Matching collapse and expansion across Matter Trapping surfaces in inhomogeneous $Λ$CDM models

In the present work we examine the MTS, for the restriction to spherical dust plus $Λ$, proving that it actually is a characteristic surface of the Cauchy problem (generated by its characteristic curves), which opens the possibility for infinite solutions. This translate as the MTS being a boundary between arbitrarily independent solutions, reminiscent of the Birkhoff theorem effects. This property is illustrated with combinations of 3 examples containing MTSs and $Λ$ ($Λ$CDM, Schwarzschild-de,Sitter, Lemaître-Tolman-Bondi-de,Sitter: LTBdS – i.e. the inhomogeneous, spherically symmetric $Λ$CDM). The LTBdS model presents a static, stable MTS for the first time.

💡 Research Summary

The paper investigates Matter Trapping Surfaces (MTS) in spherically symmetric spacetimes filled only with pressure‑less dust and a cosmological constant Λ, i.e. the ΛCDM model. Starting from the 1+1+2 decomposition, the authors introduce the timelike unit vector nᵃ (fluid flow) and the spacelike radial unit vector eᵃ, and rewrite the metric in the Generalized Painlevé‑Gullstrand (GPG) form. Two scalar invariants are central: the Misner‑Sharp mass M_ms, defined by g^{ab}∂_a r ∂_b r = 1 – 2M_ms/r, and the energy function E = (n^a∂_a r)^2 – 2M_ms/r, which coincides with the Newtonian specific energy. In GPG coordinates the lapse α(t,r) and shift β(t,r) appear, with β directly related to the radial velocity of the fluid.

Einstein’s field equations are expressed in terms of M_ms, E, the density ρ and pressure P. When the matter content reduces to dust (P=0) plus Λ, the lapse becomes a pure function of time and can be set to unity by a time rescaling. The system collapses to three first‑order quasi‑linear PDEs:  ∂_t E + β ∂_r E = 0,  ∂_t M + β ∂_r M = 0,  ∂_r M = 4π r^2 ρ, with β given algebraically by β = ±α r^2 M/r + Λ r^2/3 + E. The initial data are simply the profiles M(t₀,r) and E(t₀,r); ρ follows from the third equation.

The characteristic curves of this system are defined by the vector field ∂_s = ∂_t + β ∂_r, which coincides with the fluid flow n^a. Along each characteristic the quantities E and M are constant, so the solution is obtained by propagating the initial data along the fluid world‑lines. The MTS is identified by the kinematic condition β = 0 (equivalently L_n r = 0, i.e. the areal radius has zero time derivative) and the dynamical condition g_TOV = 0, where g_TOV = –(M/r^2 – Λ r/3) is the radial acceleration term. When both conditions hold, the surface is a characteristic (null) hypersurface for the PDE system; consequently the interior and exterior regions evolve independently, much like the Birkhoff theorem but now in a non‑vacuum, Λ‑dominated setting.

Three illustrative examples are presented. In a homogeneous ΛCDM background the MTS corresponds to the turnaround radius where cosmic expansion balances local gravity. In a Schwarzschild‑de Sitter (SdS) matching, the MTS acts as a static interface separating a black‑hole interior from an expanding de Sitter exterior. The most novel case is the inhomogeneous Lemaître‑Tolman‑Bondi‑de Sitter (LTB‑dS) model. By stitching together two distinct LTB solutions across an MTS, the authors demonstrate the existence of a static, stable MTS: β = 0 and β̇ = 0 simultaneously, implying both radial velocity and acceleration vanish. This static sphere remains unchanged in time, representing a region completely decoupled from the global expansion—a concrete realization of a bound structure that is shielded from cosmological dynamics.

The authors conclude that MTS are not merely geometric separators of expanding and collapsing domains; they are genuine characteristic hypersurfaces of the Einstein‑dust‑Λ Cauchy problem. As such, any admissible interior solution can be matched to any exterior solution across an MTS, provided the TOV‑type balance condition is satisfied. This insight opens new avenues for constructing exact inhomogeneous cosmological models, for setting boundary conditions in numerical relativity, and for understanding the dynamical isolation of astrophysical structures within an accelerating universe. Future work is suggested to extend the analysis to non‑spherical geometries, time‑varying Λ, and more complex matter components.