This paper develops a unified framework for identifying spatial and temporal boundaries of treatment effects in difference-in-differences designs. Starting from fundamental fluid dynamics equations (Navier-Stokes), we derive conditions under which treatment effects decay exponentially in space and time, enabling researchers to calculate explicit boundaries beyond which effects become undetectable. The framework encompasses both linear (pure diffusion) and nonlinear (advection-diffusion with chemical reactions) regimes, with testable scope conditions based on dimensionless numbers from physics (P\'eclet and Reynolds numbers). We demonstrate the framework's diagnostic capability using air pollution from coal-fired power plants. Analyzing 791 ground-based PM$_{2.5}$ monitors and 189,564 satellite-based NO$_2$ grid cells in the Western United States over 2019-2021, we find striking regional heterogeneity: within 100 km of coal plants, both pollutants show positive spatial decay (PM$_{2.5}$: $\kappa_s = 0.00200$, $d^* = 1,153$ km; NO$_2$: $\kappa_s = 0.00112$, $d^* = 2,062$ km), validating the framework. Beyond 100 km, negative decay parameters correctly signal that urban sources dominate and diffusion assumptions fail. Ground-level PM$_{2.5}$ decays approximately twice as fast as satellite column NO$_2$, consistent with atmospheric transport physics. The framework successfully diagnoses its own validity in four of eight analyzed regions, providing researchers with physics-based tools to assess whether their spatial difference-in-differences setting satisfies diffusion assumptions before applying the estimator. Our results demonstrate that rigorous boundary detection requires both theoretical derivation from first principles and empirical validation of underlying physical assumptions.

翻译：本文构建了一个统一框架，用于识别双重差分设计中处理效应的空间与时间边界。从基础流体动力学方程（纳维-斯托克斯方程）出发，我们推导出处理效应在空间与时间上呈指数衰减的条件，使研究者能够计算出效应变得不可检测的显式边界。该框架涵盖线性（纯扩散）与非线性（伴随化学反应的平流-扩散）两种机制，其可检验的适用范围条件基于物理学中的无量纲数（佩克莱数与雷诺数）。我们以燃煤电厂的空气污染为例展示了该框架的诊断能力。通过分析2019-2021年美国西部791个地面PM$_{2.5}$监测站与189,564个卫星NO$_2$网格单元数据，我们发现显著的区域异质性：在燃煤电厂100公里范围内，两种污染物均呈现正向空间衰减（PM$_{2.5}$：$\kappa_s = 0.00200$，$d^* = 1,153$公里；NO$_2$：$\kappa_s = 0.00112$，$d^* = 2,062$公里），验证了框架的有效性。在100公里以外，负衰减参数正确表明城市污染源占主导地位且扩散假设失效。地面PM$_{2.5}$的衰减速率约为卫星柱浓度NO$_2$的两倍，这与大气传输物理规律一致。该框架在八个分析区域中的四个区域成功诊断了其自身有效性，为研究者提供了基于物理学的工具，可在应用估计量前评估其空间双重差分设定是否满足扩散假设。我们的结果表明，严格的边界检测既需要基于第一性原理的理论推导，也需要对基础物理假设进行实证验证。