Zipoy--Voorhees spacetime with Maxwell and dilaton fields: exact solution and equatorial geodesics

Abstract

We derive a new exact solution of four-dimensional Kaluza-Klein theory by applying the uplift-boost-reduction procedure to the Zipoy-Voorhees vacuum seed. The resulting Zipoy-Voorhees-dilaton spacetime is a static, axisymmetric, asymptotically flat configuration parametrised by a seed length scale $M$, the Zipoy-Voorhees deformation parameter $k$, and a boost parameter $α$ that controls the electric charge. For $k=1$ it admits a regular black-hole interpretation; for $k \neq 1$ it inherits the naked curvature singularity of the seed at $r=2M$. We extract the ADM mass, electric charge, and dilaton charge, finding that the dilaton charge obeys an algebraic constraint independent of the quadrupole deformation. The equatorial geodesics of neutral test particles and photons are studied in detail. A compact algebraic identity locates the exterior-photon-ring threshold at $k=1/2$ for all $α$, generalising the known threshold of the vacuum Zipoy-Voorhees case. The innermost stable circular orbit is mapped over the $(k,α)$ plane, and the comparison with Schwarzschild is shown to depend qualitatively on whether seed-mass or ADM-mass units are used. We further compute the gravitational redshift, which diverges on the $r=2M$ surface and renders the near-singularity orbits effectively invisible, and the critical impact parameter setting the characteristic shadow size, which in ADM-mass units shrinks below the Schwarzschild value as the charge grows.