# 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

## 1 Answer

They're not two separate solutions. It's just that when you express them in a particular set of coordinates, the Schwarzschild coordinates, the coordinates misbehave at the horizon. There are other coordinates, such as the Kruskal-Szekeres coordinates, that don't have this problem.

The other thing to realize is that it just isn't normally possible to cover a manifold with one set of coordinates and have the coordinates be well behaved everywhere. If you impose x-y Cartesian coordinates on North America, they will end up misbehaving if you try to extend them to cover the whole globe. Latitude-longitude coordinates misbehave at the poles.

By the way, the two regions of spacetime that you have in mind are only half of the maximally extended Schwarzschild spacetime.

• Thanks Ben but didn't Schwarzschild find the interior solution some months after he found the exterior one? Why did he have to do that? So I'm a bit confused. Also, if they're not two solutions why are they always referred to that way? – dcgeorge Jan 14 '18 at 19:55
• @dcgeorge, is this the interior solution you're asking about? If so, note that it's not part of the Schwarzschild black hole solution that you seem to be asking about. – Alfred Centauri Jan 15 '18 at 14:29
• @Alfred Centauri That's the one. I see that it is a static solution. I didn't know that it doesn't apply to a BH interior (does this mean there is really no such thing as a Schwarzschild black hole solution?). But then Wikipedia says "The ... solution, taken to be valid for all r > 0, is called a Schwarzschild black hole. It is a perfectly valid solution of the ... field equations, ... The Schwarzschild coordinates ... give no physical connection between the two patches, which may be viewed as separate solutions." So thanks for the new info. I'm still confused but I'm starting to see why. – dcgeorge Jan 16 '18 at 21:47
• @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
• @Alfred Centauri, OK, it's the S-child BLACK HOLE SOLUTION as opposed to the S-child INTERIOR SOLUTION. So, in the black hole solution (zero mass everywhere), one uses the exterior solution only? To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at r = R, such as the interior Schwarzschild metric. – dcgeorge Jan 17 '18 at 15:39