In previous works, matter trapping surfaces (MTS) were defined as hypersurfaces separating cosmologically expanding regions of spacetime from regions where collapse can take place independently. In the present work we examine the MTS, for the restriction to spherical dust plus $\mathrm{\Lambda }$, 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 three examples containing MTSs and $\mathrm{\Lambda }$ ($\mathrm{\Lambda }\mathrm{CDM}$, Schwarzschild–de Sitter, $\mathrm{\Lambda }+$Lemaître–Tolman–Bondi: $\mathrm{\Lambda }\mathrm{LTB}$—i.e., the inhomogeneous, spherically symmetric $\mathrm{\Lambda }\mathrm{CDM}$). The $\mathrm{\Lambda }\mathrm{LTB}$ model presents a static, stable MTS for the first time.

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• Cosmological constant
• Fluids & classical fields in curved spacetime
• General relativity
• General relativity equations & solutions
• General relativity formalism
• Gravitation
• Large scale structure of the Universe
• Relativistic aspects of cosmology