Stealth Black Holes Overview

• Stealth black holes are GR vacuum metrics endowed with nontrivial fields that have vanishing stress–energy, ensuring an identical background geometry.
• They emerge in diverse frameworks—such as Horndeski, Proca, and nonlinear electrodynamics—often introducing secondary or primary hair with distinct thermodynamic implications.
• Subtle differences appear in perturbations, where boundary term shifts and strong coupling issues hint at potential observational signatures beyond classical GR.

A stealth black hole is a solution to a gravitational theory in which the metric is identical to a vacuum black hole of General Relativity (GR) (such as Schwarzschild, (A)dS–Schwarzschild, or Kerr), while a nontrivial additional field (“hair”) is present but has vanishing energy-momentum and thus does not backreact on the geometry at the level of the background equations. The stealth field can be a scalar, vector, or nonlinear electromagnetic field, and is designed such that, despite being nontrivial, its stress–energy tensor vanishes identically on the GR black hole background. While the spacetime geometry is thus indistinguishable from its GR counterpart, subtle distinctions can arise at the level of black hole thermodynamics, perturbations, and conserved charges, depending on the nature of the stealth field and its interactions with gravity or matter sectors.

1. Definition and Construction of Stealth Black Holes

Stealth black holes are defined as follows: Starting from an action

$$S[g,\,\psi] = \frac{1}{2\kappa}\int d^4x\sqrt{-g}(R-2\Lambda) + S_\mathrm{matter}[g,\,\psi]$$

with field equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$, a stealth solution is one for which

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$

with $\psi\neq0$. Thus, the metric $g_{\mu\nu}$ is an exact vacuum (or $\Lambda$-vacuum) black hole of GR, while the “stealth field” $\psi$ is nontrivial but its stress–energy vanishes, so the metric “cannot see” the field. This construction generalizes to scalar-tensor, vector-tensor, nonlinear electrodynamics, and higher-order gravity theories (Bakopoulos et al., 2024).

Stealth fields are either (i) algebraically constrained—entering the action such that their equation of motion leads to an algebraic relation fixing the profile (bona fide stealth), or (ii) dynamically constrained, in which the stealth field obeys a differential equation but with boundary terms adjusted to cancel all contributions to $T_{\mu\nu}$ (dynamical stealth).

2. General Theoretical Frameworks and Explicit Models

Stealth solutions have been constructed in a variety of contexts:

• Horndeski and shift-symmetric scalar-tensor theories: Stealth black holes can arise with both constant and non-constant kinetic scalar field profiles. In bona fide stealth, such as the model with $G_2(X) = -2\Lambda + 2\eta \sqrt{X}$, $G_4(X) = 1+\lambda \sqrt{X}$, the requirement $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$0 fixes $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$1 algebraically, yielding an (A)dS–Schwarzschild metric and a nontrivial scalar profile with no modification to mass or entropy (Bakopoulos et al., 2024, Bakopoulos et al., 2023).
• Kerr and rotating stealth black holes: Shift-symmetric quadratic DHOST theories (class Ia, luminal) admit stealth Kerr and Kerr–(A)dS metrics with a scalar profile of constant kinetic density $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$2 (Achour et al., 2020, Charmousis et al., 2019). The scalar often takes a Hamilton–Jacobi form, and the stealth conditions reduce theory couplings to particular algebraic loci evaluated at $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$3.
• Nonlinear electrodynamics (NLED): Certain NLED models admit nontrivial stealth field configurations (e.g., power–Maxwell or Euler–Heisenberg-type models with $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$4), allowing for electromagnetic hair that does not enter the metric equations (Smolić, 2017).
• Vector–tensor and Proca-type hair: Stealth vector fields arise in generalized Proca or vector Galileon theories with curvature couplings, as well as via the mechanism that elevates any Killing vector field of the geometry to a stealth solution of an appropriate Proca–type equation. In these models, the vector field can carry global charges without affecting the black-hole geometry (Kubiznak et al., 21 May 2026, Chagoya et al., 2023).
• Gauss–Bonnet and higher-curvature gravity: In Einstein–Gauss–Bonnet gravity with nonminimal scalar coupling at the Chern–Simons point, exact AdS planar black holes with stealth scalar hair can be constructed (Gaete et al., 2013).
• Mimetic and Aether–Scalar–Tensor theories: Mimetic gravity admits Schwarzschild stealth configurations with unit-timelike mimetic scalar fields (Gorji et al., 2019), and AeST gravity supports stealth black holes with nontrivial scalar and aether profiles, precisely matching the Schwarzschild/Reissner–Nordström spacetimes (Skordis et al., 2024).

