Entropy and The Information Paradox

[andyandy]

Ok, having just watched a horizon program on Hawkings, the below equation is referred to as leading to Hawkings' theory re the information paradox - ie that information could be lost in a black hole.

[latex]$$ S=\frac{kc^3A}{4\hbar G} $$[/latex]

Where S= entropy of a black hole k=Boltzman constant c=speed of light [latex]$\hbar$[/latex]=reduced planck's constant G=gravity constant A=surface area 4=a good old bog standard number

Now, what exactly does entropy mean in the context of a black hole? If all matter passing through the event horizon is compacted within an infinitly dense singularity, does that constitute an extremely highly ordered, or extremely highly disordered state? How could entropy increase/decrease? from that of such a singularity - ie shouldn't black hole entropy be independent of surface area?

Where does Hawking Radiation come into all this?

Why does this equation lead to a paradox?

[and many more questions later ]

you can watch the documentary here....

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[Dark Jaguar]

What does entropy have to do with order?

 

[Schneibster]

Now, what exactly does entropy mean in the context of a black hole? If all matter passing through the event horizon is compacted within an infinitly dense singularity, does that constitute an extremely highly ordered, or extremely highly disordered state? How could entropy increase/decrease? from that of such a singularity - ie shouldn't black hole entropy be independent of surface area?

Once matter or energy crosses the event horizon, it was claimed to be lost, and its entropy with it. This was the subject of Hawking's famous bet with Kip Thorne.

Where does Hawking Radiation come into all this?

If I understand correctly, Thorne won because someone managed to show that the Hawking radiation returned the entropy to the universe at large.

Why does this equation lead to a paradox?

That one I can't help you with now. I have other fish to fry, specifically a promised discourse upon quantum mechanics.

 

[~enigma~]

If I understand correctly, Thorne won because someone managed to show that the Hawking radiation returned the entropy to the universe at large.

The bet was won by John Preskill. Stephen Hawking conceded because he believed black holes "leaked." Kip Thorne was on Hawkings side but refused to concede along with Hawkings.

 

[Schneibster]

Nice, thanks. I'll have to look it up at some point.

 

[andyandy]

well this answers a couple of questions....and throws up a few more

In 1975 Hawking and Bekenstein made a remarkable connection between thermodynamics, quantum mechanics and black holes which predicted that black holes will slowly radiate away. (see Relativity FAQ Hawking Radiation). It was soon realized that this prediction created an information loss problem which has since become an important issue in quantum gravity.

In order to understand why the information loss problem is a problem, we need first to understand what it is. Take a quantum system in a pure state and throw it into a black hole. Wait for some amount of time until the hole has evaporated enough to return to its mass previous to throwing anything in. What we start with is a pure state and a black hole of mass M. What we end up with is a thermal state and a black hole of mass M. We have found a process (apparently) which converts a pure state into a thermal state. But, and here's the kicker, a thermal state is a MIXED state (described quantum mechanically by a density matrix rather than a wave function). In transforming between a mixed state and a pure state, one must throw away information. For instance, in our example we took a state described by a set of eigenvalues and coefficients, a large set of numbers, and transformed it into a state described by temperature, one number. All the other structure of the state was lost in the transformation.

In technical jargon, the black hole has performed a non-unitary transformation on the state of system. As you may recall, non-unitary evolution is not allowed to occur naturally in a quantum theory because it fails to preserve probability, ie, after non-unitary evolution, the sum of the probabilities of all possible outcomes of an experiment may be greater or less than 1.

In the face of such evolution, quantum mechanics falls apart, and we are faced with a dilemma. Do black holes really defy the tenets of quantum theory, or have we missed something in our thought experiment. Perhaps the black hole is not the same after it has evaporated to mass M as it was initially at mass M. Or perhaps there is some subtle correlation in the Hawking radiation that we are missing, but that supplies the missing information about the pure state.

