Fast and accurate analytical formulas for light propagation in general static, spherically symmetric spacetimes

TL;DR

This work addresses the problem of efficiently modeling light propagation in strong gravitational fields beyond Schwarzschild by deriving fully analytical, fast mappings between emission points and observer images for general static, spherically symmetric spacetimes. Building on Beloborodov–Poutanen-type approaches, the authors develop a small-$v$ expansion in $v^2=1-\,\cos\alpha$ and express the trajectory through metric-dependent integrals $\,\mathcal{I}_{1,2,3}$, inverting to obtain an explicit $x(y)$ relation with coefficients $B_1,B_2,B_3$ plus a metric-dependent empirical correction $oldsymbol{-\frac{e}{100}\gamma(R)y^{2}[\ln(1-y/2)+y/2]}$; the linear limit recovers a generalized Beloborodov formula. They apply the framework to three metric families—Johannsen–Psaltis (JP), Rezzolla–Zhidenko (RZ), and Einstein–Maxwell–dilaton–axion (EMDA)—and show that the full approximation achieves sub-3% accuracy across strong-field regimes, enabling rapid, accurate isoradial disk imaging and QU-diagram polarization analyses without full numerical ray tracing. The results offer a practical tool for testing gravity theories with black holes and neutron stars, providing fast analytic capabilities applicable to disk imaging, polarization, and pulsar light curves, while paving the way to extensions to axisymmetric spacetimes and higher-order imaging. Overall, the work significantly accelerates strong-field phenomenology by delivering robust semi-analytic prescriptions for light propagation in a broad class of spacetimes.

Abstract

In this article, we extend our previously presented analytical formulas (Phys.Rev.D 109 (2024) 12, 124055) for describing light rays passing near or emitted in the vicinity of compact objects to a broader class of spherically symmetric, static spacetimes, including the Johansen-Psaltis and Rezzolla-Zhidenko metric families. The generalized formulas retain the simplicity and accuracy of the original approach while allowing for more general deviations from Schwarzschild geometry. These expressions provide an approximate yet accurate mapping between emission points and the image plane of an asymptotic observer, enabling fast analytical computations of accretion disk images, polarization of the emitted radiation, luminosity curves associated with pulsars, and other related applications. As examples, we compute isoradial curves for several metric families and the Stokes parameters Q and U for a hot spot orbiting near a black hole described by one of the studied metrics, presenting the corresponding polarization (QU) curves.

Figures (9)

• Figure 1: Geometry of a light ray traveling from the vicinity of a compact object to a distant observer. The ray reaches the observer, located in the asymptotic region ($\psi = 0$), with an impact parameter $b$. The ray's point of closest approach (periastron) occurs at the radial coordinate $r = p$. At a given coordinate $r=R$, $\alpha$ represents the angle between the radial direction and the light ray’s path.
• Figure 2: This figure compares the numerical integration of the exact relation $\cos\psi$ to $\cos \alpha$ (labeled 'Exact') with the analytical approximations from Eq. \ref{['eq:our_aprox_full']} ('Approx. T') and \ref{['eq:our_aprox_belo']} ('Approx. L'). The values $r_H$ and $r_c$ denote the radial coordinates of the event horizon and photon sphere, respectively, for the Johannsen-Psaltis metric. The bottom panel shows the relative error $\delta \alpha/\alpha \%$. Solid lines indicate the error for 'Approx. T' (vs 'Exact'), while dashed lines indicate the error for 'Approx. L' (vs 'Exact').
• Figure 3: Similar to Figure \ref{['fig:johan_cospsi']}, this plot compares the numerical integration of the exact relation $\cos \psi \text{ vs}\cos \alpha$ with the approximations \ref{['eq:our_aprox_full']}\ref{['eq:our_aprox_belo']} for the Rezzolla-Zhidenko metric.
• Figure 4: Similar to Figures \ref{['fig:johan_cospsi']} and \ref{['fig:rezzo_cospsi']}, this plot provides the comparison for the Einstein-Maxwell-dilaton-axion metric.
• Figure 5: The geometry of the different coordinate frames is illustrated. The orbit of fluid element $P$ at radius $r = R$ (board), the plane containing the geodesic trajectory (blue), and the plane of the distant observer with its corresponding projection (gray) are represented. Vectors used for the study of polarization in Sec. \ref{['sec:qu_diagrams']} are also shown: the fluid velocity (purple) and the equatorial projection of the magnetic field (red).