Multi-Soliton Solutions of the Einstein Equation and the Tomimatsu-Sato Metric

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Abstract
We present a new recognition about the Tomimatsu-Sato metric through reviewing the recent study of the stationary and axially symmetric Einstein field equation. We describe some powerful methods of solving the Einstein equation; the Bäcklund transformation, the inverse scattering method and any other. These methods derive the so-called multi-soliton solution, which represents the Kerr-NUT metric or a non-linear superposition of several Kerr-NUT metrics aligned along their common rotational axis. The Tomimatsu-Sato metric of γ = N is constructed via a limiting process that the N Kerr metrics with the same mass and angular momentum approach mutually towards their complete overlapping. We investigate the space-time properties of the multi-soliton metric, by taking the two Kerr case as a typical example.

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