Why is the Schwarzschild Solution Independent of the Spatial Stresses?

The following paradox is raised and resolved: Both the energy density $T_{00}$ and the spatial stresses $T_{\mathrm{kl}}$ are (independent) sources of the tensor gravitational field in general relativity. Yet the general Schwarzschild exterior solution depends only on one parameter m, related to $T_{00}$ , so that no properties of $T_{\mathrm{ij}}$ “propagate” to affect an outside observer.