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

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).

• What's a resting observer inside a black hole? Is there any reason to assume that the laws of thermodynamics do not hold anywhere? What's a zero entropy state? Is that the thing that is explicitly forbidden by the third law of thermodynamics? Why would the universe originate from the singularity? The falling observer knows where he came from until he gets killed. Why would the observable universe expand and at what rate? Why would the exterior universe not be observable when everything that falls into it is left essentially intact? Dec 26, 2014 at 0:17
• Why do you assume a singularity that all timelike worldlines will inevitably hit in their future must be low-entropy? Physicists don't generally think a universe ending in a Big Crunch would need to have its thermodynamic arrow of time reverse when the universe began to contract. Dec 26, 2014 at 0:27
• Has this thing a name, yet? Can we call it "Hollow Earth Hypothesis-Black Hole Edition"? Dec 26, 2014 at 0:38
• Resting frame of reference is the coordinate system in which the observer in side the black hole is at rest. You can always select such a coordinate system.
– Tomi
Dec 26, 2014 at 3:29
• @Tomi - Why does the fact that the radial axis behaves like time imply that the thermodynamics of the interior should be like those of the Big Bang, as opposed to those of a Big Crunch? As for the claim that the number of microstates decreases with size, that's true for some thermodynamic systems like an ideal gas, not true for others like an Einstein solid--figuring out how multiplicity would change during a GR collapse (either inside a black hole or for the universe in a big crunch) would probably require a theory of quantum gravity. Dec 26, 2014 at 4:08

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.

• -Was your "red herring" remark meant to invalidate the notion of time & space changing places at the BH horizon (which was endorsed by Jean Eisenstaedt, Senior Researcher at France's CNRS, attached to the Paris Observatory, as recently as the 2006 publication of the English ed. of his book entitled "The Curious History of Relativity"), or to reinforce physics' effective endorsement of the "block universe" view of time, or simply to bring out the lack of potential that that notion has for adding to knowledge in the physics community (as compared to the general population)? Thanks. Nov 20, 2018 at 21:50
• @Edouard The Curious History of Relativity is a popular science book and some liberties have to be taken when explaining GR to non-physicists. The bottom line is that coordinates do not have a physical meaning - they are just a way of labelling points in spacetime. We can use coordinates to calculate things that have a physical meaning, but the coordinates themselves are just a mathematical device. This is why there is no special significance to the fact the Schwarzschild coordinates behave oddly inside the event horizon. It is just the coordinates behaving oddly, not the universe. Nov 21, 2018 at 5:20
• Actually, it turns out I'd overlooked something in Eisenstaedt--ds squared (inflinitesimal proper time squared) is mentioned by him as having a phys. meaning discernible only inside a BH, and, like you're saying, it would take quite a while to get there. Sorry for the bother. Nov 21, 2018 at 18:56