3. Thermodynamics of Stealth Black Holes

The thermodynamics of stealth black holes, as analyzed via the Euclidean action formalism, reveals two archetypal behaviors:

1. Genuine (bona fide) stealth: The stealth field contributes only through algebraic constraints and induces no extra boundary terms in the action. As a result, the mass $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$5, entropy $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$6, and free energy $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$7 coincide exactly with their GR values: $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$8 and $G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}(\psi)$9. The stealth field is entirely hidden at both the geometric and thermodynamic levels.
2. Dynamical stealth: When the stealth field obeys a differential equation and the action involves nonminimal couplings (e.g., $$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$0, $$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$1), its variation produces nontrivial boundary terms at the horizon or infinity. This produces explicit shifts:

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$2

with potential shifts $$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$3, $$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$4 of either sign, depending on the model. The free energy is also shifted:

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$5

If $$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$6, the stealth solution is thermodynamically preferred. However, the presence of these terms means that, at the level of thermodynamics, stealth black holes are distinguishable from their GR counterparts (Bakopoulos et al., 2024).

Boundary terms must be carefully defined to ensure a well-posed variational principle for the action. The extraction of $$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$7 and $$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$8 proceeds by matching boundary terms at infinity and at the horizon to the corresponding ADM mass and entropy contributions.

4. Linear Perturbations and Strong Coupling Pathology

A central feature of stealth black holes in several scalar–tensor and higher-order models is the pathology in their perturbative sector:

• Odd-parity gravitational perturbations (axial sector, Regge–Wheeler modes): The equations of motion and effective potential coincide exactly with GR. Hence, the spectrum of axial quasinormal modes is unchanged, and no new polarizations or modifications of the ringdown are expected.
• Even-parity gravitational and scalar perturbations (polar sector, Zerilli modes): Despite the stealth field’s vanishing stress–energy at the background level, linearized perturbations reveal a degenerate or singular structure. Specifically, the kinetic operator for the scalar perturbation or for the polar sector becomes nonhyperbolic, with a vanishing determinant of the effective metric for perturbations, corresponding to infinite sound speed (“strong coupling”) (Bernardo et al., 2020, Rham et al., 2019, Langlois et al., 2021, Bernardo et al., 2019). For low multipoles ($$G_{\mu\nu} + \Lambda g_{\mu\nu} = 0\,,\quad T_{\mu\nu} = 0$$9), explicit mode analysis shows that the scalar perturbations are either non-gauge (pathological) or diverge at the horizon, invalidating the linear analysis.
• Physical implications: While the background geometry remains GR, the breakdown of perturbation theory signals that exact stealth black holes in these models may not be realized in nature, or require UV completion (e.g., via “scordatura” terms or higher-derivative corrections) (Felice et al., 2022). In “approximately stealth” black holes (small but nonzero detuning), the metric deviations become negligible for realistic cutoff scales, and the strong coupling problem is evaded below the cutoff.

