Progress in relativistic gravitational theory using the inverse scattering method

Abstract

The increasing interest in compact astrophysical objects (neutron stars, binaries, galactic black holes) has stimulated the search for rigorous methods, which allow a systematic general relativistic description of such objects. This paper is meant to demonstrate the use of the inverse scattering method, which allows, in particular cases, the treatment of rotating body problems. The idea is to replace the investigation of the matter region of a rotating body by the formulation of boundary values along the surface of the body. In this way we construct solutions describing rotating black holes and disks of dust ("galaxies"). Physical properties of the solutions and consequences of the approach are discussed. Among other things, the balance problem for two black holes can be tackled.

Non-Existence of Stationary Two-Black-Hole Configurations

“…As was shown in [23,24] (see Eqs. (34), (45), (57), (58) in [24]) the result of the integration is a matrix representation of the axis values of the Ernst potential (4) :…”

“…To obtain the dependence on λ for the two-horizon problem we have to integrate the linear system (17) along the dashed line (2), (3), (10), (11). As was shown in [23,24] (see Eqs. (34), (45), (57), (58) in [24]) the result of the integration is a matrix representation of the axis values of the Ernst potential (4) :…”

The Ernst equation and ergosurfaces

“…Next, large classes of further examples are given in [9,12,[14][15][16][17]22] 7 . Some of the solutions in those references have non-trivial zero-level sets of E , with g ρρ = g zz and g tϕ smooth across E f (see in particular [12]), but smoothness of g ϕϕ is not manifest.…”

“…solving a boundary-value problem [17]. The spacetime metric is then obtained by solving ODEs for the metric functions.…”

Novel Traveling Hyperbolic Function Solution for Evolution Equation

“…Nonlinear equations are very important in describe evolution phenomena in engineering and physics. People have investigated many useful methods to obtain the evolution solutions: the inverse scattering method [1][2][3], the Hirota's bilinear method [4][5][6], the Darboux and Backlund transformation [7,8]. Sine-cosine method [9,10].…”