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Another argument that might be advanced in support of the proportionality of entropy and area comes from the holographic hypothesis[6,7], ie, the idea that the state of the part of the universe inside a spatial region can be fully specified on the boundary of that region. However, currently the primary support for this hypothesis comes from black hole thermodynamics itself. Since we are trying to account for the occurrence of thermodynamic-like laws for classical black holes it would therefore be circular to invoke this argument.

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If the entropy density on the horizon takes the form $e^\rho d{\cal A}$ , with $\rho$ a function of the Ricci scalar, one finds that $\delta Q=TdS$ implies $(2\pi/\hbar\eta)T_{ab}= R_{ab} -\nabla_a\nabla_b\rho + f g_{ab}$ for some function f [10]. Imposing the local energy conservation equation $\nabla^a T_{ab}=0$ then allows one to solve uniquely for f up to a constant.

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FIGURE 1: Spacetime diagram showing the heat flux $\delta Q$ across the local Rindler horizon $\cal H$ of a 2-surface element ${\cal P}$ . Each point in the diagram represents a two dimensional spacelike surface. The hyperbola is a uniformly accelerated worldline, and $\chi^a$ is the approximate boost Killing vector on $\cal H$ .