Toda Molecule and Tomimatsu-Sato Solution $\sim$ Towards the complete proof of Nakamura's conjecture$\sim$

TL;DR

This paper advances the proof of Nakamura's conjecture, linking Tomimatsu-Sato black hole solutions to Toda molecule equations, by rearranging parameters and proving specific cases, paving the way for a complete proof.

Contribution

It provides a partial proof of Nakamura's conjecture by explicitly proving cases and proposing a method to achieve a full proof.

Findings

Partial proof of Nakamura's conjecture achieved.

Rearrangement of the rotation parameter is key.

Method shows promise for complete proof.

Abstract

We discuss the Nakamura's conjecture stating that the Tomimatsu-Sato black hole solution with integer deformation parameter $n$ is composed of the special solutions of the Toda molecule equation at the $n$-th lattice site. From the previous work, in which the conjecture was partly analytically proved, we go further towards final full proof by rearranging the rotation parameter. The proof is explicitly performed for the highest and lowest orders. Though the proof for all orders remains still unsolved, the prospect to the full proof becomes transparent and workable by our method.