Accurate closed-form trajectories of light around a Kerr black hole using asymptotic approximants

Abstract

Highly accurate closed-form expressions that describe the full trajectory of photons propagating in the equatorial plane of a Kerr black hole are obtained using asymptotic approximants. This work extends a prior study of the overall bending angle for photons (Barlow, et al. 2017, Class. Quantum Grav., 34, 135017). The expressions obtained provide accurate trajectory predictions for arbitrary spin and impact parameters, and provide significant time advantages compared with numerical evaluation of the elliptic integrals that describe photon trajectories. To construct approximants, asymptotic expansions for photon deflection are required in various limits. To this end, complete expansions are derived for the azimuthal angle as a function of radial distance from the black hole in the far-distance and closest-approach (pericenter) limits, and new coefficients are reported for the bending angle in the weak-field limit (large impact parameter).

Accurate closed-form solution of the SIR epidemic model

“…An asymptotic approximant is a closed-form expression whose expansion in one region is exact up to a specified order and whose asymptotic equivalence in another region is enforced. The remarkable feature of asymptotic approximants is their ability to attain uniform accuracy not only in these two regions, but also at all points in-between, as demonstrated thus far for problems in thermodynamics, astrophysics, and fluid dynamics [1][2][3][4][5][6][7]. The current need to model and predict viral epidemics motivates us to extend the application of asymptotic approximants to the commonly used Susceptible-Infected-Recovered (SIR) model.…”

Asymptotic Approximant for the Falkner–Skan Boundary Layer Equation

“…Asymptotic approximants provide nearly exact closed-form solutions to the Falkner-Skan boundary layer equation for varying wedge angle. This adds to the increasing number of problems in disparate areas of mathematical physics to which asymptotic approximants have been applied successfully [12,13,14,15,18,17]. Advantages of asymptotic approximants, specifically for the Falkner-Skan problem and in general for other problems, are their simple form, ability to yield highly accurate solutions, accuracy in solution derivatives, and low computational load.…”

“…However, whereas Padé approximants are restricted to rational functions and thus have a specific asymptotic behavior about a chosen expansion point, asymptotic approximants are tailor-made to have the correct behavior in both regions of the domain. For example, asymptotic approximants are shown to accurately describe the light trajectory around a Kerr black hole [17,18] by incorporating the correct logarithmic behavior near the black hole; a Padé is incapable of representing such behavior efficiently. Asymptotic approximants are used to construct accurate solutions for boundary flows over a stationary flat plate (the Blasius problem) and for a flat plate moving through a stationary fluid (the Sakiadis problem) [15].…”

The Rayleigh collapse of two spherical bubbles

“…The evenness of ( 13) is expected, as ( 12) is invariant if the independent variable, t, is replaced with −t [6]. Figure 1 compares the numerical solution of ( 12) with the power series solution (13) for ǫ = 0.1. Note that the numerical solution was obtained via a 4 th order-accurate Runga-Kutta scheme with ∆t = 10 −5 .…”

“…Here, we assure that the approximant matches the exact power series solution as T approaches zero as well as the asymptotic behavior as T approaches the time of bubble collapse. The desirable feature of asymptotic approximants is their ability to attain uniform accuracy not only in these two regions, but also at all points in-between, as demonstrated thus far for problems in thermodynamics [11,12], astrophysics [13], fluid dynamics [10,14], and epidemiology [15,16].…”