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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.03104 |
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| _version_ | 1866911299100737536 |
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| author | Fukuyama, Takeshi |
| author_facet | Fukuyama, Takeshi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03104 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tau--Function Multilinear Hierarchy of the Tomimatsu--Sato Spacetime: A Gravitational Realization of the YTSF Integrable Structure Fukuyama, Takeshi General Relativity and Quantum Cosmology High Energy Physics - Theory 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. |
| title | Tau--Function Multilinear Hierarchy of the Tomimatsu--Sato Spacetime: A Gravitational Realization of the YTSF Integrable Structure |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2512.03104 |