Timelike Vacuum Interfaces and Entropic Completionof the Stress–Energy Tensor in General Relativity

This study develops a coarse-grained description of timelike vacuum interfaces in classical general relativity and explores how such interfaces can support an effective dust-like dark contribution without modifying Einstein's equations. The starting point is the thin-shell formalism: a timelike hypersurface \( Σ \) separating two vacuum or cosmological-constant regions is endowed with an induced metric \( hμν \), extrinsic curvature \( Kμν \) and a surface stress tensor \( Sμν \) fixed by the Israel junction conditions. To this purely geometric structure an area-based entropy \( S_Σ = k_B A_Σ/(4_p^2) \) is assigned to spacelike cross-sections of \( Σ \), motivated by the Bekenstein--Hawking area law and the area scaling of entanglement entropy, with the patch number \( N_Σ = A_Σ/(4_p^2) \) serving as a geometric control parameter for the entropic loading of the interface. After coarse graining over many interface events, the shell stress--energy acquires an entropic contribution \( Tμνₑₙₜ ρₑₙₜ u^μ u^ν \) that is well approximated by a pressureless component on large scales. In homogeneous FLRW backgrounds the entropic density obeys \( ρ̇ₑₙₜ + 3Hρₑₙₜ = 0 \) and thus follows the standard cold-matter scaling \( ρₑₙₜ ∝ a⁻³ \), providing an effective dark contribution in the Friedmann equations. In the stationary, weak-field regime the logarithmic temperature potential \( θ = ln Tgrav \) satisfies a Poisson-type equation \( ∇^2θ = -(4π G/c^2)(ρᵥᵢₛ+ρₑₙₜ) \) and yields the gravitational field via \( g = c^2∇θ \), so that \( ρₑₙₜ \) appears as an apparent halo component in clusters and galaxies. The framework organises familiar dark-matter phenomenology in terms of timelike vacuum interfaces and their entropic state, providing a classical arena for studying coarse-grained gravitational entropy on timelike surfaces and its connections to entanglement- and holography-inspired ideas, while leaving fine-grained microphysical interpretations to future work.