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 \ (\) separating two vacuum or cosmological-constant regions is endowed with an induced metric \ (h_ \), extrinsic curvature \ (K_ \) and a surface stress tensor \ (S_ \) fixed by the Israel junction conditions. To this purely geometric structure an area-based entropy \ (S_ = kB A_/ (4ₚ²) \) is assigned to spacelike cross-sections of \ (\), motivated by the Bekenstein--Hawking area law and the area scaling of entanglement entropy, with the patch number \ (N_ = A_/ (4ₚ²) \) 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 \ (T^₄₍ₓ ₄₍ₓ u^ u^ \) that is well approximated by a pressureless component on large scales. In homogeneous FLRW backgrounds the entropic density obeys \ (̇₄₍ₓ + 3H₄₍ₓ = 0 \) and thus follows the standard cold-matter scaling \ (₄₍ₓ a^-3 \), providing an effective dark contribution in the Friedmann equations. In the stationary, weak-field regime the logarithmic temperature potential \ (= T₆ₑ₀ₕ \) satisfies a Poisson-type equation \ (² = - (4 G/c²) (ₕ₈ₒ+₄₍ₓ) \) and yields the gravitational field via \ (g = c² \), so that \ (₄₍ₓ \) 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.