Slowly rotating Black Holes in DHOST Theories

Abstract: We study slowly rotating black hole solutions within Degenerate Higher Order Scalar Tensor (DHOST) theories. Starting from a static, spherically symmetric metric solution of a DHOST theory, we employ the Hartle-Thorne ansatz to model a slowly rotating spacetime. We show that the differential equation governing the frame-dragging function $ω$ (which is supposed to depend on the radial coordinate only) is integrable for any DHOST theory allowing us to obtain its explicit form. We also consider angular dependence in $ω$ and show that regularity at the horizon and at infinity forbids it, as in General Relativity. As an illustration of the formalism introduced here, we study the slowly-rotating version of black hole solutions with primary hair obtained recently, examining the influence of the rotation on the Innermost Stable Circular Orbit (ISCO) and on the circular light trajectories in the equatorial plane.

• The paper derives an analytic expression for the frame-dragging function ω(r) from a modified second-order ODE within DHOST theories.
• It demonstrates that disformal transformations map solutions from Horndeski models while preserving the integrability of the slowly rotating metrics.
• The study quantifies observable deviations in ISCO and photon ring radii due to scalar hair, offering new tests for alternative gravity.

Slowly Rotating Black Holes in DHOST Theories: An Authoritative Summary

Introduction and Motivation

The study explores slowly rotating black hole metrics within the general framework of Degenerate Higher Order Scalar Tensor (DHOST) theories, representing the most generic class of scalar-tensor extensions beyond General Relativity (GR) still propagating three degrees of freedom (2 tensor, 1 scalar). While static, spherically symmetric solutions with nontrivial scalar hair are well-established in DHOST, analytic rotating black hole solutions are rare due to increased complexity when spherical symmetry is relaxed. Existing analytic examples mainly rely on disformal transformations applied to stealth Kerr backgrounds. The practical motivation arises from the need to interpret gravitational-wave and electromagnetic observational data potentially deviating from Kerr, as new generation detectors improve probe sensitivity.

Formalism for Slowly Rotating DHOST Black Holes

The authors start from an arbitrary static solution $g_{\mu\nu}^{(0)}$ in DHOST and consider the Hartle-Thorne metric ansatz, which extends the background to linear order in angular momentum $J$:

$$ds^2 = -h(r)dt^2 + \frac{dr^2}{f(r)} + r^2 (d\theta^2 + \sin^2\theta\, d\varphi^2) - 2\omega(r) r^2 \sin^2\theta\, dt\, d\varphi + \mathcal{O}(J^2).$$

Here, the frame-dragging function $\omega(r)$ encapsulates all leading-order rotational corrections. Unlike GR, where $\omega(r) = 2J/r^3$, finding $\omega(r)$ generically in DHOST requires solving modified field equations.

A key result: the field equations for $\omega(r)$ in any quadratic or cubic DHOST theory reduce to a single, explicitly integrable second-order ODE:

$\frac{d}{dr}\left[Q(r)\, \omega'(r)\right] = 0,$

with $Q(r)$ given in terms of the background metric and scalar field profiles, involving only a subset of all DHOST action functions. The integrability arises from a hidden symmetry originating in the coordinate freedom (shift symmetry in $\omega$), which generates a conserved current for $\omega$ even in non-GR theories.

Disformal Transformations and General Solution Structure

Disformal transformations, $$g_{\mu\nu} \rightarrow \tilde{g}_{\mu\nu} = C(X)g_{\mu\nu} + D(X)\nabla_\mu\phi\nabla_\nu\phi$$, preserve the integrability property and relate solution families of different DHOST actions. Analysis demonstrates that under such transformations, the radial profile of $\omega$ in the new metric is linked algebraically to the original—if the coordinate redefinition is invertible.

The upshot is that slowly rotating metrics in shift-symmetric DHOST theories can often be mapped from known Horndeski or "beyond Horndeski" backgrounds, retaining simple forms for $\omega(r)$ whenever disformal factors $C$ approach unity asymptotically (ensuring asymptotic flatness).

Explicit Examples and Geodesic Structure

Focusing on recently constructed quadratic DHOST black holes with primary scalar hair (i.e., scalar charge independent of mass), the paper analyzes the influence of rotation on geodesics. In particular, the metrics are such that $\omega(r) = 2J/r^3$—as in Kerr—even though $h(r), f(r)$, and the form of the scalar profile $\phi=qt+\psi(r)$ depart significantly from Schwarzschild.

Time-like (ISCO) and light-like (photon ring) circular orbits are examined as a function of primary hair parameter $\xi_2$, revealing how deviations from Kerr alter their radii in response to slow rotation. Notably, the ISCO and photon ring radius shifts can be substantial for particular scalar hair configurations.

Figure 1: The non-rotating ISCO radius and BH horizon as functions of the primary hair parameter $\xi_2$, alongside the retrograde ISCO shifts for $a = 0.1$ and $a = 0.2$ from the linearized slow-rotation approximation; all radii in units of BH mass $M$.

Figure 2: The circular light orbit (photon ring) radius as a function of $\xi_2$, with corresponding modifications under slow rotation for retrograde orbits and the behavior of $\partial r^-_{\rm LCO}/\partial a$; units in $M$.

The figures illustrate that both ISCO and photon ring locations are sensitive to scalar hair and rotational corrections, providing potentially observable deviations from standard Kerr predictions.

Angular Dependence and No-Hair Properties

The possibility of $\omega = \omega(r, \theta)$, i.e., angular dependent frame dragging, is addressed. By analyzing the partial differential equations governing $\omega$ and imposing regularity at the horizon and infinity, it is shown that solutions with $\theta$-dependence necessarily violate differentiability or boundedness constraints. This recovers the non-existence of such dependence in GR (by no-hair theorem) and extends the result to the full class of DHOST black holes under consideration.

Implications and Future Prospects

The paper demonstrates that slowly rotating black holes in DHOST theories generically lead to frame-dragging profiles which can be expressed via explicit integrals. In prominent subclasses, the frame-dragging correction matches Kerr even though the background metric and scalar field are distinct, leading to geodesic structures that can differ observably from GR. These results have practical implications for the modeling and interpretation of astrophysical signals from compact objects in strong-field regimes. The analytic tractability of these solutions provides a valuable toolkit for further theoretical investigations in alternative gravity, particularly regarding perturbative and stability analyses.

The method's reliance on a symmetry and integrability property potentially extends to other classes of modified gravity. The framework lays groundwork for systematically classifying rotational corrections in non-GR theories and for comparison with higher-precision observational data anticipated from future gravitational wave and horizon imaging experiments.

Conclusion

This work provides a comprehensive analytic approach for constructing slowly rotating black hole solutions in DHOST theories, clarifies the special role of symmetries in integrability, and quantifies departures from GR in orbital structure. The findings suggest DHOST models can generate observable modifications in strong gravity phenomenology without violating regularity or symmetry constraints, motivating further studies of rotating compact object solutions and their astrophysical consequences.