Gravitational Atoms from Topological Stars

Abstract

We study the bound states of a massive scalar field around a topological star, and show that these are strictly normal modes. This yields a genuine gravitational atom, sharply distinguishing horizonless objects from black holes. It is shown that the modes are controlled by the field's Compton wavelength compared to the size of the star. When the Compton wavelength is large, the field forms a cloud with a hydrogen-like spectrum, while in the opposite regime it localizes along timelike trajectories. When the two scales are comparable the spectrum becomes richer, and we characterize it in detail allowing the field to carry electric charge and Kaluza--Klein momentum.

Figures (14)

• Figure 1: Schematic energy spectrum of normal modes at fixed overtone, $E_{\ell}=-\sqrt{\mu^{2}-\omega_{\ell}^{2}}$, as a function of the harmonic multipole $\ell$. Red (blue) dots correspond to inner (cloud) modes. Cloud modes are associated to stable circular geodesics, and terminate approximately at the mode associated to the ISCO, indicated with dashed thin lines. Inner modes instead fluctuate close to the star and are sensitive to its structure.
• Figure 2: Schematic representation of the spatial distribution of neutral (left panel) and charged (right panel) modes. Both states carry angular momentum in the $z$ direction, in the neutral case due to orbital motion, and in the charged one by the Thomson dipole effect.
• Figure 3: Surface $\Sigma_{t}$ in a Carter--Penrose diagram of a TS. The diagram represents the quotient by the $y$-circle and the 2-sphere.
• Figure 4: Schematic representation of the scalar potential $V_{eff}(\tilde{r})$ in \ref{['eq:potEq']}, and the classification of normal modes. Left panel: Case $\Lambda> \Lambda_{-}$. Given a normal mode energy level $\epsilon<0$, this can be either a TS, mixed or cloud mode. Dashed and dotted lines correspond to two illustrative configurations, where classically allowed regions are indicated with thicker markers. The curve $V_{eff}=0$ is the ionization threshold, since modes $\epsilon>0$ are not bound. Right panel: The same, but for $\Lambda< \Lambda_{-}$, where only TS modes exist.
• Figure 5: Binding energy, $\gamma$, \ref{['eq:nsEnCondk01']}, of massive, scalar field modes as a function of $\ell$, in the TS case, for $\eta=1.25$, $\tilde{\mu}=7$, $\tilde{k}=0$, $N=0$, evaluated numerically for the $n=0,\,1,\,2,\,3$ radial overtones in cyan disks, orange diamonds, pink pentagons and green six-pointed stars, respectively. Log plot for clarity. Filled markers are pure TS modes. Empty markers correspond to cloud modes, localized outside of the timelike geodesics ISCO. Half-filled markers indicate mixed modes. The respective regions, where our analysis specifies they can exist are shaded differently for clarity. The dashed green line for the eikonal approximation, \ref{['eq:eikonalApprox']}, and the dashed-dotted magenta line for the large scalar mass approximation, \ref{['eq:largeMuApprox']}, are for $n=0$ only. In black, $\gamma_g=\sqrt{\tilde{\mu}^2\,(1-\mathcal{E}^2)}$ with $\mathcal{E}$ from \ref{['eq:circOrbits']}, represents the geodesic approximation.