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 \( \Sigma \) separating two vacuum or cosmological-constant regions is endowed with an induced metric \( h_{\mu\nu} \), extrinsic curvature \( K_{\mu\nu} \) and a surface stress tensor \( S_{\mu\nu} \) fixed by the Israel junction conditions. To this purely geometric structure an area-based entropy \( S_\Sigma = k_B A_\Sigma/(4\ell_p^2) \) is assigned to spacelike cross-sections of \( \Sigma \), motivated by the Bekenstein--Hawking area law and the area scaling of entanglement entropy, with the patch number \( N_\Sigma = A_\Sigma/(4\ell_p^2) \) 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 \( T^{\mu\nu}_{\mathrm{ent}} \simeq \rho_{\mathrm{ent}} u^\mu u^\nu \) that is well approximated by a pressureless component on large scales. In homogeneous FLRW backgrounds the entropic density obeys \( \dot{\rho}_{\mathrm{ent}} + 3H\rho_{\mathrm{ent}} = 0 \) and thus follows the standard cold-matter scaling \( \rho_{\mathrm{ent}} \propto a^{-3} \), providing an effective dark contribution in the Friedmann equations. In the stationary, weak-field regime the logarithmic temperature potential \( \theta = \ln T_{\mathrm{grav}} \) satisfies a Poisson-type equation \( \nabla^2\theta = -(4\pi G/c^2)(\rho_{\mathrm{vis}}+\rho_{\mathrm{ent}}) \) and yields the gravitational field via \( \mathbf{g} = c^2\nabla\theta \), so that \( \rho_{\mathrm{ent}} \) 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.