No abstract

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.