Scalarised Black-Hole Solutions

• Scalarised black-hole solutions are nontrivial configurations in scalar-tensor theories, featuring a dynamic scalar field that distinguishes them from standard GR cases.
• The model employs a complex boson field with specific frequency and boundary conditions to demonstrate a clear bifurcation between boson stars and ordinary hairy black holes.
• These solutions exhibit enriched ergoregion structures and may exceed Kerr limits, offering potential observational signatures for testing gravity beyond General Relativity.

Scalarised black-hole solutions are nontrivial configurations in alternative theories of gravity—principally, scalar-tensor frameworks—where the black hole is endowed with scalar hair through dynamical gravitational or matter-induced mechanisms. Unlike the Kerr or Schwarzschild solutions of General Relativity (GR), which are uniquely characterised by mass and angular momentum (and electric charge, in the case of Kerr–Newman or Reissner–Nordström), scalarised black holes possess an additional nontrivial scalar field profile. In the presence of a complex scalar field coupled via scalar–tensor interactions, these solutions represent a physically distinct branch that is not continuously connected to the GR vacuum, appearing only under certain parameter and boundary conditions. Scalarised black holes can violate familiar GR constraints, such as the Kerr angular-momentum bound or the uniqueness theorems, and demonstrate new structures in their ergoregions and global properties (Kleihaus et al., 2015).

1. Existence Conditions and Domain of Scalarised Solutions

The scalarisation mechanism for black holes in scalar–tensor theories with a complex boson field is determined by the existence of a nontrivial scalar field (often denoted ϕ or Φ) that is "excited" in conjunction with a horizon-supporting metric. Solutions are constructed using a metric parameterisation with horizon coordinate r_H and a boson field ansatz:

$$\Psi(r, \theta, t, \phi) = \psi(r,\theta)\,e^{i\omega t+in\phi}$$

where the frequency–quantum number relation $\omega = n\Omega_H$ (with $\Omega_H$ the horizon angular velocity) is enforced.

The physical parameters, such as mass $M$, angular momentum $J$, and Noether charge $Q$, are extracted from the asymptotic behaviour of the metric:

$$f_0 \to 1 - \frac{2\mu}{r}, \qquad f_3 \to \frac{2J}{r^3}$$

and the total mass is given by $M = \mu + (r_H/2)$ in coordinates where $r = \sqrt{\bar{r}^2 - \bar{r}_H^2}$.

Scalarised solutions exist in a domain of the rescaled parameter space, typically in $(M/M_0, \omega/\omega_0)$, that is bounded by

• the branch of scalarised rotating boson stars (horizonless, globally regular solutions) from above,
• and the branch of ordinary hairy black holes (solutions with trivial gravitational scalar field) from below.

The nontrivial scalar field is supported when the boson frequency $\omega$ lies within a window:

$\omega_{\rm cr}^2 < \omega < \omega_{\rm cr}^1$

where $\omega_{\rm cr}^1$ signals the bifurcation from ordinary boson stars (scalarisation onset) and $\omega_{\rm cr}^2$ marks the point where the solution merges back with the GR branch (scalar “turn-off”). The parameter $\omega_{\rm cr}^1$ is nearly independent of the rotational quantum number $n$ (Kleihaus et al., 2015).

2. Relationships with Boson Stars and GR Hairy Black Holes

Three principal classes of solutions arise in the same theory:

• Scalarised Rotating Boson Stars: Regular, horizonless objects which bifurcate from ordinary boson stars at $\omega_{\rm cr}^1$ and merge back at $\omega_{\rm cr}^2$. Their energetic favourability depends on $n$; for low $n$ the ordinary branch is preferred, for larger $n$ scalarised stars may attain a global maximal mass not found in the ordinary sector.
• Ordinary Hairy Black Holes: Solutions with a nonzero horizon parameter but a trivial (zero) gravitational scalar field, existing in the GR framework.
• Scalarised Hairy Black Holes: "Dressed" black holes carrying nontrivial scalar hair, defined within the domain bounded by the aforementioned boson star branch and the ordinary hairy black holes. This scalarised sector extends to lower boson frequencies (for small $n$) and to higher mass and charge (for large $n$), thus augmenting the solution space with configurations unavailable in GR.

The inclusion of the gravitational scalar field modifies the energy landscape and stability properties, introducing physically distinct configurations that can be energetically or dynamically favoured over their GR analogues.

