There and back again: Outspiraling motion in non-Kerr compact objects

TL;DR

This work investigates circular equatorial orbits in generic spinning spacetimes to identify conditions under which GW-driven dissipation can cause an inspiral to transition into an outspiral, a hallmark of non-Kerr geometries. The authors develop a general formalism based on an effective potential V(R,E,L) for equatorial geodesics and analyze energy loss via quadrupole radiation, deriving a closed-form expression for the radial drift dR/dt that depends on metric functions. They demonstrate, via engineered geometries and spinning boson stars, that a degeneracy C(R_C)=0 can trigger a prograde-to-retrograde transition at R_C, producing outspirals whose endpoints can be a stationary light ring, a stable light ring, or an outward plunge, with GW signatures including a backward chirp. Numerical GW calculations show the backward chirp is robust, though the outspiral phase often yields a diminished GW amplitude, highlighting the potential of these features as smoking guns for non-Kerr spacetimes. The results motivate targeted GW searches for non-monotonic metric signatures and guide future work on eccentric or more realistic energy flux models.

Abstract

In Keplerian dynamics, a test body orbiting a point particle in circular motion has a monotonically increasing frequency, with decreasing radius. If a dissipative channel is introduced, such as gravitational wave (GW) emission, under the quadrupole approximation, the corresponding GW strain has an ever-increasing frequency with time. A similar statement holds for equatorial motion of a test particle on the Kerr manifold, except such inspiral is cut off at the ISCO, wherein stable circular orbits cease to exist and a plunge is expected. We analyze circular timelike orbits in generic spinning spacetimes and study the conditions in which exotic motion can occur, due to the presence of non-Kerr features. In particular, we derive conditions under which an inspiral towards a compact object is naturally followed by an outspiral motion, and give concrete examples as well as the corresponding GW phenomenology. This analysis serves both as a theoretical exploration of non-Kerrness as well as an example of a concrete smoking gun of exotic spacetimes.

Figures (8)

• Figure 1: Illustration of the secondary motion with inspiral and outspiral. (Left) The spinning primary is envisioned as a core (black) and a surrounding environment (orange). Darker tones represent regions with higher density. A central (green) region has no circular orbits and a static ring (blue dashed line) is assumed for retrograde orbits. The arrows in red (green) represent qualitatively the evolution of the angular velocity for prograde (retrograde) orbits $\Omega_+$ ($\Omega_-$), becoming degenerate at $C=0$ (yellow arrows), where $\Omega_+=\Omega_-$. (Right) Radial dependence of the circular orbits' energy. The yellow dot represents the radius at which $C=0$ and $E_+=E_-$. The secondary initially inspirals along prograde orbits, and then continues to lose energy by transitioning to a retrograde orbit at $R_C$ outspiraling towards the SR.
• Figure 2: Toy model results. (Left) Radial profiles of $g_{tt}$ and $g_{t\varphi}$ in \ref{['toy_metric']} are parameterized by $a_1=4$, $k_1=3$ for $g_{tt}$; $a_2=2$, $k_2=4$ for $(g_{t\varphi})_a$; and $a_2=2$, $k_2=4$ for $(g_{t\varphi})_b$. (Right) Radial profile of $\Omega$ for the three distinct $g_{t\varphi}$ profiles. The continuous (dotted) lines correspond to prograde (retrograde) orbits, while the dots mark $C=0$. The inset plot shows the curves zoomed near $R=R_C$
• Figure 3: Structure of prograde (left) and retrograde (right) equatorial circular orbits for rotating BSs in the first branch. The black vertical dashed line on the right panel corresponds to $\omega/\mu=0.658$, from which the endpoint switches from a SR to a stable LR.
• Figure 4: Properties of the rotating BSs. (Left) Emergence of ergoregion from light point orbit, where an ergosurface, a stable LR and the SR coincide. The horizontal dashed line corresponds to $\omega/\mu=0.658$, with the yellow dot representing the light point orbit. (Right) Plot of the radial coordinate where $|\Psi|$ is maximum with the SR and $C=0$.
• Figure 5: Evolution of $R /m$ and $\Omega m$ as a function of time for the engineered geometries. The parameters associated with the $g_{tt}$, $(g_{t\varphi})_a$, and $(g_{t\varphi})_b$ profiles of the underlying geometries can be found in the caption of Fig. \ref{['fig:toymetric']}.