The arrow of time and the cosmology of a black hole interior

Question

When solving the Einstein field equations in Schwarzschild metric for an observer falling into a black hole the radial coordinate r of the black hole and time t switch roles in the equations when r<2M.

If we transform into the resting coordinate system of an observer inside a black hole, the timelike geodesics will be along the radial dimension of the black hole. Would an observer inside the event horizon of a spherically symmetric black hole observer the radial dimension of the black hole as time? If yes, is it safe to assume that the laws of thermodynamics would hold inside the black hole, in which case the singularity of the black hole would as a zero entropy state be in the past along the radial "time" axis and the high entropy event horizon would be in the future along the same?

What would the cosmology of a spherically symmetric black hole look like from the perspective of an observer inside the black hole. It seems to me that from the perspective of an observer within the event horizon of the black hole:

• The observable universe originates from a singularity (black hole singularity)
• The observable universe expands along the radial (time) dimension
• The exterior of the black hole is not observable from within the black hole
• There would be future boundary conditions defining the faith of the interior (event horizon)
• The interior in other than radial dimension would be relatively uniform for a static black hole

How does time behave inside a black hole from the perspective of an observer inside the black hole? Could such an observer see the interior of the black hole as a universe relatively similar to ours (assuming the arrow of time would be along the radial axis of the black hole).

Answer 1

The coordinates you are using are called the Schwarzschild coordinates, and they are the coordinates that match measurements made by an observer at an infinite distance from the black hole. That is, if you're an infinite distance from the black hole then the Schwarzschild $t$ coordinate matches what you'd measure on your clock and the $r$ coordinate matches what you'd measure with your ruler. Obviously the physical relevance of the coordinates is why Schwarzschild chose them (actually he originally chose slightly different coordinates, but that's another story :-).

But the coordinates we use don't have to have a physical interpretation, e.g. Kruskal-Szekeres coordinates are frequently used for black holes, and coordinates that have a simple physical interpretation in some parts of spacetime don't necessarily have a simple physical interpretation in all parts of the spacetime.

And this last point is what happens here. If you're a Schwarzschild observer and you measure the time taken for something to fall into the event horizon you find it takes an infinite time to reach the event horizon. That means the whole of your time coordinate all the way up to $t = \infty$ only describes what happens up to, but not including, the event horizon and everything inside it.

So the $t$ coordinate inside the event horizon does not have the simple physical interpretation people think it does, and the apparent weirdness of time becoming space and space becoming time is a red herring. It just means the coordinate system you're using is more complicated than you think.

There's nothing wrong with using Schwarzschild coordinates inside the event horizon provide you are careful what you calculate and how you interpret it. For example we can calculate the time someone falling into the black hole would measure on a clock they are carrying - this is called the proper time and is very different from the Schwarzschild time. You find the traveller falls through the horizon and hits the singularity in a finite (and very short!) time. In fact the falling observer would see nothing weird about the spacetime in their vicinity in the few milliseconds of life left to them after crossing the event horizon. Looking outwards they would see some visual distortion, but they could still see the external universe. Looking inwards they would see an apparent horizon retreating before them - in fact they would never see themselves crossing an event horizon.