AutoTortoise.jl

AutoTortoise.jl is a small Julia package prototype for horizon-aware tortoise-coordinate construction for static spherically symmetric black-hole metrics whose metric function is of Laurent-polynomial form

$$f(r)=\sum_n a_n r^n,$$

where the integer exponent n may be negative, zero, or positive.

The package is intended as infrastructure for QNM, CSM, and CLD calculations. It automates the pipeline

MetricFunction -> Horizons -> TortoiseMap -> InverseMap -> V(r(r_*))

for Laurent-polynomial metric functions.

Current scope

Version 0.1.4 supports:

• Laurent-polynomial metric functions.
• Conversion to a rational integral via P(r)=r^m f(r).
• Polynomial root finding with a companion matrix.
• Positive-real horizon detection.
• Simple-pole partial fractions for 1/f(r).
• Real tortoise maps on static intervals.
• Numerical inverse maps by bracketed bisection.
• Schwarzschild, Schwarzschild--dS, and Schwarzschild--AdS helper constructors.
• An experimental complex inverse map for CSM-style continuation.

Near-extremal or exactly degenerate horizons are detected but not yet treated as a fully reliable repeated-pole problem.

Installation

] add AutoTortoise

Basic usage

using AutoTortoise

f = schwarzschild_ds(M=1.0, Lambda=0.08)
h = horizons(f)
I = static_region(h; prefer=:finite)
tm = tortoise_map(f, I)

r = (I.left + I.right)/2
x = tortoise(tm, r)
r_back = inverse_tortoise(tm, x)

For a custom Laurent-polynomial metric:

f = MetricFunction(-1 => -2.0, 0 => 1.0, 2 => -0.08/3)

This corresponds to

$$f(r)=1-\frac{2}{r}-\frac{0.08}{3}r^2.$$

AdS-like finite boundary convention

For Schwarzschild--AdS,

f = schwarzschild_ads(M=1.0, L=10.0)
h = horizons(f)
I = static_region(h; prefer=:exterior)
tm = tortoise_map(f, I; convention=:boundary_zero)

The convention :boundary_zero normalizes the finite tortoise-coordinate boundary at infinity to r_* = 0 for standard AdS-like rational metrics.

Mathematical structure

For

$$f(r)=\sum_n a_n r^n,$$

choose

$$m=\max(0,-n_{\min}), \qquad P(r)=r^m f(r).$$

Then

$$\frac{1}{f(r)}=\frac{r^m}{P(r)}.$$

For simple roots r_i of P(r), the pole residues are

$$A_i=\frac{r_i^m}{P'(r_i)}=\frac{1}{f'(r_i)}$$

when r_i is a nonzero simple horizon. The tortoise coordinate is represented as

$$r_*(r)=\sum_i A_i \log(r-r_i)+\int^r Q(\bar r)\,d\bar r+C,$$

where Q is the polynomial quotient produced by rational division.

On a real physical interval, real horizon terms are evaluated using the real branch log(abs(r-r_i)).

Roadmap

Planned extensions:

• Fully reliable repeated-pole support for extremal and near-extremal horizons.
• BigFloat and ArbNumerics support for near-degenerate roots.
• Explicit branch tracking for complex-scaled rays.
• Grid generation and potential composition utilities for CSM calculations.
• Documentation examples for Reissner--Nordström, SdS, and AdS cases.