Tomimatsu--Sato $\delta =2$ Solution in Kaluza--Klein Theory

Abstract

We construct the Kaluza-Klein electric dressing of the Tomimatsu-Sato $\delta =2$ vacuum spacetime by uplifting to five dimensions, boosting along the compact direction and reducing to Einstein-Maxwell-dilaton theory with coupling $a=\sqrt{3}$. 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 $g=2-{\tanh }^2\alpha$ 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 $M_2(\alpha )=\frac{1}{2}(1+{\cosh }^2\alpha )M_2^{(0)}$, so the $\delta =2$ quadrupole excess $\Delta M_2=-\frac{1}{4}{p}^2{m}^{2}M$ survives every boost. In the Kodama-Hikida picture the Weyl segment $\rho =0$, $|z|<\kappa$ is a singular rod with degenerate Killing horizons at its endpoints. There the boost function equals $\cosh \alpha$, 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 $\cosh \alpha$, and the electric potential is $\frac{1}{2}\tanh \alpha$. The electric flux through any closed surface is $\sinh \alpha \cosh \alpha$ 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 $M-{\Phi }_{e}Q=m$ follows. The ergosurface stays at $f=0$; a new curvature-singular boundary appears at $f={\coth }^2\alpha$, 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.

arXiv comment27 pages, 13 figures