on an initial pre-collapse surface $\Sigma_I$ to be exactly spherically symmetric. The (large) Lorentzian proper-time interval between $\Sigma_{I}, \Sigma_{F}$, measured at spatial infinity, is denoted by $T$. The complexified classical boundary-value problem is expected to be well-posed, provide that the time interval $T$ has been rotated into the complex: $T\to{\mid}T{\mid}\exp(-i\theta)$, for $0<\theta\leq{\pi}/2$. We calculate the second-variation classical Lorenztian action $S ^{(2)}_{\rm class}$. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as $\theta\to 0_+$ of the semi-classical amplitude $\exp(iS^{(2)}_{\rm class})$. The boundary data for $ s=1$ involve the Maxwell magnetic field; the data for $s=2$ involve the magnetic part of the Weyl curvature tensor. The magnetic boundary conditions are related to each other and to the natural $s={1 \over 2}$ boundary conditions by supersymmetry
This paper continues earlier work on the quantum evaporation of black holes. This work has been concerned with the calculation and understanding of quantum amplitudes for final data perturbed slightly away from spherical symmetry on a space-like hypersurface Σ F at a late Lorentzian time T . For initial data, we take, for simplicity, sphericallysymmetric asymptotically-flat data for Einstein gravity with a massless scalar field on an initial surface Σ I at time t = 0 . Together, such boundary data give a quantum analogue of classical Einstein/scalar gravitational collapse to a black hole, perhaps starting from a diffuse, early-time configuration. Quantum amplitudes are calculated following Feynman's approach, by first rotating: T → |T | exp(−iθ) into the complex, where 0 < θ ≤ π/2 , then solving the corresponding complex classical boundary-value problem, which is expected to be well-posed provided θ > 0 , and computing its classical Lorentzian action S class and corresponding semi-classical quantum amplitude, proportional to exp(iS class ). For a locally-supersymmetric Lagrangian, describing supergravity coupled to supermatter, any loop corrections will be negligible, provided that the frequencies involved in the boundary data are well below the Planck scale. Finally, the Lorentzian amplitude is recovered by taking the limit θ → 0 + of the semi-classical amplitude. In the black-hole case, by studying the linearised spin-0 or spin-2 classical solutions in the above (slightly complexified) case, for the corresponding classical boundary-value problem with the given perturbative data on Σ F , one can compute an effective energy-momentum tensor < T µν > EF F , which has been averaged over several wavelengths of the radiation, and which describes the averaged extra energy-momentum contribution in the Einstein field equations, due to the perturbations. In general, this averaged extra contribution will be spherically symmetric, being of the form of a null fluid, describing the radiation (of quantum origin) streaming radially outwards. The corresponding space-time metric, in this region containing radially outgoing radiation, is of the Vaidya form. This, in turn, justifies the treatment of the adiabatic radial mode equations, for spins s = 0 and s = 2 , which is used elsewhere in this work.
In a previous Letter, we outlined an approach to the calculation of quantum amplitudes appropriate for studying the black-hole radiation which follows gravitational collapse. This formulation must be different from the familiar one (which is normally carried out by considering Bogoliubov transformations), since it yields quantum amplitudes relating to the final state, and not just the usual probabilities for outcomes at a late time and large radius. Our approach simply follows Feynman's +iǫ prescription. Suppose that, in specifying the quantum amplitude to be calculated, initial data for Einstein gravity and (say) a massless scalar field are specified on an asymptotically-flat space-like hypersurface Σ I , and final data similarly specified on a hypersurface Σ F , where both Σ I and Σ F are diffeomorphic to R 3 . Denote by T the (real) Lorentzian proper-time interval between Σ I and Σ F , as measured at spatial infinity. Then rotate: T → |T | exp(−iθ) , 0 < θ ≤ π/2 . The classical boundary-value problem is then expected to become well-posed on a region of topology I × R 3 , where I is the interval 0, |T | . For a locally-supersymmetric theory, the quantum amplitude is expected to be dominated by the semi-classical expression exp(iS class ), where S class is the classical action. Hence, one can find the Lorentzian quantum amplitude from consideration of the limit θ → 0 + . In the usual approach, the only possible such final surfaces Σ F are in the strong-field region shortly before the curvature singularity; that is, one cannot have a Bogoliubov transformation to a smooth surface 'after the singularity'. In our complex approach, however, one can put arbitrary smooth gravitational data on Σ F , provided that it has the correct mass M ; thus we do have Bogoliubov transformations to surfaces 'after the singularity in the Lorentzian-signature geometry' -the singularity is simply by-passed in the analytic continuation (see below). In this Letter, we consider Bogoliubov transformations in our approach, and their possible relation to the probability distribution and density matrix in the traditional approach. In particular, we find that our probability distribution for configurations of the final scalar field cannot be expressed in terms of the diagonal elements of some density-matrix distribution.
Electrodeposited multilayered nanowires grown within a polycarbonate membrane constitute a new medium in which giant magnetoresistance (GMR) perpendicular to the plane of the multilayers can be measured. These structures can exhibit a perpendicular GMR of at least 22% at ambient temperature. We performed detailed studies both of reversible magnetization and of irreversible remanent magnetization curves for CoNiCu/Cu/CoNiCu multilayered and CoNiCu pulse-deposited nanowire systems with Co:Ni ratios of 6:4 and 7:3 respectively in the range 10 - 290 K, allowing the magnetic phases of these structures to be identified. Shape anisotropy in the pulse-deposited nanowire and inter-layer coupling in the multilayered nanowire are shown to make important contributions to the magnetic properties. Dipolar-like interactions are found to predominate in both nanowire systems. Magnetic force microscope (MFM) images of individual multilayered nanowires exhibit a contrast consistent with there being a soft magnetization parallel to the layers. Switching of the magnetic layers in the multilayered structure into the direction of the MFM tip's stray field is observed.
