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Isotropic Coordinates for Generalized Schwarzschild-like Solutions

Zeyu Zeng, Elena Kopteva

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.

Isotropic Coordinates for Generalized Schwarzschild-like Solutions

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.
Paper Structure (57 sections, 8 theorems, 206 equations, 2 tables)

This paper contains 57 sections, 8 theorems, 206 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Setup and Conventions
  3. General setup
  4. Parameter range in the generalized Kiselev-type ansatz
  5. Assumptions and Domain
  6. General Isotropic-Coordinate Transformation
  7. Corollaries and Special Cases
  8. Schwarzschild limit
  9. Reissner--Nordström solution
  10. Series representation and consistency with the general integral
  11. Köttler metric
  12. Isotropic coordinates via elliptic functions
  13. Consistency with the general transformation formula
  14. Differential check.
  15. Series viewpoint.
  16. ...and 42 more sections

Key Result

Theorem 3.1

Under assumptions (A1)--(A3) and the domain definitions specified in §sec:domain, any static spherically symmetric metric of the form admits an isotropic representation where the isotropic radius is defined implicitly by with Here $F(\rho)=f(r(\rho))$ and $\Phi(\rho)^2=r(\rho)/\rho$. Moreover, $\rho$ is strictly increasing on $\mathcal{S}=(r_+,r_\uparrow)$, and therefore admits a unique inverse

Theorems & Definitions (19)

  • Theorem 3.1: Main Result
  • proof
  • Remark 3.1
  • Proposition 3.1: Multi-index expansion for generalized Kiselev-type backgrounds
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • ...and 9 more