3. Global Geometric and Physical Properties

Scalarised rotating hairy black holes show global features analogous—yet distinct—to those of their GR cousins, marked by:

• Exceedance of the Kerr Bound: Both ordinary and scalarised hairy black holes may violate the GR Kerr bound $m_{\rm Pl}^2 J/M^2 \leq 1$. The dimensionless ratio $m_{\rm Pl}^2 J/M^2$ can exceed unity for scalarised solutions; this is evidenced by explicit parameter-plane diagrams in Fig. 3 of (Kleihaus et al., 2015).
• Ergo-Saturn Configurations: The ergoregion structure is notably enriched. The ergosurface (determined by the vanishing of the norm of $\chi = \xi + \Omega_H \eta$) can split into a standard ergosphere (spherical topology) and a distinct ergoring (toroidal), together constituting an "ergo-Saturn". This complex structure, presented in contour diagrams [Fig. 4, (Kleihaus et al., 2015)], illustrates that the distribution of energy and angular momentum can produce topologically intricate regions where the Killing horizon is approached.
• Conserved Charges and Quantities: Quantities such as Komar mass, angular momentum, and Noether charge are defined using asymptotic expansions and global phase symmetries of the underlying boson field, directly influenced by the coupling to the gravitational scalar.

4. Implications for Scalar–Tensor Theories and Extensions of GR

The existence and properties of scalarised black-hole solutions have significant ramifications for the status of strong-field deviations from GR:

• Strong-Field Scalarisation: Scalar-tensor theories permit nonperturbative scalarisation effects in the strong-gravity regime, analogous to spontaneous scalarisation in neutron stars, but arising for rotating compact objects.
• Novel Physical Phenomena and Observability: The enhancements in mass, particle number, and angular momentum relative to GR maxima may provide observational discriminants between scalar-tensor models and pure Einstein gravity. The presence of ergoregion structures such as ergo-Saturns introduces possible new environments for dynamical instabilities or superradiant phenomena, potentially with astrophysical signatures.
• Theoretical Insights: The dynamical scalar field plays the role of an effective, spacetime-dependent gravitational coupling. The spectrum, stability, and bifurcation behaviour of these solutions supply essential information for efforts to test, constrain, or extend gravity theories beyond GR through both mathematical and observational probes.

5. Visualization and Parameter Space Structure

Key features and properties of scalarised black holes are lucidly illustrated in multi-dimensional parameter space plots:

• Phase Diagrams: Mass vs. boson frequency, angular momentum vs. mass, and particle number vs. frequency diagrams explicitly visualise bifurcations, domain boundaries, and the interconnectedness of boson stars, ordinary, and scalarised black holes.
• Contour Plots: Energy densities, particle number densities, and ergoregion boundaries (e.g., the juxtaposition of ergosphere and ergoring) are mapped in the vicinity of the black hole horizon, conveying the spatial distribution of field configurations and global charges.

These figures (Figs. 1–4 in (Kleihaus et al., 2015)) are essential for identifying the parameter regimes where scalarised solutions exist and for understanding their relation to both horizonless compact objects and traditional GR black holes.

6. Summary and Outlook

Scalarised black-hole solutions in scalar–tensor theories exemplify the rich phenomenology available once the assumption of a trivial gravitational scalar is relaxed. The emergence of these solutions is driven by a coupling to a complex boson field, whose frequency–rotation relationships and boundary conditions define sharply delimited domains of existence, bounded by physically meaningful thresholds ($\omega_{\rm cr}^1$ and $\omega_{\rm cr}^2$). Scalarised black holes can possess angular momentum in excess of the Kerr limit and feature multi-component ergoregions such as the ergo-Saturn, all while being embedded in an enlarged phase space compared to GR.

Their paper not only clarifies the properties of compact objects in scalar-tensor and extended gravity theories but also exposes novel avenues for confronting these theories with empirical observations—especially via deviations in gravitational wave signatures, superradiant phenomena, and high-curvature astrophysical processes. The explicit mapping of domains and the detailed characterisation of global properties (mass, angular momentum, ergoregion morphology, thermodynamic indicators) underscore the importance of scalarised solutions in the broader effort to understand and test the gravitational interaction beyond General Relativity (Kleihaus et al., 2015).