Two Spinning Boson Stars (2sBSs)

• Two spinning boson stars (2sBSs) are horizonless, equilibrium solutions in Einstein gravity coupled to a massive complex scalar field, characterized by dual rotating condensates with opposite phase symmetry.
• They exhibit a unique double-torus energy density profile and equatorial antisymmetry, with properties that depend critically on a repulsive quartic self-interaction.
• The 2sBS configuration serves as a solitonic parent for balanced scalar-hairy black holes, offering insights into force balance and the interplay between gravity and nonlinear scalar dynamics.

Searching arXiv for recent and contextual papers on two spinning boson stars. Search query: "two spinning boson stars synchronized scalar hair parity-odd boson stars" Two spinning boson stars (2sBSs) are horizonless, stationary, asymptotically flat solutions of Einstein gravity coupled to a massive complex scalar field, characterized by two separated rotating scalar lumps arranged on a common symmetry axis and by opposite phases across the equatorial plane. In the formulation developed for self-interacting, synchronised scalar hair, they are the solitonic parent configurations from which balanced two-black-hole hairy equilibria are obtained by inserting horizons at the centers of the two components. Their defining physics combines axial rotation, equatorial antisymmetry, and a two-center scalar condensate supported by gravity and scalar self-interaction rather than by any material strut (Liang et al., 19 May 2026).

1. Conceptual position within boson-star theory

In the broader boson-star taxonomy, 2sBSs belong to the intersection of spinning axially symmetric boson stars, parity-odd or axially antisymmetric configurations, and multicomponent or multi-lump bound states. The relevant review literature emphasizes that parity-odd spinning boson stars are antisymmetric under $\theta \to \pi-\theta$, possess an angular node at the equatorial plane, and can be interpreted as out-of-phase bosonic lumps whose scalar repulsion is balanced by gravity. In that sense, the 2sBS construction is the sharp asymptotically flat realization of a stationary, phase-sensitive, gravity-supported two-constituent boson-star configuration rather than a simple orbiting binary (Shnir, 2022).

This distinction is important because the phrase “two spinning boson stars” is used in several nearby but non-identical ways in the literature. In one direction, rotating multistate boson stars constructed from two coexisting complex scalar fields are also “two-state” spinning boson stars, but those solutions describe coexistence of scalar-field states rather than two separated matter concentrations on the symmetry axis (Li et al., 2019). In another direction, generic initial data for binary boson stars concerns dynamical inspiral and merger configurations obtained by superposing isolated boson stars and solving the Einstein constraints; these are binary systems in evolution, not stationary two-center solitons (Siemonsen et al., 2023).

A common misconception is therefore to identify 2sBSs with generic binary boson stars. The dedicated 2sBS solutions are instead regular, equilibrium scalar environments with two separated rotating condensates and equatorial phase opposition. This makes them conceptually closer to parity-odd multicomponent boson-star constellations than to quasi-circular binaries.

2. Einstein–Klein–Gordon model and stationary ansatz

The self-interacting 2sBSs are formulated in the Einstein–Klein–Gordon system with quartic self-interaction,

$$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$

with

$$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$

The field equations are the Einstein equations and the nonlinear Klein–Gordon equation,

$$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$

$\left(\nabla^2-\frac{dU}{d|\Psi|^2}\right)\Psi=0,$

and the global $U(1)$ invariance $\Psi \to e^{i\alpha}\Psi$ yields the Noether current

$$j^\mu=\frac{i}{2}\left(\Psi^*\partial^\mu\Psi-\Psi\partial^\mu\Psi^*\right),\qquad Q=\int_\Sigma j^t\sqrt{-g}\,d^3x.$$

For the 2sBS sector, the standard stationary, axisymmetric metric ansatz is

$$ds^2=-e^{2F_0}dt^2+e^{2F_1}(dr^2+r^2d\theta^2)+e^{2F_2}r^2\sin^2\theta\,(d\varphi-Wdt)^2,$$

with scalar field

$$\Psi=\psi(r,\theta)e^{i(m\varphi-\omega t)},\qquad m=1 \text{ in the study.}$$

The solutions are regular at the origin, asymptotically flat, regular on the axis, and $$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$0-symmetric across the equatorial plane. The equatorial antisymmetry is imposed through

$$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$1

so that the scalar field changes sign across the equator.