This, then, is the black hole information loss problem. The fact that information is lost is reflected in the thermal nature of the emitted radiation. But any thermal system can be assigned an entropy via the Gibbs law dE = S dT. Thus, we can calculate the black hole entropy by dint of the fact that we can calculate the black hole temperature (by dint of the fact that the quantum radiation is thermal). This is, I think, what people are getting at when they say that black hole entropy is responsible for the information loss. I would say it the other way, that black hole information loss is responsible for black hole entropy. The simple fact of the matter is that they are the same thing in slightly different terms.

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/info_loss.html

 

[Schneibster]

Deeply into it. My head's into QM just now; it's a major gear-change to think about relativity. But I'll try to have an informed opinion once I'm done thinking about quantum weirdness.

 

[andyandy]

ok let's see....

black holes are apparently maximal entopy objects, and as such the entropy within a given region of space is [latex]$\le$[/latex] the entropy of the largest black hole which could fit in that region.....and this is known as the Bernstein bound.
but apparently there's a problem, insofar as entropy is normally proportional to volume - but for black holes it's proportional to the surface area.....
so this might mean that all entropy is actually proportional to surface area - and that volume is in actuality illusory, and the universe is just a hologram

ha

 

[fuelair]

What does entropy mean to me, anyway?!

 

[Cuddles]

ok let's see....

black holes are apparently maximal entopy objects, and as such the entropy within a given region of space is [latex]$\le$[/latex] the entropy of the largest black hole which could fit in that region.....and this is known as the Bernstein bound.
but apparently there's a problem, insofar as entropy is normally proportional to volume - but for black holes it's proportional to the surface area.....
so this might mean that all entropy is actually proportional to surface area - and that volume is in actuality illusory, and the universe is just a hologram

ha

Of course, the problem with the holographic view of the universe is that while the universe could just be a projection, there must be dimensions for it to be projected into. An ordinary hologram is encoded in 2 dimensions, but is projected into 3. If the third dimension didn't exist then the hologram wouldn't exist, only the information would. If the holographic idea is true then it needs to explain where the volume for the projection to be projected into comes from and why it is empty rather than being full of universe.

 

[casebro]

What does entropy mean to me, anyway?!

In my pea brain, I lump entropy, relativity, quantum and strings all together in a group that will do nothing to practically influence my life. YMMV.

I suspect that other than the professors whose job it is to discuss those things, very few people are affectd by them.

Now some egg heads will tell me how I am wrong:

 

[andyandy]

Of course, the problem with the holographic view of the universe is that while the universe could just be a projection, there must be dimensions for it to be projected into. An ordinary hologram is encoded in 2 dimensions, but is projected into 3. If the third dimension didn't exist then the hologram wouldn't exist, only the information would. If the holographic idea is true then it needs to explain where the volume for the projection to be projected into comes from and why it is empty rather than being full of universe.

yes it does seem a rather extraordinary theory....without too much proof

The whole explanation of how information can be smeared on the event horizon of a black hole to ensure that there is no information loss does seem quite bizarre too...and that would be [if i understand it] in effect a mapping of 3 dimensional information onto 2 dimensions.....very weird....

 

[Beth]

A relevant article on Sciencedaily today. This stuff passes beyond my understanding though.

http://www.sciencedaily.com/releases/2007/02/070227105134.htm

 

[Apathia]

yes it does seem a rather extraordinary theory....without too much proof

The whole explanation of how information can be smeared on the event horizon of a black hole to ensure that there is no information loss does seem quite bizarre too...and that would be [if i understand it] in effect a mapping of 3 dimensional information onto 2 dimensions.....very weird....

"Information smeared on the event horizon," "multiple unkverses some with a given black hole, others without it:" It all seems a kind of desperate fudge making.

It seems physics keeps banging its head against a paradox of many disguises, or a lump in the carpet that you can move under the couch, till you need to move the couch.

