Einstein Equations From Holographic Thermodynamics and Holographic Entropy

We derive the Einstein field equations and black hole entropy from the first law of thermodynamics on a holographic time-like screen. Because of the universality of gravity, the stress tensor on the screen must be independent of the details of matter fields, so it should be a pure geometric quantity. For simplicity, we assume that the stress tensor on the screen depends on surface Ricci curvature and extrinsic curvature linearly. Then we prove that the surface stress tensor is just the Brown-York stress tensor plus terms which do not affect the field equations of gravitation and the entropy of the system. By assuming a generalized "Fine first law of thermodynamics" or the usual universal first law of thermodynamics on the screen, we can derive the matter field equations as well.