The ADM charges are extracted from the asymptotics

$$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$2

and obey the standard quantization relation

$$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$3

For the numerical construction, the fields are rescaled so that one may work in units where $$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$4 (Liang et al., 19 May 2026).

3. Geometric and physical structure of the solutions

Geometrically, a 2sBS consists of two spinning scalar condensates placed symmetrically above and below the equatorial plane. Because the scalar amplitude changes sign across the equator and because $$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$5, the amplitude vanishes both on the equatorial plane and on the symmetry axis. The energy density $$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$6 displays two peaks above and below the equator, while the full three-dimensional matter distribution is a double torus (Liang et al., 19 May 2026).

This double-torus structure is central to the interpretation of the solutions. They are not two isolated boson stars joined by an external mechanism; rather, they are a single regular soliton with a nontrivial nodal structure. The equatorial node encodes the phase inversion, and the two matter peaks represent two spatially separated condensate cores within one globally stationary field configuration.

A useful diagnostic is the proper distance $$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$7 between the two matter concentrations, introduced as a proxy for how close the two components are in the strong-gravity regime. Along the left side of the mass curve, corresponding to the strong-gravity branch, the proper distance between the two components approaches zero. As $$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$8 increases, the configuration becomes more Newtonian and the ADM mass decreases. For the couplings shown, the distance at maximum mass is around $$S=\int d^4x\,\sqrt{-g}\left(\frac{R}{16\pi G}-g^{\mu\nu}\Psi^*_{,\mu}\Psi_{,\nu}-U(|\Psi|^2)\right),$$9 (Liang et al., 19 May 2026).

The optical structure is also nontrivial. The solutions possess up to four relevant light-ring structures, although the detailed light-ring analysis was deferred. This suggests a strong-field geometry rich enough to support nontrivial photon dynamics, even before horizons are introduced.

4. Domain of existence and the role of quartic self-interaction

The 2sBS family is constructed numerically using finite-difference Newton–Raphson methods on compactified coordinates, with reported relative errors below $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$0. Its domain of existence in the $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$1 plane is spiraling, similar to ordinary boson stars, but the quartic term $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$2 reorganizes the family in several robust ways (Liang et al., 19 May 2026).

First, the quartic term is repulsive, and the maximum ADM mass increases with $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$3. The full 2sBS domain of existence is pushed to larger $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$4. This parallels the behavior of ordinary spinning boson stars with repulsive self-interaction, but in the two-center case it has an additional interpretation: for fixed separation $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$5, the ADM mass increases with $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$6, which is interpreted as the self-interaction not only opposing gravitational collapse of each individual lump, but also strengthening the effective repulsion between the two components of the double-torus configuration.

Second, increasing $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$7 dilutes the scalar profile. Both the scalar amplitude $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$8 and the energy density $$U(|\Psi|^2)=\mu^2|\Psi|^2+\lambda|\Psi|^4,\qquad \lambda>0.$$9 decrease in magnitude and become more spread out. In the weak-hair or large-frequency limit, the scalar field is diluted and self-interaction effects become small. In the opposite strong-gravity regime, the quartic interaction strongly reshapes the geometry and matter distribution.

Third, the branch structure acquires a clear physical reading in terms of separation. The strong-gravity side corresponds to small inter-core distance and large relativistic effects; the more weakly bound side corresponds to larger separation and a more Newtonian configuration. This provides a geometric interpretation of the spiral that goes beyond the usual mass–frequency diagram.

5. Ergosurfaces, topology change, and strong-field behavior

The ergosurface is defined by the vanishing of

$$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$0

For 2sBSs, ergoregions appear only in the strong-gravity part of the branch, where the two scalar lumps are sufficiently close. The principal qualitative result is that the quartic self-interaction changes the topology of the ergoregion: for smaller $$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$1, the ergoregion is a single torus; as $$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$2 increases, it undergoes a transition to a double torus (Liang et al., 19 May 2026).

The interpretation given is geometric. The repulsive self-interaction suppresses the scalar density in the region between the two components, modifying the local metric so that the $$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$3 surface disconnects along the symmetry axis. In other words, the matter self-repulsion reorganizes the strong-field frame-dragging region into two disconnected toroidal components.

This behavior should be viewed against the general rotating-boson-star literature, where ergoregions are associated with potential ergoregion instability and with superradiant decay channels in rapidly rotating relativistic configurations (Shnir, 2022). No dedicated dynamical instability analysis of the 2sBS ergoregion was given here, but the presence of ergoregions only on the strong-gravity branch already identifies the most relativistic sector of the family.