Here, quantum amplitudes for s = 2 linearised gravitational-wave perturbations of a spherically-symmetric Einstein/massless-scalar background, describing gravitational collapse to a black hole, are treated by analogy with the previous treatment of s = 1 linearised Maxwell-field perturbations. As with the spin-1 case, the spin-2 perturbations split into parts with odd and even parity. Their detailed angular behaviour is analysed, as well as their behaviour under infinitesimal coordinate transformations and their (linearised vacuum Einstein) field equations. In general, we work in the Regge-Wheeler gauge, except that, at a certain point, it becomes necessary to make a gauge transformation to an asymptoticallyflat gauge, such that the metric perturbations have the expected falloff behaviour at large radii. As with the s = 1 treatment, so in this s = 2 case we isolate suitable 'coordinate' variables which can be taken as boundary data on a final space-like hypersurface ΣF . (For simplicity of exposition, we take the data on the initial surface ΣI to be exactly spherically-symmetric.) The (large) Lorentzian proper-time interval between ΣI and ΣF , measured at spatial infinity, is denoted by T . We then consider the classical boundary-value problem and calculate the secondvariation classical Lorentzian action S (2) class , on the assumption that the time interval T has been rotated into the complex: T → |T | exp(−iθ) , for 0 < θ ≤ π/2 . This complexified classical boundary-value problem is expected to be well-posed, in contrast to the boundary-value problem in the Lorentzian-signature case (θ = 0), which is badly posed, since it refers to hyperbolic or wave-like field equations. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as θ → 0+ of the semi-classical amplitude exp(iS (2) class ) . The boundary data for s = 2 involve the magnetic part of the Weyl tensor, just as the data for s = 1 involve the usual (Maxwell) magnetic field. These relations are also investigated, using 2-component spinor language, in terms of the Maxwell field strength φAB = φ (AB) and the Weyl spinor ΨABCD = Ψ (ABCD) .
In recent papers, we have studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat spacetime and weak radiation at a very late time. In this recent work, we have been concerned with evaluating quantum amplitudes (not just probabilities) for transitions from initial to final states. In a general asymptotically flat context, one may specify a quantum amplitude by posing boundary data on (say) an initial space-like hypersurface ΣI and a final space-like hypersurface ΣF. To complete the specification, one must also give the Lorentzian proper-time interval between the two boundary surfaces, as measured near spatial infinity. We have assumed that the Lagrangian contains Einstein gravity coupled to a massless scalar field ϕ, plus possible additional fields; there is taken to be a ‘background’ spherically symmetric solution (γμν, Φ) of the classical Einstein/scalar field equations. For bosonic fields, the gravitational and scalar boundary data can be taken to be gij and ϕ on the two hypersurfaces, where gij (i, j = 1, 2, 3) gives the intrinsic 3-metric on the boundary, and the 4-metric is gμν (μ, ν = 0, 1, 2, 3), the boundary being taken locally in the form {x0 = const}. The classical boundary value problem, corresponding to the calculation of this quantum amplitude, is badly posed, being a boundary value problem for a wave-like (hyperbolic) set of equations. Following Feynman's +iϵ prescription, one makes the problem well-posed by rotating the asymptotic time interval T into the complex: T → ∣T ∣ exp(−iθ), with 0 < θ ⩽ π/2. After calculating the amplitude for θ > 0, one then takes the ‘Lorentzian limit’ θ → 0+. Such quantum amplitudes have been calculated for weak s = 0 (scalar), s = 1 (photon) and s = 2 (graviton) anisotropic final data, propagating on the approximately Vaidya-like background geometry, in the region containing radially outgoing black-hole radiation. In this paper, we treat quantum amplitudes for the case of fermionic massless spin-½ (neutrino) final boundary data. Making use of boundary conditions originally developed for local supersymmetry, we find that this fermionic case can be treated in a way which parallels the bosonic case. In particular, we calculate the classical action as a functional of the fermionic data on the late-time surface ΣF; the quantum amplitude follows straightforwardly from this.
SUMMARY A new SEM technique for imaging uncoated non‐conducting specimens at high beam voltages is described which employs a high‐pressure environment and an electric field to achieve charge neutralization. During imaging, the specimen surface is kept at a stable low voltage, near earth potential, by directing a flow of positive gas ions at the specimen surface under the action of an electric bias field at a pressure of about 200 Pa. In this way charge neutrality is continuously maintained to obtain micrographs free of charging artefacts. Images are formed by specimen current detection containing both secondary electron and backscattered electron signal information. Micrographs of geological, ceramic, and semiconductor materials obtained with this method are presented. The technique is also useful for the SEM examination of histological sections of biological specimens without any further preparation. A simple theory for the charge neutralization process is described. It is based on the interaction of the primary and emissive signal components with the surrounding gas medium and the resulting neutralizing currents. Further micrographs are presented to illustrate the pressure dependence of the charge neutralization process in two glass specimens which show clearly identifiable charging artefacts in conventional microscopy.
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