Jean-Philippe Nicolas

SCATTERING OF MASSLESS DIRAC FIELDS BY A KERR BLACK HOLE

For the massless Dirac equation outside a slow Kerr black hole, we prove asymptotic completeness. We introduce a new Newman–Penrose tetrad in which the expression of the equation contains no artificial long-range perturbations. The main technique used is then a Mourre estimate. The geometry near the horizon requires us to apply a unitary transformation before we find ourselves in a situation where the generator of dilations is a good conjugate operator. The results are eventually re-interpreted geometrically to provide the solution to a Goursat problem on the Penrose compactified exterior.

Conformal scattering and the Goursat problem

We work on a class of non-stationary vacuum space-times admitting a conformal compactification that is smooth at null and timelike infinity. Via a conformal transformation, the existence of a scattering operator for field equations is interpreted as the well-posedness of a Goursat problem on null infinity. We solve the Goursat problem in the case of Dirac and Maxwell fields. The case of the wave equation is also discussed and it is shown why the method cannot be applied at present. Then the conformal scattering operator is proved to be equivalent to an analytical scattering operator defined in terms of classical wave operators.

A nonlinear Klein–Gordon equation on Kerr metrics

Abstract We consider the nonlinear Klein–Gordon equation □u+m2u+λ|u|2u=0, with λ⩾0, outside a Kerr black hole. We solve the global Cauchy problem for large data with minimum regularity. Then, using a Penrose compactification, we prove, in the massless case, the existence of smooth asymptotic profiles and Sommerfeld radiation conditions, at the horizon and at null infinity, for smooth solutions.

Non linear Klein-Gordon equation on Schwarzschild-like metrics

We solve the global Cauchy problem for a non linear Klein-Gordon equation outside a spherical Black Hole. At the horizon of the Black Hole, the solution satisfies T. Damour’s impedance condition. In the case of an asymptotically flat space-time, massless fields satisfy Sommerfeld’s condition at infinity.

Global Results for the Rarita–Schwinger Equations and Einstein Vacuum Equations

We prove global existence and uniqueness of solutions to the Rarita-Schwinger evolution equations compatible with the constraints. We use a gauge fixing for the Rarita-Schwinger equations for helicity 3/2 fields in curved space that leads to a straightforward Hilbert space framework for their study. We explain how these results might be applied to the global analysis of the full Einstein vacuum equations and provide a complete analysis as a basis for such applications. These and a programme for developing a scattering/inverse scattering transform for the full Einstein equations are discussed. 1991 Mathematics Subject Classification: 83C60, 35Q75, 83C05, 35L45.

The Conformal Approach to Asymptotic Analysis

Albert Einstein’s general theory of relativity is a geometric theory of gravity, using the framework of Lorentzian geometry: an extension of Riemannian geometry in which space and time are united in a real 4-dimensional manifold endowed with an indefinite metric of signature (1, 3) or (3, 1). The metric allows to distinguish between timelike and spacelike directions in an intrinsic manner and, provides a description of gravity via its curvature. The introduction by Minkowski in 1908 of the notion of spacetime was a decisive change of viewpoint which opened the road for Einstein to develop the geometrical framework for the fully covariant theory he was after. Instead of discussing the history of this development and the crucial influence of Riemannian geometry through the help of Marcel Grossmann, this essay explores Roger Penrose’s approach to general relativity which bears a remarkable kindred of spirit with Einstein’s and perpetuates the geometrical view of the universe initiated by Riemann and Einstein. More specifically, Penrose’s approach to asymptotic analysis in general relativity, which is based on conformal geometric techniques, is presented through historical and recent aspects of two specialized topics: conformal scattering and peeling. Other essays in this volume are related to general relativity: Jacques Franchi [15] discusses relativistic analogues of the Brownian motion on various Lorentzian manifolds; Andreas Hermann and Emmanuel Humbert [23] discuss the positive mass theorem, which is closely related to the Yamabe problem in Riemannian geometry; Marc Mars [28] presents some local intrinsic ways of characterizing a spacetime.

Conservation laws and evolution schemes in geodesic, hydrodynamic and magnetohydrodynamic flows

Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamiltons principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike what has been previously thought, this approach can also accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron description of ideal magnetohydrodynamics. When Noether symmetries associated with Killing vectors or tensors are present in geodesic flows, they lead to constants of motion polynomial in the momenta. We generalize these concepts to hydrodynamic flows. Moreover, the Hamiltonian descriptions of ideal magnetohydrodynamics allow one to cast the evolution equations into a hyperbolic form useful for evolving rotating or binary compact objects with magnetic fields in numerical general relativity. In this framework, Ertels potential vorticity theorem for baroclinic fluids arises as a special case of a conservation law valid for any Hamiltonian system. Moreover, conserved circulation laws, such as those of Kelvin, Alfven and Bekenstein-Oron, emerge simply as special cases of the Poincare-Cartan integral invariant of Hamiltonian systems. We use this approach to obtain an extension of Kelvins theorem to baroclinic (nonisentropic) fluids, based on a temperature-dependent time parameter. We further extend this result to perfectly or poorly conducting baroclinic magnetoflows. Finally, in the barotropic case, such magnetoflows are shown to also be geodesic, albeit in a Finsler (rather than Riemann) space.

Superradiance on the Reissner–Nordstrøm metric

In this article, we study the superradiance of charged scalar fields on the sub-extremal Reissner-Nordstrom metric, a mechanism by which such fields can extract energy from a static spherically symmetric charged black hole. A geometrical way of measuring the amount of energy extracted is proposed. Then we investigate the question numerically. The toy-model and the numerical methods used in our simulations are presented and the problem of long time measurement of the outgoing energy flux is discussed. We provide a numerical example of a field exhibiting a behaviour analogous to the Penrose process: an incoming wave packet which splits, as it approaches the black hole, into an incoming part with negative energy and an outgoing part with more energy than the initial incoming one. We also show another type of superradiant solution for which the energy extraction is more important. Hyperradiant behaviour is not observed, which is an indication that the Reissner-Nordstrom metric is linearly stable in the sub-extremal case.

Dirac Fields on Asymptotically Simple Space-Times

The notion of global hyperbolicity of a space-time was introduced by Jean Leray in 1953 [11]. Intuitively, a space-time is globally hyperbolic if the Cauchy problem for the wave equation is well posed. This is a fundamental notion in General Relativity since globally hyperbolic space-times are the only class of space-times on which it is meaningful to study the global evolution of fields.