Velocity rotation curves in the gravimagnetic dipole spacetime

Technical Summary: Velocity Rotation Curves in the Gravimagnetic Dipole Spacetime

Problem Statement
This work investigates the geodesic motion of relativistic massive and massless test particles within the gravimagnetic dipole spacetime. This specific solution to the Einstein field equations describes an axisymmetric, asymptotically flat, and stationary metric consisting of two counter-rotating NUT (Newman-Unti-Tamburino) black holes of equal mass but opposite NUT charges. These black holes are connected by a Misner string. The study focuses on the configuration where the separation between the black holes is precisely tuned such that the Misner string is tensionless, placing the system in equilibrium. The primary objective is to derive the exact velocity rotation curves for circular orbits in the equatorial plane of this spacetime and to compare these exact results with previous approximate analytic expressions derived from a gravito-electromagnetic approximation in the weak-field regime.

Methodology
The authors employ a Hamiltonian formalism to analyze particle dynamics. Starting from the first-order action for a free point particle, they utilize the stationarity and axisymmetry of the metric to identify conserved quantities: the energy $E$ and the angular momentum $L$.

1. Metric Characterization: The gravimagnetic dipole metric is defined in Weyl coordinates. The condition for a tensionless Misner string is applied to eliminate the separation parameter $k$ in favor of the NUT charge $\nu$ and the mass scale $m$. The system is subsequently adimensionalized by setting $m=1$, leaving the NUT parameter $\nu$ as the sole governing parameter.
2. Effective Potential: An effective potential $V(\rho, b)$ is derived for motion in the equatorial plane ($z=0$), where $b = cL/E$ is the impact parameter. Circular orbits are identified by the conditions $V(\rho) = 0$ and $V'(\rho) = 0$.
3. Orbit Stability and Existence: The stability of these orbits is determined by the sign of the second derivative of the potential, $V''(\rho)$. The authors analyze the number of solutions for circular orbits as a function of the NUT parameter $\nu$ and particle energy $E$, constructing bifurcation diagrams to visualize how orbits coalesce and disappear as $\nu$ increases.
4. Velocity Calculation: The coordinate velocity $v_t$ (as measured by an asymptotic observer) is computed for both massive and massless particles. This involves solving for the energy $E$ corresponding to a specific radius $\rho$ using the condition for circular orbits ($dp_\rho/du = 0$) and the mass-shell constraint. The proper spatial distance is also defined via the radar distance method to relate coordinate radii to physical distances.

Key Contributions and Results

• Exact Velocity Curves: The paper provides exact analytical and numerical expressions for the rotation velocities of massive and massless particles in circular orbits. These curves are plotted against both the coordinate radius $\rho$ and the proper radius $\bar{\rho}$ for various values of the NUT parameter $\nu$.
• Orbital Bifurcation: The analysis reveals that the number of possible circular orbits (ranging from zero to four) depends critically on $\nu$ and the particle's energy. As $\nu$ increases, prograde and retrograde orbits coalesce and eventually vanish. The locations of these saddle-node bifurcations are numerically identified (e.g., for massless particles, bifurcations occur at $\nu \approx 0.27$ and $\nu \approx 0.79$).
• Photon Orbits and Discontinuities: The study identifies the existence of photon orbits (massless particle circular orbits) which act as boundaries for the existence of massive particle orbits. In regions where photon orbits exist, the energy required for a massive particle to maintain a circular orbit becomes infinite or imaginary, leading to discontinuities in the velocity curves.
• Comparison with Approximations: The exact results are compared with the approximate velocity rotation curve derived in a previous study [6] using a gravito-electromagnetic approximation. In the regime where the NUT parameter approaches unity ($\nu \approx 1$), corresponding to the tensionless string limit, the exact results show close agreement with the approximate analytic expressions.
• Stability Analysis: The authors map the stability of orbits, distinguishing between stable ($V'' < 0$) and unstable ($V'' > 0$) configurations based on the second derivative of the effective potential.

Significance and Claims
The paper claims to establish a general method for computing velocity rotation curves for any axisymmetric, stationary, and asymptotically flat metric, which is then specifically applied to the gravimagnetic dipole spacetime. The significance lies in providing exact results that validate and extend previous weak-field approximations. By demonstrating that roughly flat rotation curves can emerge in this specific vacuum solution without invoking dark matter (a finding noted in the context of the approximate study [6]), the work offers a rigorous general relativistic framework for exploring such phenomena. The authors modestly note that the current analysis is confined to the equatorial plane and suggest that future work could explore non-equatorial oscillatory behaviors or the inclusion of matter distributions to better model realistic astrophysical scenarios.