Tau--Function Multilinear Hierarchy of the Tomimatsu--Sato Spacetime: A Gravitational Realization of the YTSF Integrable Structure

TL;DR

The Tomimatsu–Sato spacetimes are algebraically intricate; this work reframes the stationary axisymmetric vacuum equations using the Ernst potential as a tau-ratio, $\mathcal{E}=\tau_1/\tau_0$. The Ernst numerator splits into a cubic sector containing all second derivatives and a universal gradient envelope, so the nonlinear kernel is governed by the cubic part (integrability in the Ernst sense). The cubic core is recast in terms of $Z_3$-symmetric trilinear Hirota derivatives into a Yu–Toda–Sasa–Fukuyama (YTSF)–type kernel, verifying the $\delta=2$ TS case explicitly with the known polynomials $A(\xi,\eta)$ and $B(\xi,\eta)$. This places the TS geometries in a two-layer integrable hierarchy: bilinear Toda dynamics in the discrete soliton index $\delta$ and a continuous, trilinear Ernst kernel on the Weyl base, and it outlines a programmable path to extend to $\delta>2$ and to broader stationary axisymmetric seeds.

Abstract

The Tomimatsu--Sato (TS) family generalizes the Kerr black hole to higher multipole order $δ$ and has long been regarded as algebraically complicated without any clear integrability. We show instead that stationary axisymmetric vacuum Einstein equations, when the Ernst potential is written as a $τ$--ratio $\mathcal{E}=τ_1/τ_0$, admit a universal decomposition of the Ernst numerator into a cubic part containing all second derivatives and a quartic \emph{gradient envelope}. The cubic sector can be written in terms of $Z_3$--symmetric trilinear Hirota operators, revealing a hidden integrable structure. For $δ=2$, using the explicit Tomimatsu--Sato polynomials, we verify that this trilinear sector coincides with a Yu--Toda--Sasa--Fukuyama (YTSF) equation-type kernel. Thus the TS geometry forms a gravitational realization of a multilinear $τ$--function hierarchy in stationary axisymmetric general relativity.

Figures (1)

• Figure 1: Two complementary views of the Tomimatsu-Sato hierarchy. Left: standard Toda/Nakamura description, where the TS family $\delta=1,2,3,...$ is encoded in the bilinear Toda molecule equation Toda1967Toda1Toda2Toda3 for the tau tower $\{\tau_n\}$. Right: the present work, where after projecting to the two tau functions $\tau_0,~\tau_1$ entering the Ernst potential, the cubic part of the Ernst numerator reveals a YTSF-type trilinear kernel, while the quartic gradient envelope remains passive.