Circular geodesics in a New Generalization of q-metric

TL;DR

This work generalizes the static q-metric by embedding it in an external, quadrupole gravitational field using Weyl methods, yielding a three-parameter family described by the total mass $M$, deformation $\alpha$, and distortion $\beta$. The authors derive the generalized metric with explicit external-field contributions and formulate the geodesic problem via an effective potential, focusing on equatorial circular orbits and the innermost stable circular orbit (ISCO). They show that the external quadrupole shifts ISCO locations and can even produce bound photon orbits for certain negative $\beta$, highlighting rich strong-field phenomenology beyond the standard Schwarzschild or pure q-metric spacetimes. The results have potential implications for modeling accretion disks, quasi-periodic oscillations, and gravitational-wave sources in non-Kerr, deformed environments, and point to future work on stationary extensions, off-equatorial geodesics, and matching to realistic external matter distributions.

Abstract

This paper introduces an alternative generalization of the static solution with quadrupole moment, the $\rm q$-metric, that describes a deformed compact object in the presence of the external fields characterized by multipole moments. In addition, we also examine the impact of the external fields up to quadrupole on the circular geodesics and the interplay of these two quadrupoles on the place of the innermost stable circular orbit (ISCO) in the equatorial plane.

Figures (9)

• Figure 1: Plots of $V_{\rm Eff}$ for different values of $\alpha$ and $\beta$. In both plots, the dot-dashed line in the middle is Schwarzschild $(\alpha=\beta=0)$, the solid line under the Schwarzschild corresponds to values $(\alpha=0.5, \beta=0)$, and the solid line above the Schwarzschild corresponds to $(\alpha=-0.4, \beta=0)$. The dotted line under the Schwarzschild is for the values $(\alpha=0.5, \beta=-0.00001)$, and the dotted line above the Schwarzschild is plotted for $(\alpha=-0.4, \beta=-0.00001)$. The dashed line under the Schwarzschild is for values $(\alpha=0.5, \beta=0.00001)$, and the dashed line above the Schwarzschild is plotted for $(\alpha=-0.4, \beta=0.00001)$.
• Figure 2: The dashed lines are $\mathnormal{l}_1$\ref{['curve1']} and $\mathnormal{l}_2$\ref{['curve2']}. The solid lines are the plots of $\beta$\ref{['qsol']} for $\alpha=1$, noted as $\beta^{\alpha=1}$. Minimum of $\beta^{\alpha=1}$ is $-0.0047632$ at $x=5.94338$, and maximum of $\beta^{\alpha=1}$ is $0.0000659$ at $x=13.38972$. Moreover, $\beta=0$ at $x=10.35890$, so this is the place of ISCO for $\alpha=1$ with vanishing $\beta$ in the q-metric.
• Figure 3: The dot-dashed lines depict $\mathnormal{l}_1$\ref{['curve1']} and $\mathnormal{l}_2$\ref{['curve2']}. The solid lines present plots of $\beta$\ref{['qsol']} for $\alpha=-0.4$, noted as $\beta^{\alpha=-0.4}$. Minimum of $\beta^{\alpha=-0.4}$ is $-0.0805014$ at $x=1.55038$, and maximum of $\beta^{\alpha=-0.4}$ is $0.0011090$ at $x=3.47165$. Also, $\beta=0$ at $x=2.69443$, so this $x$ shows the value of ISCO for $\alpha=-0.4$ with $\beta=0$ in the q-metric.
• Figure 4: Around $\alpha= - 0.5$ curve $\beta$ begins to intersect with curve 1 \ref{['curve1']}. Before this value, the minimum is determined by the minimum of $\beta$ itself, and after that, the minimum of $\beta$ is obtained by the intersection of these two curves.
• Figure 5: The dotted line is $\mathnormal{l}_1$\ref{['curve1']}, and solid line is $\beta$\ref{['qsol']} for $\alpha=-0.526$. In this interval $[-0.5528,-0.5)$ the minimum is determined by the minimum of the curve $\beta$\ref{['qsol']} itself.