Analytic and accurate approximate metrics for black holes with arbitrary rotation in beyond-Einstein gravity using spectral methods

TL;DR

This work develops a general, spectrally driven framework to construct analytic, closed-form rotating black-hole solutions in beyond-Einstein EFTs to leading order in small couplings. By deforming the Kerr solution and expanding both the scalar field and the metric corrections in spectral bases, the authors transform the coupled EFT field equations into algebraic problems for spectral coefficients, solvable at each spin value and then fitted to yield global analytic expressions in spin. Applied to sGB, dCS, and AD gravity, the approach delivers near machine-precision accuracy up to a = 0.9–0.999, outperforming traditional slow-rotation series, and enables precise computations of observables such as surface gravity, horizon angular velocity, ISCO, and photon ring. The resulting analytic solutions enable efficient, high-fidelity modeling for gravitational-wave and black-hole imaging tests of general relativity, and set the stage for quasinormal-mode analyses and extreme-mass-ratio inspiral studies in beyond-Einstein theories.

Abstract

A key obstacle for theory-specific tests of general relativity is the lack of accurate black-hole solutions in beyond-Einstein theories, especially for moderate to high spins. We address this by developing a general framework--based on spectral and pseudospectral methods--to obtain analytic, closed-form spacetimes representing stationary, axisymmetric black holes in effective-field-theory extensions of general relativity to leading order in the coupling constants. The approach models the spacetime (and extra fields) as a stationary, axisymmetric deformation of the Kerr metric in Boyer-Lindquist-like coordinates, expands metric deformations as a spectral series in radius and polar angle, and converts the resulting beyond-Einstein field equations into algebraic equations for the spectral coefficients. For any given spin, these equations are solved via standard linear-algebra methods; the coefficients are then fitted as functions of spin with non-linear functions, yielding fully analytic metrics for rotating black holes in beyond-Einstein theories. We apply this to quadratic gravity theories--dynamical Chern-Simons, scalar-Gauss-Bonnet, and axi-dilaton gravity--obtaining solutions valid for any spin, including near-extremal cases with errors below machine precision for $a \leq 0.9$ and $\lesssim 10^{-8}$ for $a \leq 0.999$. We show that existing slowly-rotating solutions break down at $a \sim (0.2, 0.6)$, depending on approximation order and chosen accuracy. We then use our metrics to compute observables, such as the surface gravity, horizon angular velocity, and the locations of the innermost circular orbit and the photon ring. The framework is general and applicable to other effective-field-theory extensions for black holes of any spin.

Figures (16)

• Figure 1: Scalar field backward modulus difference $\mathcal{B}_{\vartheta}(N)$, defined by Eq. (\ref{['eq:BWD_scalar_field']}) in the main text, of the sGB and dCS scalar field solution as a function of spectral order $N$ computed at different values of spin $a$.
• Figure 2: The absolute error of the sGB and dCS scalar field equation against spectral order $N$. Each solid (dashed) line represents the trends of dCS (sGB) scalar field at different values of spin. The circles (squares) on each line mark the spectral order at which the residual error of the spectral dCS (sGB) scalar field becomes lower than that of the series-in-$a$ solution up to ${\cal{O}}(a^{15})$.
• Figure 3: The absolute error of the sGB (dashed lines) and dCS (solid lines) scalar field equation computed for the spectral and series-in-$a$ solution. The $N = 45$ spectral solutions (blue) are taken to compare with the series-in-$a$ solutions up to ${\cal{O}}(a^{15})$ (orange) and ${\cal{O}}(a^{40})$ (green).
• Figure 4: Meridional cross-section of the scalar field profiles for sGB (right) and dCS gravity (left) at $a = 0.9$, computed at spectral order $N = 50$. The color bars indicate the value of the scalar fields, while the contour lines illustrate their multipolar structure. Observe that the dCS scalar field is odd in parity, while the sGB scalar field is even in parity, as expected.
• Figure 5: The total backward modulus difference of the metric corrections $\mathcal{B}_T(\mathcal{N})$ in dCS (left) and sGB (right) gravity as a function of spectral order $\mathcal{N}$ computed at different values of spin $a$.