The Information Paradox of Black Holes

We need you here. For the physics universe side of increasing entrophy and the information paradox in black holes.

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As an side, there has been some interesting work that shows this is a non-paradox. The conservation of information ālawā in physics appears to be a bit of a mirage.

This paradox is going to be solved. And the person who solves it will be remembered with Shannon and Hawkins

I agree that there is no paradox. The entropy of a black hole is undefined.

ETA: The information of a black hole is also undefined.

ok. Like a singularity. Letās talk about something else. How about transubstantiation? During transubstantiation does entropy increase or decrease?

That Iām not sure is true. I think it is an open question with some intriguing theories in play.

So where is the experiment that validates Boltzmannās equation where information is equal to entropy. If you look at the original definition of second law of thermodynamics

Clausius statement
The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.[26] His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:

Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.[27]

The statement by Clausius uses the concept of āpassage of heatā. As is usual in thermodynamic discussions, this means ānet transfer of energy as heatā, and does not refer to contributory transfers one way and the other.

Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.
And here a description equation describing entropy
Clausius statement
The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.[26] His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:

Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.[27]

The statement by Clausius uses the concept of āpassage of heatā. As is usual in thermodynamic discussions, this means ānet transfer of energy as heatā, and does not refer to contributory transfers one way and the other.

Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.

I am pulling from old mans memory at this point but the second law of thermodynamics that discusses entropy has a very different experimental validation involving transfer of heat and not information.

We went form experimentally validated science to an abstract concept(Boltzmann) and so what foundation does Hawkings work really have?

Itās a nonsensical question. The entropy of transubstantiation is undefined. The information of transubstantiation is undefined. Does the entropy of what increase of decrease during transubstantiation. You donāt say. Nonsensical.

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One of the issues I have with all this that Thermodynamic ālawsā are not precisely laws. They are just stochastic averages. They are not laws in the way we normally think about them. We have already found violations of the 2nd law, for example, in precisely the places we would expect them.

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Thanks. My other point though is there is experimental validation to Clausius hypothesis. What about Boltzmannās hypothesis or description?

There absolutely is validation of Boltzmanās distribution.

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Posting for reference:

However, Boltzmannās paradigm was an ideal gas of N identical particles, of which are in the i-th microscopic condition (range) of position and momentum. For this case, the probability of each microstate of the system is equal, so it was equivalent for Boltzmann to calculate the number of microstates associated with a macrostate. W was historically misinterpreted as literally meaning the number of microstates, and that is what it usually means today. W can be counted using the formula for permutations

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There is a lot of misconceptions in this thread.

The entropy of a black hole is well defined. What is still under investigation is the physical states that go into the equation for the entropy, but given your favorite theory, the entropy is well defined. Note that entropy is the logarithm of the number of states.

Here are the most popular ideas of what the states are:

1. The states are the physical, internal states of matter and gravity in the black hole. This is in direct analogy to both normal matter and electromagnetic fields.
2. The entropy is the entanglement entropy between states that are separated by the event horizon.
3. The states are gravitational states on the horizon.

Singularities, especially in general relativity, is well defined.

I think this is a joke. In case it is not, entropy is only defined for physical states, and is in fact ill-defined for metaphysical events.

The generalized 2nd law of thermodynamics (2nd law + black hole entropy) is necessary to keep the 2nd law from being violated on average. Without it, even systems that have spend enough time to relax to a thermodynamic equilibrium can violate the 2nd law.

Note that there is often a lot of confusion between the thermodynamics of a black hole (generalized 2nd law, etc) and the information paradox. They are closely related, but not identical.

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Help me understand this. 2nd law of thermodynamics clearly applies in many contexts, but has well known violations in boundary cases. How do we know that black holes are just another one of those boundary cases?

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The adage is that the 2nd law is statistical. Violations of the 2nd law happen in predictable boundary cases where the statistical assumption breaks down. This happens, for example, in systems away from thermodynamic equilibrium, as follows from the Fluctuation Theorem.

However, if our idea of black hole entropy is wrong (for example, if it turns out that black holes have no entropy whatsoever), then the 2nd law breaks down even statistically. This is not part of the predictable edge cases where we expect to see the 2nd law being violated.

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I donāt see itā¦that just means that is a different sort of entity that violates the assumptions we had when we started. I suppose this might conflict with the āarrow of timeā arguments, is that why it is a big deal?

Black holes are clearly a boundary case. Iām not sure we know how things will work in a boundary case like this. All we can really appeal to is the heuristic of beauty, but that isnāt enough to be sure. What am I missing?

Two things:

1. Typical violations of the 2nd law happens because they violate an assumption that is required to derive the 2nd law. E.g. not large enough statistical sample. Black holes do not violate this assumption.
2. This generalized thermodynamics are most famously applicable to black holes. However, it actually applies to more general causal horizons in general relativity. In particular, it applies to cosmological horizons, and the ubiquitous āpersonalā horizons you create every time you accelerate. Black holes are exotic objects, but the problems are caused by causal horizons, not black holes, and causal horizons are pretty mundane.

They violate a different assumption though. Right? They violate the assumption that we have to consider the whole system. If the black holes are part of the system we intrinsically cannot observe, we do not expect the 2nd law to hold. The universe outside the black holes is all we can observer, and black holes are entropy sinks. I donāt see a contrdiction here.

Yes, and that is where some recent results show an absence of a firewall. Fairly unsurprisingly to me. Though I canāt find the paper now! And what do I know any ways. . Iām sure you are going to show me the error of my ways any moment now.

This is the right line of thinking. However, this is a very different violation of the assumptions that goes in the 2nd law than the statistical assumption. This is because we know that the 2nd law is restored in a very predictable way if we expand the system. Pushing this to its logical conclusion actually gives us the notion that black holes possess entropy.

Imagine the mundane situation in which I have a closed box. I define my system to be everywhere except within the box. Whenever I calculate my entropy, I compute the entropy of everywhere but inside the box. I can drop stuff into the box, thus reducing the entropy of my system. However, for the 2nd law to be true, then I must agree that there is an entropy associated with the box. Indeed, if I include the box in my system, the entropy of the system must increase after I drop stuff into the box.

The argument showing that black holes do possess entropy is exactly the same, but replacing box with black holes. If black holes do not possess entropy, then the 2nd law is false even statistically. Note that in this argument, the important feature of the black hole is the horizon which plays the part of the āboxā, and there is nothing special about it that warrants a different treatment than my literal box.

As a relativist, to me the Einstein Equivalence Principle is close to a de fide sacred dogma. As such, I found firewalls abhorrent.

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Iām just not sure why I see the problem with this. All it does is violate a āBeautyā heuristic. It just means that we canāt measure the whole system and that black holes are just another type of boundary condition.

Also iām not sure that the horizon is the problem. It is the singularity that is what creates the confusion, but we donāt really know if the singularity is a reality any way. As Sabine puts it:

In his paper, he also points out correctly that ā from a strictly logical point of view ā thereās nothing to worry about because the information that fell into a black hole can be kept in the black hole forever without any contradictions. I am not sure why he doesnāt mention this isnāt a new insight either ā itās what goes in the literature as a remnant solution. Now, physicists normally assume that inside of remnants there is no singularity because nobody really believes the singularity is physical, whereas Maudlin keeps the singularity, but from the outside perspective thatās entirely irrelevant.
Sabine Hossenfelder: Backreaction: A Philosopher Tries to Understand the Black Hole Information Paradox

It seems the real challenge is the singularity, if and only if it exists, which we canāt really know for sure yet. The event horizon is doesnāt seem to be such a big problem. RIght?