5. Conserved Charges, Hair Structure, and Distinguishing Features

The stealth field can endow black holes with nontrivial hair:

• Secondary (nongravitating) hair: In bona fide stealth, the field is “secondary hair”—it labels a one-parameter family of stealth profiles that do not affect the geometry or local observables. For example, the integration constant $\psi\neq0$0 in Horndeski models parameterizes a family of linearly time-dependent scalar fields with fixed Schwarzschild geometry (Bakopoulos et al., 2023, Bakopoulos et al., 2024).
• Primary stealth hair and global charges: In classes with a stealth electromagnetic field, the hair can carry a physical (Komar) charge. For instance, in nonlinear electrodynamics, the stealth electromagnetic configurations can support black holes with primary magnetic stealth hair, which contributes to the Komar charge but does not modify the metric (Smolić, 2017).
• Vector–tensor and Proca hair: The mechanism that promotes any Killing vector to a stealth Proca configuration provides an infinite family of charged and magnetized stealth black holes in all dimensions, carrying independent electric and magnetic parameters, with nontrivial physical effects for charged test particles (Kubiznak et al., 21 May 2026). In vector–tensor theories, stealth black holes can exhibit modifications to the Bekenstein–Hawking entropy through classical logarithmic corrections proportional to the stealth charge (Chagoya et al., 2023).
• Thermodynamic distinguishability: Nontrivial shifts in mass and entropy from boundary terms, or the presence of hidden hair, provide avenues for distinguishing stealth black holes from GR solutions, at least in principle (Bakopoulos et al., 2024, Chagoya et al., 2023).

6. Extensions and Physical Contexts

Stealth black hole solutions have been constructed in various contexts beyond static black holes in four dimensions:

• Planar and higher-dimensional black holes: Exact stealth solutions are known in maximally symmetric (AdS) backgrounds and higher dimensions, including planar and topological black holes in Einstein–Gauss–Bonnet gravity (Gaete et al., 2013), new massive gravity in three dimensions (Hassaine, 2013), and higher-order Maxwell–Einstein theories (Minamitsuji, 12 Jun 2025).
• Time-dependent and cosmological settings: By conformally mapping stealth black holes to new backgrounds, it is possible to construct solutions with black holes embedded in expanding FLRW universes, providing exact models for rotating black holes in cosmological backgrounds (Babichev et al., 2023).
• Aether–Scalar–Tensor and mimetic gravity: Such models furnish stealth black holes with vector and/or scalar fields whose profiles are regular across the event horizon and may be joined smoothly to cosmological configurations, enabling astrophysically realistic strong-field solutions consistent with observed black holes (Skordis et al., 2024, Gorji et al., 2019).
• Astrophysical constraints and relevance: Approximately stealth black holes in higher-order scalar–tensor theories remain credible EFT solutions if the cutoff is above $\psi\neq0$1 GeV, and the field accretion rate is slow enough to escape observational bounds (Felice et al., 2022). In AeST the $\psi\neq0$2 branch allows matching to an FLRW cosmology, making these holes viable astrophysical objects (Skordis et al., 2024).

7. Summary Table: Stealth Black Hole Classes and Key Properties

Stealth Field Type	Metric Class	Hair Structure	Thermodynamic Shifts	Perturbation Pathology
Horndeski Scalar	Schwarzschild, Kerr	Secondary	Possibly none	Strong coupling (polar)
NLED (power-law)	Schwarzschild, Kerr	Primary (magnetic)	None	None
Vector–Tensor (Proca)	Kerr, Myers–Perry	Primary (vector)	Log-corrected entropy	None for gravitational modes
AeST (scalar+aether)	Schwarzschild/RN	Secondary	None	Extra modes possible
Mimetic Scalar	Schwarzschild	Secondary	None	None
Higher-order Maxwell	Schwarzschild/RN	Primary (EM)	None	None

Thermodynamic shifts refer to genuine corrections to $\psi\neq0$3, $\psi\neq0$4 due to stealth boundary terms; “primary” hair denotes hair with associated Gauss-type global charge; “secondary” is nongravitating.

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The detailed structure and properties of stealth black holes depend sensitively on the action and couplings, the algebraic versus dynamical character of the stealth field’s equations, and the boundary conditions. While the canonical example remains a Schwarzschild or Kerr metric “dressed” with a non-backreacting field, the implications for astrophysical observability, stability, and quantum thermodynamics are model-dependent and, in some cases, constrained by fundamental limitations of EFT or instability of perturbations (Bakopoulos et al., 2024, Felice et al., 2022, Chagoya et al., 2023, Kubiznak et al., 21 May 2026).