Scattering laws for interfaces in self-gravitating matter flows

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.

💡 Research Summary

The paper develops a comprehensive theoretical framework for describing sharp interfaces that arise in self‑gravitating matter flows, such as cosmological bounces or phase‑transition fronts in relativistic fluids. The authors introduce the notion of a “scattering map” prescribed on two distinct classes of hypersurfaces: (i) a gravitational singularity hypersurface, where the spacetime metric and any coupled scalar field become singular, and (ii) a fluid‑discontinuity hypersurface, where the fluid variables (density, velocity, phase) jump. By analysing the causal structures generated by the light cone (gravity) and the acoustic cone (fluid), they formulate a local evolution problem for the Einstein–Euler system that remains well‑posed only when appropriate junction conditions—encoded by the scattering maps—are supplied.

The first part of the paper revisits the authors’ earlier work on quiescent (velocity‑dominated) singularities in Einstein–scalar systems. Using a Gaussian foliation around the singular hypersurface, they expand the induced metric (g_{ab}(s)), the second fundamental form (K_{ab}(s)), and the scalar field (\phi(s)) in powers of the proper time or proper distance (s). The leading asymptotic profiles involve Kasner exponents (k_i) (eigenvalues of the normalized extrinsic curvature) and scalar coefficients (\phi_0,\phi_1). The Einstein constraints reduce to algebraic relations among these quantities. A scattering map (S) takes the incoming data ((g^-,K^-,\phi^-_0,\phi^-_1)) to outgoing data ((g^+,K^+,\phi^+_0,\phi^+_1)) while preserving the reduced constraints. Under three physically motivated axioms—general covariance, constraint compatibility, and ultra‑locality (the outgoing state at a point depends only on the incoming state at that point)—the admissible maps fall into two families.

1. Anisotropic ultra‑local scattering ((\gamma\neq0)). The traceless part of the extrinsic curvature rescales as (\dot K^+ = \epsilon, (r^+/r^-),\dot K^-), where (\epsilon=\operatorname{sgn}(\gamma)) and (r^\pm) are scalar invariants built from the Kasner exponents and the scalar field gradient. The scalar sector undergoes a canonical (symplectic) transformation (\Phi) that satisfies a signed Jacobian condition, possibly depending on a finite set of scalar invariants (\chi_m=\operatorname{tr}(\dot K^-/r^-)^m) (with (3\le m\le d)). The metric after the bounce is obtained from the metric before the bounce by an anisotropic scaling expressed as a polynomial in (\dot K^-) whose coefficients are functions of (\Phi) and the invariants.

2. Isotropic ultra‑local scattering ((\gamma=0)). Here the traceless part vanishes, (\dot K^+=0), forcing the extrinsic curvature to be purely isotropic, (K^+=\delta/d). The scalar field’s leading coefficient is fixed to (\phi^+_0=\pm\sqrt{(d-1)/d}) while (\phi^+_1) remains free. The outgoing metric can be any positive‑definite rescaling of the incoming metric, encoded by a scalar factor (\Delta) that is a polynomial in the invariants (\chi_m). This class represents maximal loss of anisotropic information at the macroscopic level.

An explicit example is given for a string‑theory motivated pre‑Big‑Bang bounce, where the scalar momentum transforms via a Möbius‑type rule, fixing the post‑bounce anisotropy amplitude.

The second major component of the work introduces a two‑phase relativistic fluid model on a fixed globally hyperbolic background. Each phase ( \iota\in{I,II}) obeys a perfect‑fluid stress‑energy tensor (T^{\alpha\beta}\iota = \mu u^\alpha u^\beta + p\iota(\mu)(g^{\alpha\beta}+u^\alpha u^\beta)) with a subluminal sound speed (k_\iota=\sqrt{p’\iota(\mu)}\le1). A phase selector field (\iota(x)) indicates which equation of state applies at each spacetime point, and a sharp interface corresponds to a discontinuity of (\iota). The authors focus on “under‑compressive” interfaces, where the acoustic cone intersects the hypersurface transversally, ensuring that causal observers can cross the interface. For such interfaces, the Einstein–Euler system again requires supplementary junction conditions. By extending the scattering‑map formalism, they relate the incoming fluid variables ((\mu^-,u^-\alpha,K^-{ab},\dots)) to the outgoing ones ((\mu^+,u^+\alpha,K^+_{ab},\dots)) in a way that respects energy‑momentum conservation, the propagation of the Hamiltonian and momentum constraints, and the ultra‑locality principle. The resulting relations mirror those found for the scalar‑gravity case: the extrinsic curvature’s traceless part rescales according to a universal factor, while the pressure law contributes model‑dependent scalar invariants that may affect the map.

A central philosophical point is that ultra‑locality reflects the physical expectation that microscopic processes (e.g., quantum gravity effects, micro‑scale phase‑transition dynamics) act only at the interface point, not along the interface. Consequently, the space of admissible scattering maps is dramatically reduced to a finite‑dimensional family parameterised by (i) the canonical scalar transformation (\Phi), (ii) a finite set of curvature invariants (\chi_m), and (iii) the specific functional form of the pressure laws in each phase. These parameters can, in principle, be derived from a more fundamental microscopic theory (e.g., string theory, loop quantum cosmology, kinetic theory of phase transitions).

In the concluding discussion the authors outline a program for a full classification of admissible junction prescriptions. They stress that while the present work establishes the universal algebraic structure of the scattering maps, concrete physical models will select particular members of the families described. The framework thus provides a rigorous macroscopic description that bridges relativistic fluid dynamics, general relativity, and cosmological bounce scenarios, offering a solid foundation for future analytical and numerical studies of self‑gravitating matter flows with sharp interfaces.