Melvin-Zipoy-Voorhees Spacetime and Circular Orbits

TL;DR

This paper constructs an exact magnetized generalization of the Zipoy-Voorhees spacetime by applying the Harrison transformation to a static, quadrupole-deformed seed, producing a solution that interpolates between the Zipoy-Voorhees geometry and the Melvin magnetic universe. By deriving the associated Ernst potentials and metric functions, it analyzes curvature, Petrov type, and the structure of the external magnetic field, showing a generic Petrov type I classification and a purely magnetic field for static observers. It then studies equatorial geodesics, revealing a Lorentz shift that inwardly shifts the ISCO for massive charged particles while magnetization rescale the metric, and an outward shift of the photon ring for null geodesics. These results provide a controlled analytic setting to explore how external magnetization and quadrupole deformations shape accretion dynamics and shadow observables in non-spherical spacetimes, guiding future work on ray tracing, stability, and quasi-normal modes.

Abstract

We construct an exact magnetized generalization of the Zipoy-Voorhees spacetime by applying the magnetic Harrison transformation to a static seed with quadrupolar deformation parameter $k$. The resulting Melvin-Zipoy-Voorhees metric is a solution to the Einstein-Maxwell equations that interpolates between the unmagnetized Zipoy-Voorhees geometry and the Melvin magnetic universe. We analyze the algebraic structure, finding the spacetime to be generically of Petrov type I, and investigate the equatorial dynamics of charged test particles and photons. Our analysis reveals that the external magnetic field $b$ induces a ``Lorentz shift'' in the effective angular momentum, suppressing the potential barrier and causing the Innermost Stable Circular Orbit (ISCO) to migrate inward. In contrast, the radius of the photon ring shifts slightly outward with increasing magnetization.

Figures (5)

• Figure 2.1: Logarithmic-linear plot of the dimensionless squared Riemann tensor $K^* = K M^4$ on the equatorial plane of the Melvin-Zipoy-Voorhees spacetime for deformation parameters $k \in \{0.5, 1, 2\}$ and magnetic field strengths $b M \in \{0, 1\}$. Black curves correspond to $b = 0$; red curves to $b M = 1$. Line styles denote different deformation parameters: $k = 0.5$ (dotted), $k = 1$ (solid), $k = 2$ (dash-dotted). For fixed radial coordinate $r$, the curvature $K$ increases with $k$, while magnetization ($b M = 1$) suppresses the equatorial curvature amplitude throughout this radial range. All curves exhibit monotonic decay with increasing $r$.
• Figure 3.1: Polar magnetic field measured by a static observer. Angular profiles of dimensionless $B^{*\hat{\theta}}(\theta)=M B^{\hat{\theta}}(\theta)$ at fixed radius $r=4M$ for $k\in\{0.5,1,2\}$ with $bM=0.5$. The field is evaluated in the static orthonormal frame [Eq. \ref{['eq:OrthoStatic']}] using the Harrison-magnetized potential [Eq. \ref{['eq:harrison_A']}]. From Eq. \ref{['eq:BhatCOMP']} the profile is symmetric about the equator and vanishes on the axis.
• Figure 3.2: Radial magnetic field measured by a static observer. Angular profiles of dimensionless $B^{*\hat{r}}(\theta)=M B^{\hat{r}}(\theta)$ at fixed radius $r=3M$ for $k\in\{0.5,1,2\}$ with $bM=0.5$. The field is evaluated in the static orthonormal frame [Eq. \ref{['eq:OrthoStatic']}] using the Harrison-magnetized potential [Eq. \ref{['eq:harrison_A']}]. From Eq. \ref{['eq:BhatCOMP']}, $B^{\hat{r}}$ is antisymmetric about the equator and vanishes at $\theta=\pi/2$, while remaining finite and of opposite sign on the two axes. Its amplitude is modulated by $k$ through the factors $f^{1-k}$, $\Lambda^{-2}$, and $\mathrm{e}^{-\gamma}$.
• Figure 3.3: Electromagnetic invariant on the equatorial plane. We plot $I_1(r,\theta=\pi/2)$ for the Melvin-Zipoy-Voorhees spacetime, where $I_1=2\mathbf{B}^2\ge0$ for static observers [cf. Eq. \ref{['eq:invariantsPos']}]. Panel (a) uses a logarithmic axis to resolve the sharp growth just outside $r=2M$; panel (b) displays the full exterior decay on a linear scale.
• Figure 4.1: Equatorial effective potential on a semi-log scale. Shown is $U_{\rm eff}(r; L_z, \eta)$ for fixed angular momentum $L_z=4$ and charge parameter $\eta=0.10$. Curves vary the deformation parameter $k\in\{0.5,1,2\}$ (line style) and the magnetic parameter $bM\in\{0,0.20,0.40\}$ (color). The black curves ($b=0$) show the unmagnetized Zipoy--Voorhees reference. Increasing $bM$ (blue and red curves) significantly lowers the potential barrier, a direct consequence of the electromagnetic interaction term $-\eta A_\phi$ shifting the effective angular momentum (the Lorentz shift discussed in Eq. \ref{['eq:Radial_Eq_Ueff']}).