The two-centre problem in general relativity concerns the axisymmetric Majumdar–Papapetrou spacetime describing two extreme (extremal) Reissner–Nordström black holes held in electrostatic equilibrium at finite separation. Chandrasekhar (1989) analysed its axial (odd-parity) perturbations, reducing them to coupled partial differential equations, constructing a four-channel scattering matrix for incident gravitational and electromagnetic radiation, and deriving energy-conservation identities from a flux integral. Chandrasekhar and Ferrari (1990) then derived the corresponding conserved flux integral for polar (even-parity) perturbations of static Einstein–Maxwell space-times, explicitly earmarking it for 'the setting up of a scattering matrix' for this same two-black-hole problem, and a sibling flux integral for reformulating the non-radial oscillations of a star as a scattering problem. The sibling applications were subsequently completed — the stellar reformulation and the axial stellar modes by the authors' own 1991 series, and the relation to variational symplectic currents by Burnett and Wald (1990) — whereas the polar two-centre scattering problem itself has never been constructed or analysed. The task is to do for polar perturbations what the 1989 analysis did for axial ones: reduce the coupled polar Einstein–Maxwell perturbation equations of the two-centre spacetime, determine the asymptotic behaviours at infinity and at the two horizons, set up the multi-channel scattering matrix via the 1990 flux integral, derive the attendant energy-conservation identities, and analyse the resulting scattering, including graviton–photon conversion and its dependence on the black-hole separation.

Background

An extreme (extremal) Reissner–Nordström black hole is a charged, non-rotating black hole whose charge equals its mass in geometrized units (Q=MQ=M), so that its inner and outer horizons coincide. The Majumdar–Papapetrou solutions of the Einstein–Maxwell equations describe arbitrary static arrangements of such extreme black holes, each held in place by the balance between gravitational attraction and electrostatic repulsion. The simplest non-spherical member places two of them a finite distance apart on a symmetry axis; because the two centres make it the relativistic analogue of the Coulomb two-centre problem of molecular physics, the natural chart is prolate spheroidal coordinates — the coordinates adapted to two foci, in which the coordinate surfaces are ellipsoids and hyperboloids sharing the two black holes as common foci.

Axisymmetric perturbations (azimuthal number m=0m=0) of a static, axially symmetric space-time split into two sectors of opposite reflection parity. Axial (odd-parity) perturbations have metric amplitudes odd under reflection in the equatorial plane; polar (even-parity) perturbations have even parity and, in electrovacuum (a region containing an electromagnetic field but no matter), generically couple the metric perturbations to perturbations of the electromagnetic field. In charged backgrounds the two kinds of radiation interconvert: incident gravitational radiation scatters partly into electromagnetic radiation and vice versa (photon–graviton conversion).

A flux integral, in the sense introduced by Chandrasekhar and Ferrari, is a quadratic, time-independent expression in the perturbation amplitudes whose divergence (or, after Fourier decomposition in time, whose radial derivative) vanishes as a consequence of the linearized field equations alone. Evaluated at the boundaries of a scattering region, it expresses conservation of energy and yields algebraic identities among scattering amplitudes, playing the role that the Wronskian plays in one-dimensional Schrödinger scattering.

Chandrasekhar (1989) carried out this programme for the axial perturbations of the two-extreme-Reissner–Nordström spacetime: the analysis reduced to a pair of coupled partial differential equations (no separation into ordinary differential equations is available in the two-centre geometry); the asymptotic behaviours were determined at infinity and at the two horizon singularities, where gravitational and electromagnetic radiation prevail as coupled photon–graviton waves sharing a common amplitude and phase; and the attendant flux integral allowed the formulation of a four-channel scattering matrix — incidence of gravitational or of electromagnetic radiation from infinity, and incidence of coupled photon–graviton waves on the horizons of the two holes — together with four identities among the amplitudes expressing conservation of energy. The analysis was explicitly restricted to axial perturbations.

Chandrasekhar and Ferrari (1990) then showed how the corresponding flux integral can be derived directly from the linearized equations for polar perturbations of static space-times with either a Maxwell source or a perfect-fluid source, and stated two intended applications. First, the Einstein–Maxwell flux integral would be 'particularly useful … (as in the scattering of radiation by two extreme Reissner-Nordström black-holes) and in the setting up of a scattering matrix' — that is, for extending the 1989 two-centre analysis from axial to polar modes. Second, the perfect-fluid flux integral would apply to the non-radial oscillations of a star with accompanying gravitational-radiation emission, 'enabl[ing] its reformulation as a problem in scattering theory'.

Both sibling applications were in fact completed soon after. The stellar reformulation was carried out by the same authors in 'On the non-radial oscillations of a star' (1991) and its sequels (Papers II–V, 1991–1995), which develop the complete scattering theory of stellar polar and axial oscillations, including the complex quasi-normal frequencies that encode radiation damping; Ferrari's 1992 review presents it under exactly that title. The relation to the general variational theory was settled in the same setting by Burnett and Wald (1990), who constructed the symplectic current — the conserved current that the variational structure of the linearized field equations defines for any pair of perturbations — of the Einstein–Maxwell theory from its Lagrangian variational principle and showed that it reproduces the Chandrasekhar–Ferrari current for axisymmetric polar perturbations of static space-times (with later developments by Torres del Castillo and collaborators). What no later work has done is return to the first intended application: the polar-mode two-centre scattering problem itself.

Problem Statement

For the Majumdar–Papapetrou spacetime describing two extreme Reissner–Nordström black holes at finite separation — the two-centre problem in general relativity — construct and analyse the scattering of axisymmetric polar (even-parity) gravitational and electromagnetic radiation, completing for the polar sector what Chandrasekhar's 1989 analysis did for the axial sector. Specifically:

(a) formulate the coupled linearized Einstein–Maxwell equations governing polar perturbations of the two-centre metric (in prolate spheroidal coordinates, with harmonic time dependence), fix the gauge and variable conventions, and reduce the system as far as possible;

(b) determine the asymptotic behaviours of the polar perturbations at infinity and near the two horizon singularities, where gravitational and electromagnetic perturbations are expected to combine into coupled photon–graviton waves;

(c) using the Chandrasekhar–Ferrari flux integral for polar perturbations of static Einstein–Maxwell space-times, define the scattering channels (incident gravitational or electromagnetic radiation from infinity; coupled photon–graviton waves incident on the two horizons) and set up the scattering matrix of the attendant multi-channel problem;

(d) derive the energy-conservation identities among the scattering amplitudes that the flux integral guarantees;

(e) analyse the resulting scattering — reflection, transmission, and gravitational-to-electromagnetic conversion amplitudes, together with their dependence on the horizon separation — and, if feasible, the dependence on the angular-momentum index labelling the asymptotic modes (in the 1989 axial analysis the coupled waves at each end are expressed for each specified angular momentum) and any resonances or quasi-normal modes — free oscillations of the configuration damped by radiation emission — of the two-black-hole system.

The analysis centres on the reflection, transmission and graviton–photon conversion amplitudes and their dependence on the horizon separation, and extends to the dependence on the angular-momentum index and to any resonances or quasi-normal modes of the two-black-hole system. The completion boundary is a scattering analysis of the polar sector whose energy balance is certified by the flux integral, in the same sense as the completed 1989 axial analysis, and the analysis is not restricted to a particular separation, frequency regime or method.

References

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