 

[andyandy]

A relevant article on Sciencedaily today. This stuff passes beyond my understanding though.

http://www.sciencedaily.com/releases/2007/02/070227105134.htm

from the link

In the mid 1970s, Stephen Hawking showed that black holes eventually evaporate away in a steady stream of featureless radiation containing no information. But if a black hole has completely evaporated, where has the information about it gone? This long standing question is known as the black hole information paradox.

Now, Professor Braunstein and Dr Pati have ruled out the possibility that information might escape from the black hole but be somehow hidden in correlations between the Hawking radiation and the black hole’s internal state. Braunstein and Pati’s result demonstrates that the black hole information paradox is even more severe than previously believed.

Dr Pati said: "Our result shows that either quantum mechanics or Hawking’s analysis must break down, but it does not choose between these two possibilities."

Professor Braunstein said: "The no-hiding theorem provides new insight into the different laws governing classical and quantum information. It shows that there’s got to be new physics out there."

interesting stuff.....

 

[Orangutan]

"It all seems a kind of desperate fudge making.

I Loled at the visuals that came into my brain when I read that sentence.

AndyAndy, "The Fabric Of The Cosmos" (Greene) http://www.amazon.com/Fabric-Cosmos-Space-Texture-Reality/dp/0375412883
Has a large amount of it devoted to talking about entropy. I think the surface-area-proportional-to-entropy thing you mention is the basis for some of the "Holographic" universe Ideas that have been going around. I recommend this book.

 

[andyandy]

I Loled at the visuals that came into my brain when I read that sentence.

AndyAndy, "The Fabric Of The Cosmos" (Greene) http://www.amazon.com/Fabric-Cosmos-Space-Texture-Reality/dp/0375412883
Has a large amount of it devoted to talking about entropy. I think the surface-area-proportional-to-entropy thing you mention is the basis for some of the "Holographic" universe Ideas that have been going around. I recommend this book.

you know, the sad thing is that i've read it a couple of times.....and for about 10 minutes after i've finished, the universe makes some sense....give it a week and everything's hazed over again....
maybe i should start from the beginning every time i finish it....and that way it'll slowly stick

 

[Orangutan]

you know, the sad thing is that i've read it a couple of times.....and for about 10 minutes after i've finished, the universe makes some sense....give it a week and everything's hazed over again....
maybe i should start from the beginning every time i finish it....and that way it'll slowly stick

Whatever you do, don't ever get the audio version of a book like this. A friend let me borrow one of his Hawkings audio books, I have never had so many close calls driving. You just can't concentrate on this material and drive at the same time.

 

[Cuddles]

It seems physics keeps banging its head against a paradox of many disguises, or a lump in the carpet that you can move under the couch, till you need to move the couch.

You say that like it's a bad thing. When it comes down to it, that's what science is. You test what you can and ignore the rest of it in the hope that you'll understand it later, or that once you've understood one part the rest of it will suddenly make sense. At the moment things like the origin of the universe and its "true" nature are generally hidden under the sofa because we simply can't answer those questions at the moment. Some people propose theories about them, but these are never going to be satisfactory either to scientists or the layperson because we can't tell if they are actually right.

I wonder if I can get a t-shirt with "Science is the process of desperate fudge-making" on it.

 

[Apathia]

I said:

It seems physics keeps banging its head against a paradox of many disguises, or a lump in the carpet that you can move under the couch, till you need to move the couch.

You say that like it's a bad thing. When it comes down to it, that's what science is. You test what you can and ignore the rest of it in the hope that you'll understand it later, or that once you've understood one part the rest of it will suddenly make sense. At the moment things like the origin of the universe and its "true" nature are generally hidden under the sofa because we simply can't answer those questions at the moment. Some people propose theories about them, but these are never going to be satisfactory either to scientists or the layperson because we can't tell if they are actually right.

I wonder if I can get a t-shirt with "Science is the process of desperate fudge-making" on it.

Not bad at all. I like the game, and I like fudge. Where can I get the T-shirt?
Paradoxes do keep popping up. Science is a game of whack-a-mole!