Isotropic Coordinates for Generalized Schwarzschild-like Solutions

TL;DR

This paper develops a coordinate transformation to isotropic coordinates for a broad class of static, spherically symmetric solutions with multiple anisotropic fluids, improving the analysis and simulation of such spacetimes.

Contribution

It derives a general transformation from Schwarzschild-like to isotropic coordinates for complex, multi-fluid solutions, facilitating better numerical and analytical studies.

Findings

Coordinate transformation removes horizon coordinate pathologies.

Isotropic form clarifies geometric quantities like ADM mass.

Enables well-posed initial data for numerical relativity.

Abstract

We consider a broad class of static, spherically symmetric generalized Schwarzschild-like solutions with multiple non-interacting anisotropic fluid sources and derive the coordinate transformation from Schwarzschild-like (curvature) to isotropic coordinates with conformally flat spatial slices. The isotropic form removes spatial-sector coordinate pathologies at the horizon, clarifies geometric quantities (e.g., ADM mass and curvature invariants), and enables the construction of well-posed initial data on t=const hypersurfaces, suitable for the Hamiltonian and conformal formulations of numerical relativity and for perturbation theory. The backgrounds in isotropic coordinates we develop make it straightforward to separate environmental effects from intrinsic strong-gravity signals and meet the growing interest in non-vacuum black hole phenomenology across scattering, lensing, and waveform modeling.