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Adiabatic Anisotropic Gravitational Collapse in Painlevé-Gullstrand Coordinates: A Geometric Analysis

G. Abellán, N. Bolívar, A. Alexandrova, I. Vasilev

TL;DR

The paper develops a complete analytical model of adiabatic, anisotropic gravitational collapse using Painlevé-Gullstrand coordinates throughout interior and exterior regions, ensuring a coordinate-unified description and avoiding boundary artifacts. By choosing a self-consistent density profile inspired by Oppenheimer–Snyder, it closes Einstein's equations for an anisotropic fluid and applies Israel junction conditions to obtain closed-form surface evolution, horizon formation, and causal structure. A striking feature is the appearance of a double apparent-horizon phase inside the matter, with the exterior event horizon stabilizing at the Schwarzschild radius $r=2M_0$, and a critical initial-compactness relationship governing immediate horizon formation. The model exhibits systematic violations of standard energy conditions, highlighting the limitations of idealized anisotropic matter and marking the boundary where classical descriptions may fail, while offering exact analytic benchmarks and geometric insights into horizon dynamics useful for both theoretical and numerical studies.

Abstract

We present a detailed geometric analysis of adiabatic, anisotropic gravitational collapse formulated in a single Painlevé-Gullstrand coordinate system that covers both the interior and exterior, thereby eliminating cross-chart matching artifacts. Building on the Oppenheimer-Snyder framework with a phenomenologically motivated energy-density profile, we enforce the Israel junction conditions and obtain closed-form surface evolution. Within this unified chart we derive exact solutions for the complete collapse process, characterize the causal structure, and track horizon formation and evolution. In particular, we identify and analyse a double apparent-horizon phase inside the matter and show that the event horizon stabilizes at the Schwarzschild radius. We further obtain critical parameter relations that govern the dynamics, including a threshold linking initial compactness to immediate horizon formation. The model is geometrically self-consistent within Einstein's equations but exhibits violations of the standard point-wise energy conditions, highlighting known limitations of idealized anisotropic matter models and delineating the boundary where classical descriptions become inadequate. Together, these results provide geometric insights, compact analytic benchmarks and a didactic, coordinate-uniform perspective on collapse and horizon dynamics.

Adiabatic Anisotropic Gravitational Collapse in Painlevé-Gullstrand Coordinates: A Geometric Analysis

TL;DR

The paper develops a complete analytical model of adiabatic, anisotropic gravitational collapse using Painlevé-Gullstrand coordinates throughout interior and exterior regions, ensuring a coordinate-unified description and avoiding boundary artifacts. By choosing a self-consistent density profile inspired by Oppenheimer–Snyder, it closes Einstein's equations for an anisotropic fluid and applies Israel junction conditions to obtain closed-form surface evolution, horizon formation, and causal structure. A striking feature is the appearance of a double apparent-horizon phase inside the matter, with the exterior event horizon stabilizing at the Schwarzschild radius , and a critical initial-compactness relationship governing immediate horizon formation. The model exhibits systematic violations of standard energy conditions, highlighting the limitations of idealized anisotropic matter and marking the boundary where classical descriptions may fail, while offering exact analytic benchmarks and geometric insights into horizon dynamics useful for both theoretical and numerical studies.

Abstract

We present a detailed geometric analysis of adiabatic, anisotropic gravitational collapse formulated in a single Painlevé-Gullstrand coordinate system that covers both the interior and exterior, thereby eliminating cross-chart matching artifacts. Building on the Oppenheimer-Snyder framework with a phenomenologically motivated energy-density profile, we enforce the Israel junction conditions and obtain closed-form surface evolution. Within this unified chart we derive exact solutions for the complete collapse process, characterize the causal structure, and track horizon formation and evolution. In particular, we identify and analyse a double apparent-horizon phase inside the matter and show that the event horizon stabilizes at the Schwarzschild radius. We further obtain critical parameter relations that govern the dynamics, including a threshold linking initial compactness to immediate horizon formation. The model is geometrically self-consistent within Einstein's equations but exhibits violations of the standard point-wise energy conditions, highlighting known limitations of idealized anisotropic matter models and delineating the boundary where classical descriptions become inadequate. Together, these results provide geometric insights, compact analytic benchmarks and a didactic, coordinate-uniform perspective on collapse and horizon dynamics.

Paper Structure

This paper contains 33 sections, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Spacetime Geometry
  4. Matter Description
  5. Interior and Exterior Solutions
  6. Interior Solution ($r<R(t)$)
  7. Exterior Solution ($r\geq R(t)$)
  8. Israel Junction Conditions and Surface Dynamics
  9. First Fundamental Form and Surface Radius $R$
  10. Second Fundamental Form
  11. Null Geodesics and Horizon Formation
  12. Null Geodesics in the Collapse Geometry
  13. Apparent Horizon Condition
  14. Interior Apparent Horizon Evolution
  15. Exterior Apparent Horizon
  16. ...and 18 more sections

Figures (4)

  • Figure 1: (left) Evolution of the metric function $\beta$ vs. $r$ for several values of time $t$. The difference inside matter (growing towards negative values) and outside matter (decaying towards zero) is clearly seen. (right) Energy density for various values of $t$. Parameters used, $M_0=1$. The selected values are $t=0$ (solid line), $t=0.3 t_c$ (dashed line), $t=0.6t_c$ (dot--dashed line), $t=0.7t_c$ (short--dashed line).
  • Figure 2: (left) The formation of two apparent horizons is observed at $t=0$. When the apparent horizon $r^{(+)}_{\text{AH}}$ (short dashed line) at $t=t_H$ reaches the surface of the distribution (solid line), it acquires the critical value $2M_0$ which coincides with the Schwarzschild radius. The apparent horizon $r^{(-)}_{\text{AH}}$ (long dashed line) is always within the distribution. (right) Several null geodesics are plotted. The dashed line indicates trapped geodesics. The solid line indicates escaping geodesics. A critical geodesic (horizontal dot--dashed line) is also observed right at the $2M_0$ value. The curve for the surface is included for comparison.
  • Figure 3: (left) Radial Pressure. (center) Tangential pressure. (right) Anisotropy factor. Parameters used, $M_0=1$. The selected values are $t=0$ (solid line), $t=0.3 t_c$ (dashed line), $t=0.6t_c$ (dot--dashed line), $t=0.7t_c$ (short--dashed line).
  • Figure 4: Energy conditions. Aside from $\rho\geq 0$, all the constraints for the energy conditions are violated. Parameters used, $M_0=1$. The selected values are $t=0$ (solid line), $t=0.3 t_c$ (dashed line), $t=0.6t_c$ (dot--dashed line), $t=0.7t_c$ (short--dashed line).