6. Parent role for synchronized hairy black holes

Within the same solution space, 2sBSs are the horizonless parents of two distinct black-hole descendants. A single spinning black hole with synchronized quadrupolar scalar hair (1sBH) is obtained by inserting one horizon into the center of a 2sBS-like scalar environment, with synchronization enforced by

$$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$4

A two-black-hole configuration with synchronized scalar hair (2sBH) is obtained by placing two aligned horizons along the symmetry axis inside the scalar environment created by the 2sBS (Liang et al., 19 May 2026).

The two-horizon construction generalizes the Bach–Weyl two-rod picture to a spinning, scalar-hairy, asymptotically flat setting. The key physical picture is that the 2sBS double-torus scalar distribution creates a gravitational potential with three extrema on the symmetry axis: two stable minima and one unstable maximum. Horizons can bifurcate from either the stable points or the unstable one, giving rise to three classes of solution sequences. As the self-interaction becomes stronger, the bifurcation structure changes, and the pair of type-(i) sequences can merge into one type-(ii) and one type-(iii) sequence.

This parent–descendant relation also clarifies the force-balance problem. In vacuum, two stationary black holes require a conical singularity, or strut, to remain in equilibrium. Here, the scalar environment provided by the 2sBS replaces the strut by genuine matter-supported balance. The conical excess or deficit is monitored by a parameter $$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$5, and equilibrium requires $$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$6. The numerical 2sBH solutions achieve $$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$7, i.e. they are essentially balanced and free of conical singularities.

The same analysis yields a notable restriction on horizon properties. Repulsive quartic self-interactions can make the solutions “hairier,” but they cannot make the horizons heavier; horizons carrying a larger mass fraction are obtained only when attractive self-interactions are considered. This places the 2sBS family at the center of a continuous solution space linking regular two-center scalar solitons, one-horizon quadrupolar hairy black holes, and balanced two-horizon hairy configurations.

7. Interpretive significance and open issues

The principal significance of 2sBSs is that they provide explicit asymptotically flat, regular, two-center spinning scalar backgrounds in which force balance is achieved without conical defects. They are therefore more than a curiosity within boson-star phenomenology: they are the solitonic infrastructure that makes possible balanced binaries of spinning black holes with synchronized scalar hair (Liang et al., 19 May 2026).

Several interpretive points follow from the existing literature. First, 2sBSs should not be conflated with the parity-odd zero-angular-momentum “pair of boson stars” emphasized in the review literature, although both rely on phase opposition and gravity-supported balance. The review identifies that pair solution as a saddle point of the Einstein–Klein–Gordon model and stresses the delicate stability of multi-lump equilibria (Shnir, 2022). A plausible implication is that the stationary two-center sector is structurally rich but dynamically delicate, especially once ergoregions are present.

Second, 2sBSs are distinct from rotating multistate boson stars built from two scalar fields or two coexisting states. Those systems realize a different notion of multiplicity—coexisting internal states such as the $$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$8 and $$R_{\mu\nu}-\frac12 g_{\mu\nu}R-8\pi G\,T_{\mu\nu}=0,$$9 families—rather than two separated condensate cores on the symmetry axis (Li et al., 2019).

Third, 2sBSs are distinct from generic binary boson-star initial data for nonlinear evolution. Dynamical binaries constructed in the conformal thin-sandwich framework describe inspiral, merger, and scattering, including spinning and misaligned-spin cases, whereas 2sBSs are exact stationary solutions. This suggests a useful division of labor between the two lines of work: 2sBSs probe equilibrium geometry and bifurcation structure, while binary initial-data studies probe nonequilibrium dynamics and waveform generation (Siemonsen et al., 2023).

Open issues remain explicit. The detailed light-ring analysis was left for future work. No full nonlinear dynamical stability analysis of 2sBSs was presented. The ergoregion topology transition from a single torus to a double torus is established, but its implications for instability timescales or observational signatures remain unresolved. Even so, the existing results already place 2sBSs among the most structured stationary solutions of the Einstein–Klein–Gordon system: double-torus matter distributions, phase-odd symmetry, nontrivial photon dynamics, and direct continuity with balanced two-black-hole scalar-hairy equilibria.