The very best coordinates to use are the Kruskal-Szekeres coordinates $u$ and $v$ because they make the causal structure immediately obvious. In particular it becomes hard to match the description with our intuitive feel for what happens. This allows us to describe what happens inside the horizon, but at the expense of simplicity. So what to do? Well in cases like this we have to choose a set of coordinates that don't correspond directly to anything seen by an observer. The trouble is that inside the event horizon it's impossible to hover at fixed $r$, so we can't use shell coordinates either. observers hovering at a fized distance from the horizon. Outside the event horizon we can use shell observers i.e. When they turn on the torch the light just speeds off at $c$ as usual. The trouble with this is that for any freely falling observer spacetime is locally flat, so (ignoring tidal forces) the astronaut thinks they are motionless in flat space. We could try using the coordinate system of the astronaut falling in. So we need to look for a better coordinate system. Obviously we're going to struggle to describe what happens to light from a torch inside the event horizon if it takes an infinite time for the torch to even reach the event horizon, let alone cross it. This means time slows to a stop at the event horizon, and it's the source of the common claim that nothing can ever cross the event horizon. The trouble is that as you approach the event horizon the time gets dilated by a factor:Īnd at the event horizon, where $r = 2M$, this factor goes to zero. These are just the radial distance as measured with your rulers and the time measured on your clock so they have a nice simple interpretation.

Far from the black hole we use the Schwarzschild coordinates $t$ and $r$ (we'll ignore the angular coordinates).

One of the problems with describing any situation in general relativity is choosing an appropriate set of coordinates.

Are we slowly going into a black hole