# Why are two solutions to the field equations necessary to get the full Schwarzschild metric?

For a long time I've wondered why it was/is necessary to have separate solutions to the field equations for the interior and exterior metrics of a Schwarzschild black hole. Is there something weird going on at the event horizon that makes a single solution mathematically impossible? Has anyone ever found a single solution? It just seems odd that it would be necessary to find two separate solutions and then join them at the horizon. I'm not a mathematician so I would appreciate a general, non-technical answer if that's possible.

• Note that we start out looking for a spherically symmetric, static, vacuum solution to EFEs but it turns out that there isn't a static solution for the entire spacetime. The Schwarzschild black hole solution is static only for $r > 2M$, the geometry for $r < 2M$ is dynamic. – Alfred Centauri Jan 14 '18 at 19:39

• @dcgeorge, the Schwarzschild exterior solution is a static vacuum solution while the Schwarzschild interior solution is not; within the interior region is a static, uniform mass density, i.e., this no black hole, no event horizon. The Schwarzschild black hole solution, on the other hand, has zero mass density everywhere and is static only in the region $r > 2M$. – Alfred Centauri Jan 17 '18 at 0:27