Velocity rotation curves in the gravimagnetic dipole spacetime

TL;DR

This work derives an exact framework to compute the velocities of both massive and massless particles on circular orbits in the gravimagnetic dipole spacetime, the axisymmetric system of two counter-rotating NUT black holes connected by a Misner string. Using a Hamiltonian formulation, the authors define an effective potential and extract circular-orbit conditions, characterizing how the NUT parameter $ν$ and the (tensionless) string configuration shape the number and stability of orbits. They provide detailed analyses of massive and photon orbits, including bifurcation diagrams and velocity curves, and they compare exact results with the weak-field gravito-electromagnetic approximation, validating the approach in the pertinent parameter domain. The study offers a rigorous method to quantify rotation curves in a curved, nontrivial spacetime and suggests directions for extending the model to non-equatorial motion and more realistic matter content with potential astrophysical implications.

Abstract

The gravimagnetic dipole spacetime consists of two counter-rotating black holes of equal mass connected by a Misner string. For a particular distance in between them, the string is tensionless with the black holes at equilibrium with each other. The geodesics of relativistic massive, or massless particles are considered, leading to the identification of circular rotation trajectories. The velocities of these trajectories are computed.

Figures (12)

• Figure 1: This graph shows a cut in the XZ plane of the spacetime. The plain blue lines represent the static limit and the orange dots the black holes. The dashed grey lines represent the horizons. Various quantities ($\alpha_\pm$, the limits of the horizons on the vertical axis) as well as the separation of the black holes $2k^T_+$ are also shown, for a gravimagnetic dipole spacetime with NUT parameter $\nu = 0.4$.
• Figure 2: Cut of the spacetime along the $XZ$ plane for $\nu = 0.1$. The orange dots represent the black holes. The blue lines represent the static limit. The dashed grey lines represent the horizon rods.
• Figure 3: Cut of the spacetime along the $XZ$ plane for $\nu = 0.8$. The orange dots represent the black holes. The blue lines represent the static limit. The dashed grey line represent the horizon rods.
• Figure 4: The left column has a NUT parameter $\nu = 0.2$ and the right column $\nu = 0.8$. The first (resp. second) row presents the first (resp. second) derivative $V'$ (resp. $V"$) of the potential as functions of $\rho$ and evaluated at $b = b_\pm$. Two values of the energy are represented ($E = 2$ and $E = 500$) for the prograde and retrograde directions. The zeroes of the first derivative of the potential indicate the radii at which circular orbits are possible, and the value of the second derivative at these radii indicate if the corresponding orbit is stable or unstable. For example, for $\nu = 0.2$, with an energy of $E = 2$ (green dashdotted line on the left column), there is a stable prograde orbit at $\rho \approx 0.5$.
• Figure 5: For different values of the energy $E$, as functions of the parameter $\nu$, the radii of the possible circular orbits $\rho_\pm$ for massive particles. Depending on the value of the NUT parameter $\nu$ and the energy $E$, there is from $0$ to $4$ possible orbits.