Spatial and Temporal Boundaries in Difference-in-Differences: A Framework from Navier-Stokes Equation

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.