Slowly rotating Black Holes in DHOST Theories

TL;DR

This work analyzes slowly rotating black holes in Degenerate Higher Order Scalar–Tensor (DHOST) theories using the Hartle–Thorne expansion to first order in the angular momentum $J$. It derives a general, integrable equation for the frame-dragging function $\omega(r)$, yielding $\omega(r)=k\int dr/Q(r)$ once the static background is fixed, and shows this mirrors the Kerr result in GR when appropriate limits are taken. A shift-symmetry argument explains the integrability via a conserved current, and the analysis is extended to shift-symmetric DHOSTs, with disformal transformations clarifying how $\omega$ transforms under metric redefinitions. The authors demonstrate that, under regularity at the horizon and infinity, $\omega$ cannot depend on the polar angle $\theta$ for the considered theories, mirroring GR’s no-hair behavior at first order in rotation. They apply the formalism to hairy black holes with primary hair, examining ISCOs and photon orbits, and discuss implications for observational tests of modified gravity in astrophysical settings.

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.

Figures (2)

• Figure 1: Plot of the non-rotating ISCO (thick continuous line) as a function of the parameter $\xi_2$ compared with the BH horizon (dotted line). Both radii are expressed in units of the BH mass $M$. Above the non-rotating ISCO, two thin lines indicate the modified ISCO for $a=0.1$ and $a=0.2$ (in the linear approximation) for retrograde orbits. More generally, the derivative $\partial r^-_{\rm ISCO}/\partial a$ is plotted with a dashed line. For all plots, we take $\lambda=M$.
• Figure 2: Plot of the light circular orbit radius $r_{\rm LCO}$ in the non-rotating solution (thick continuous line), as a function of the parameter $\xi_2$ compared with the BH horizon (dotted line). Both radii are expressed in units of the BH mass $M$. Above the non-rotating case, two thin lines indicate the modified light radius for $a=0.1$ and $a=0.2$ (in the linear approximation) for retrograde orbits. More generally, the derivative $\partial r^-_{\rm LCO}/\partial a$ is plotted with a dashed line. For all plots, we take $\lambda=M$.