Tomimatsu--Sato $δ=2$ Solution in Kaluza--Klein Theory

Abstract

We construct the Kaluza-Klein electric dressing of the Tomimatsu-Sato $δ=2$ vacuum spacetime by uplifting to five dimensions, boosting along the compact direction and reducing to Einstein-Maxwell-dilaton theory with coupling $a=\sqrt3$. It carries a nontrivial dilaton, unlike the known Einstein-Maxwell charged and magnetized Tomimatsu-Sato solutions. At leading order it shares the mass-charge-dilaton relations and gyromagnetic ratio $g=2-\tanh^2α$ of the boosted Kerr black hole; the seed first shows up in the quadrupole. Using Mayerson's non-vacuum Geroch-Hansen multipoles, we derive the exact law $M_2(α)=\frac12(1+\cosh^2α)M_2^{(0)}$, so the $δ=2$ quadrupole excess $ΔM_2=-\frac14p^2m^2M$ survives every boost. In the Kodama-Hikida picture the Weyl segment $ρ=0$, $|z|<κ$ is a singular rod with degenerate Killing horizons at its endpoints. There the boost function equals $\coshα$, so the near-horizon geometry is a constant rescaling of the seed's: the horizons stay degenerate and non-rotating, their areas and Komar angular momenta scale by $\coshα$, and the electric potential is $\frac12\tanhα$. The electric flux through any closed surface is $\sinhα\coshα$ times the enclosed seed Komar mass, so the horizons and the ring singularity are uncharged and the rod carries all the charge; the boost-invariant identity $M-Φ_eQ=m$ follows. The ergosurface stays at $f=0$; a new curvature-singular boundary appears at $f=\coth^2α$, where the Kaluza-Klein circle becomes null. As the seed potential $f$ is unbounded above near the ring, it exists for every nonzero boost: unlike for Kerr, no Kaluza-Klein boost of TS$_2$ is a spacelike reduction everywhere. The excised region is compact, has the ring on its boundary and never reaches the horizons. Charging pointwise strengthens the causality violation near the rod.