What Does the Schwarzschild Coordinate Speed of Light Measure?

The radial coordinate speed of light in Schwarzschild spacetime,{equation*}dr/dt=c(1-2GM/rc^2),{equation*}is explicitly not the speed measured by a local observer. Itsoperational meaning, however, is often left less transparent,particularly when the same coordinate relation is used to calculatethe experimentally observed Shapiro time delay. This paper develops a local-to-global operational picture of theSchwarzschild coordinate description. Starting from a frequency-basedmapping between local operational intervals and Schwarzschildcoordinate intervals,{equation*}dτ=μ(r)\,dt,=dr/μ(r),{equation*}with{equation*}μ^2(r)=1-2GM/rc^2,{equation*}the global coordinates are interpreted as a continuous assembly ofmeasurements performed in successive local regions. The familiarrelation{equation*}dr/dt=cμ^2(r){equation*}then follows from the position-dependent conversion of locallymeasured spatial and temporal intervals onto a common coordinaterecord, while the locally measured speed of light remains exactlyc throughout. The same construction gives the coordinate propagation time{equation*}t_2-t_1=∫r_1r_2dr/c\,μ^2(r),{equation*}and provides an operational interpretation of the resultinggravitational delay. A reference atomic clock in the asymptoticallyflat μ=1 region provides a physical realization of the coordinatetime, whereas the spatially varying measurement standards encounteredalong the light path produce the difference from the uniformflat-spacetime result. The analysis offers a pedagogical picture connecting local specialrelativity, Schwarzschild coordinate propagation, and gravitationaltime delay without interpreting the coordinate speed as a locallyreduced physical speed of light.