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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.

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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).

Tomi
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    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? – CuriousOne Dec 26 '14 at 00:17
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    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. – Hypnosifl Dec 26 '14 at 00:27
  • Has this thing a name, yet? Can we call it "Hollow Earth Hypothesis-Black Hole Edition"? – CuriousOne Dec 26 '14 at 00:38
  • 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 '14 at 03:29
  • If we assume that radial axis behaves like time inside the black hole, then the thermodynamics of interior of the black hole would be similar to the thermodynamics of the early universe and big bang. In statistical physics, as the physical size of the system goes smaller, the number of possible micro states drops and so does the entropy (which depends on the number of micro states). At singularity the the size drops to zero and there would be only one micro state representing the system. The entropy of only one micro state is zero. Entropy at the event horizon is a well known measure. – Tomi Dec 26 '14 at 03:53
  • @CuriousOne If the radial axis can be interpreted as time when inside the event horizon, then I was proposing choosing the arrow of time to be to the direction where entropy always increases. At one end of the radial time axis we have the singularity of the black hole and on the other end we have the event horizon. The universe along the radial axis expands from the singularity when moving towards the event horizon. Because our hypothesis are that the radial dimension is the time dimension and the arrow of time is away from singularity, our universe inside the black hole expands with time. – Tomi Dec 26 '14 at 04:01
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    @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. – Hypnosifl Dec 26 '14 at 04:08
  • What is the resting observer at rest against? The only physical solutions inside the black hole go trough the singularity. You can always assume anything, but that doesn't get you anywhere. Physics is very tolerant against "garbage in, garbage out", and quite frankly, that's what you are doing here. Your choice of entropies is completely unphysical. Infalling matter heats up, its entropy increases (satisfying the second law). I have no idea where this assumption comes from that the singularity is in a well ordered state. I am sympathetic to the general idea, but your execution is suspect. – CuriousOne Dec 26 '14 at 04:08
  • The real problem here is that it's not clear that GR can make any reliable predictions about the "interior" of black holes since it breaks (at the very least) thermodynamics right away by making the event horizon a surface with T=0. Our semiclassical approximations can fix that, somewhat, but what really happens after that, there is just no theory to predict that. If we are going by at least borderline physical assumptions, then maybe something like the holographic principle may shed some light on this (and that's a fairly strong concession from a guy who thinks little of string theory). – CuriousOne Dec 26 '14 at 04:23
  • @Hypnosifl - The direction of the time of arrow is not given by any known theory of physics. We have two choices, one that originates from a big bang and another that leads into big crunch. My hypothesis was to choose the arrow of time that would conserve the maximum entropy principle. In general in Big Bang cosmology, the early universe is considered to be a lower entropy state and entropy is considered to increase as the universe expands. My other hypothesis was that this would also be true for the "universe" of the event horizon interior. – Tomi Dec 26 '14 at 04:23
  • @Tomi: The direction of the time arrow is given both by observation (which is the only physics that counts) and by thermodynamics which is very hard to argue with as there is no evidence against it. Your "choice" of time arrow is not yours to make. It has to have some physical meaning and your choice flies in the face of what little one can assume based on GR (which is probably not a reliable theory in this regime), but we don't have anything better. – CuriousOne Dec 26 '14 at 04:26
  • @CuriousOne - The event horizon entropy is predicted by Bekenstein–Hawking formula S = (kA)/(4l^2) where k is Boltzmann's constant, A is the area of the event horizon and l is the plank length. I was proposing as the first hypothesis of choosing the time coordinate in such a way that in the resting frame of reference of an observer within the black hole the negative element of the metric tensor would be called time and the three positive elements would be called the space. I was proposing as second hypothesis of choosing the arrow of time to be towards maximum entropy. – Tomi Dec 26 '14 at 04:42
  • @Tomi: A resting coordinate system in which nothing can rests in the very dynamic that you use to define it is not a physical proposition. Bekenstein-Hawking, by the way is a semiclassical approximation that goes beyond GR. In GR the temperature of the event horizon is trivially zero, thus violating the third law. So if one can't even analyze the event horizon correctly in GR, what makes you believe that it can be used below the level on which it breaks down, already? The time arrow inside the black hole is pointing towards the max. entropy: it's pointing at the hot singularity. – CuriousOne Dec 26 '14 at 04:53
  • @CuriousOne - https://en.wikipedia.org/wiki/Rest_frame – Tomi Dec 26 '14 at 05:18
  • @Tomi: I have PhD in physics. You can probably do better than sending me to Wikipedia. Or can you? :-) – CuriousOne Dec 26 '14 at 05:22
  • Let us continue this discussion in chat. – Tomi Dec 26 '14 at 05:36

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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.

John Rennie
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  • -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. – Edouard Nov 20 '18 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. – John Rennie Nov 21 '18 at 05:20
  • 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. – Edouard Nov 21 '18 at 18:56