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Scattering laws for interfaces in self-gravitating matter flows

Bruno Le Floch, Philippe G. LeFloch

TL;DR

This work introduces a scattering-law framework for sharp interfaces in self-gravitating matter, coupling phase-transition dynamics with bouncing cosmology via ultra-local junction maps across fluid and geometric singularities. It develops two complementary channels: (i) Einstein--scalar geometric singularities, yielding universal rescaling laws for anisotropy and two ultra-local scattering families (anisotropic and isotropic), and (ii) a hyperbolic two-phase relativistic fluid model with Rankine--Hugoniot and kinetic-type scattering across under-compressive interfaces, plus a parallel treatment of geometric singularities in self-gravitating fluids. The paper then extends these ideas to Einstein--Euler settings, deriving bounded-geometry junction conditions and classifying the admissible scattering maps for quiescent singularities, with explicit forms and invariant-based characterizations. The overarching aim is to provide a consistent, local macroscopic description of evolution with sharp interfaces, enabling a systematic classification of admissible interface conditions under general covariance, causality, and constraint compatibility, with potential applications in astrophysical contexts and numerical relativity. The framework opens avenues for incorporating more complex matter models, viscosity, capillarity, and stability analyses while connecting cosmological bounce scenarios to realistic fluid dynamics in curved spacetime.

Abstract

We consider the evolution of self-gravitating matter fields that may undergo phase transitions, and we connect ideas from phase transition dynamics with concepts from bouncing cosmology. Our framework introduces scattering maps prescribed on two classes of hypersurfaces: a gravitational singularity hypersurface and a fluid-discontinuity hypersurface. By analyzing the causal structures induced by the light cone and the acoustic cone, we formulate a local evolution problem for the Einstein-Euler system in the presence of such interfaces. We explain how suitable scattering relations must supplement the field equations in order to ensure uniqueness and thus yield a complete macroscopic description of the evolution. This viewpoint builds on a theory developed in collaboration with G. Veneziano for quiescent (velocity-dominated) singularities in solutions of the Einstein equations coupled to a scalar field, where the passage across the singular hypersurface is encoded by a singularity scattering map. The guiding question is to identify junction prescriptions that are compatible with the Einstein and Euler equations, in particular with the propagation of constraints. The outcome is a rigid set of universal relations, together with a family of model-dependent parameters. Under physically motivated requirements (general covariance, causality, constraint compatibility, and ultra-locality), we aim to classify admissible scattering relations arising from microscopic physics and characterizing, at the macroscopic level, the dynamics of a fluid coupled to Einstein gravity.

Scattering laws for interfaces in self-gravitating matter flows

TL;DR

This work introduces a scattering-law framework for sharp interfaces in self-gravitating matter, coupling phase-transition dynamics with bouncing cosmology via ultra-local junction maps across fluid and geometric singularities. It develops two complementary channels: (i) Einstein--scalar geometric singularities, yielding universal rescaling laws for anisotropy and two ultra-local scattering families (anisotropic and isotropic), and (ii) a hyperbolic two-phase relativistic fluid model with Rankine--Hugoniot and kinetic-type scattering across under-compressive interfaces, plus a parallel treatment of geometric singularities in self-gravitating fluids. The paper then extends these ideas to Einstein--Euler settings, deriving bounded-geometry junction conditions and classifying the admissible scattering maps for quiescent singularities, with explicit forms and invariant-based characterizations. The overarching aim is to provide a consistent, local macroscopic description of evolution with sharp interfaces, enabling a systematic classification of admissible interface conditions under general covariance, causality, and constraint compatibility, with potential applications in astrophysical contexts and numerical relativity. The framework opens avenues for incorporating more complex matter models, viscosity, capillarity, and stability analyses while connecting cosmological bounce scenarios to realistic fluid dynamics in curved spacetime.

Abstract

We consider the evolution of self-gravitating matter fields that may undergo phase transitions, and we connect ideas from phase transition dynamics with concepts from bouncing cosmology. Our framework introduces scattering maps prescribed on two classes of hypersurfaces: a gravitational singularity hypersurface and a fluid-discontinuity hypersurface. By analyzing the causal structures induced by the light cone and the acoustic cone, we formulate a local evolution problem for the Einstein-Euler system in the presence of such interfaces. We explain how suitable scattering relations must supplement the field equations in order to ensure uniqueness and thus yield a complete macroscopic description of the evolution. This viewpoint builds on a theory developed in collaboration with G. Veneziano for quiescent (velocity-dominated) singularities in solutions of the Einstein equations coupled to a scalar field, where the passage across the singular hypersurface is encoded by a singularity scattering map. The guiding question is to identify junction prescriptions that are compatible with the Einstein and Euler equations, in particular with the propagation of constraints. The outcome is a rigid set of universal relations, together with a family of model-dependent parameters. Under physically motivated requirements (general covariance, causality, constraint compatibility, and ultra-locality), we aim to classify admissible scattering relations arising from microscopic physics and characterizing, at the macroscopic level, the dynamics of a fluid coupled to Einstein gravity.
Paper Structure (35 sections, 59 equations, 2 figures)

This paper contains 35 sections, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scattering maps for Einstein--scalar geometric singularities
  3. Notion of singularity scattering map.
  4. Spacelike singularities.
  5. Timelike singularities.
  6. Classification of scattering maps for Einstein--scalar gravitational singularities.
  7. An explicit example.
  8. Hyperbolic model of a two-phase relativistic fluid
  9. Proposed model.
  10. Example 1. Strictly hyperbolic pressure laws.
  11. Example 2. The stiff limit $k_{\boldsymbol \iota}\equiv 1$.
  12. Example 3. Pressureless matter.
  13. Example 4. Discontinuous pressure laws.
  14. Example 5. The low/high two-phase model.
  15. Klein--Gordon scalar fields.
  16. ...and 20 more sections

Figures (2)

  • Figure 2.1: Light cones (dashed/dotted lines) crossing a singularity hypersurface (full line) transversally: (a) spacelike hypersurface, (b) timelike hypersurface.
  • Figure 4.1: Right-hand compressive (left figure) versus under-compressive (middle and right figures) interfaces. The dashed (resp. dotted) lines represent characteristic curves of the right-hand (resp. left-hand) family, and the thick line denotes the interface with speed $z$. Characteristics with speed $\lambda_{\boldsymbol \iota}^0$ are omitted to avoid clutter.

Theorems & Definitions (7)

  • Definition 3.1
  • Definition 4.1
  • Definition 4.2
  • Definition 5.1
  • Claim A.1: Rankine--Hugoniot (1)
  • Claim A.2: Rankine--Hugoniot (2)
  • Claim A.3: Rankine--Hugoniot (3)