Thermodynamics and mechanical laws of causal-diamond horizons

We develop a unified thermodynamic and mechanical framework for causal-diamond horizons in Minkowski spacetime. Euclidean regularity fixes the temperature $T_D=1\mathrm{/}(\pi \alpha )$, where $2\alpha$ is the temporal extent of the diamond. The regulated Gibbons–Hawking–York contribution gives $I_D^E=-A_0\mathrm{/}(4G)$, with $A_0$ the area of the horizon bifurcation surface, and the vanishing equilibrium geometric energy then yields the entropy–area relation $S_D=A_0\mathrm{/}(4G)$. For weak matter perturbations crossing a localized portion of the future horizon, a Raychaudhuri analysis relates the associated Killing-energy flux to the area response, yielding a local physical-process first law at leading order near the bifurcation surface and, for classical matter satisfying the null energy condition, a local second law. Uniform conformal surface gravity establishes the zeroth law, while zero temperature is approached only in the limit $\alpha \to \mathrm{\infty }$, in which the finite causal horizon disappears. These results show that a coherent horizon-law structure can arise within a causal diamond, a fundamental building block of the causal structure of Minkowski spacetime. The derivation is explicit in $(2+1)$ dimensions, while the physical results retain the same form for spherical causal diamonds in $d\ge 3$.