If Brian Greene has it right (and he’s not a cosmologist, so maybe he hasn’t), the maximum information content of a black hole is proportional to the surface area of the event horizon, and this same limit applies to all space, not just black holes.

My understanding is that this result depends on the following two assumptions:
1) Matter inside a black hole is compressed to a region of infinite density
2) The 2nd law of thermodynamics is applicable under all situations, even inside black holes.

I have several questions on this:

- Has Brian Greene got it right?

- Is this hypothesis widely accepted?

- Is my understanding in points 1) and 2) correct?

- Why is it assumed that matter is compressed to infinite density inside a Black Hole, when there is no observational evidence for this, and it is incompatible with many hypotheses about the behaviour of matter at that scale?

- What mechanism could prevent space having much more information than this? In particular, if you had a spherical lump of very dense matter at just pre-black hole stage, why would the maximum information content of this lump not be proportional to its volume, rather than its surface area, and why would this information content not be vastly greater than that of a black hole with an event horizon of the same size?

1) Matter inside a black hole is compressed to a region of infinite density

no, that is just a pop-sci view because of the “singularity”.

I’m not sure why what happens to the matter after it crosses the event horizon is relevant to the information density/distribution at the event horizon.

But as ygritte says, “you know nothing, John Snow”.

i’d guess the information thing between a BH and a not quite a BH would be the time dilation at the EH. i believe that is how the information is “stored”. at lightspeed the time dilation is infinite.

of course this is a pop-sci view and probably as wrong as rubber sheets and sultana puddings.

What ‘information’ are we talking about?

what you got up to last weekend. that sort of stuff.

information is basically: In physics, physical information refers generally to the information that is contained in a physical system. Its usage in quantum mechanics (i.e. quantum information) is important, for example in the concept of quantum entanglement to describe effectively direct or causal relationships between apparently distinct or spatially separated particles. wiki.

ChrispenEvan said:

what you got up to last weekend. that sort of stuff.

information is basically: In physics, physical information refers generally to the information that is contained in a physical system. Its usage in quantum mechanics (i.e. quantum information) is important, for example in the concept of quantum entanglement to describe effectively direct or causal relationships between apparently distinct or spatially separated particles. wiki.

Thanks.

The Rev Dodgson said:

- Why is it assumed that matter is compressed to infinite density inside a Black Hole, when there is no observational evidence for this, and it is incompatible with many hypotheses about the behaviour of matter at that scale?

This is a direct result of GR

not sure if you have seen this page, but it may help…

Black Hole Thermodynamics

diddly-squat said:

The Rev Dodgson said:

- Why is it assumed that matter is compressed to infinite density inside a Black Hole, when there is no observational evidence for this, and it is incompatible with many hypotheses about the behaviour of matter at that scale?

This is a direct result of GR

OK, but why would we assume that GR is applicable at scales and densities where there is excellent grounds to believe that it is not applicable?

ChrispenEvan said:

1) Matter inside a black hole is compressed to a region of infinite density

no, that is just a pop-sci view because of the “singularity”.

Well that’s not how Greene presented it. He presented it as the basis of a calculation by Hawking that has profound implications for the basic understanding of Physics. It’s possible that he was just presenting an over-simplified picture for what is admittedly a pop-sci publication (The Hidden Reality), but he is normally good about stating when he presenting a simplistic version of a complicated theory, and providing notes with a less simplistic version and further references; and he didn’t do any of that.

The Rev Dodgson said:

diddly-squat said:

The Rev Dodgson said:

- Why is it assumed that matter is compressed to infinite density inside a Black Hole, when there is no observational evidence for this, and it is incompatible with many hypotheses about the behaviour of matter at that scale?

This is a direct result of GR

OK, but why would we assume that GR is applicable at scales and densities where there is excellent grounds to believe that it is not applicable?

I don’t think any actually believes that the singularity is truly ‘point like’ IRL, we just don’t have a quantum gravity model is all

diddly-squat said:

not sure if you have seen this page, but it may help…

Black Hole Thermodynamics

I hadn’t seen that, but it essentially says the same as Greene, although less explicit about the mass within the hole having zero entropy (and hence all the entropy must be located at the event horizon).

diddly-squat said:

The Rev Dodgson said:

diddly-squat said:

This is a direct result of GR

OK, but why would we assume that GR is applicable at scales and densities where there is excellent grounds to believe that it is not applicable?

I don’t think any actually believes that the singularity is truly ‘point like’ IRL, we just don’t have a quantum gravity model is all

But if that is the case why we would we take seriously a result that is entirely based upon the true point like nature of the mass within a black hole?

The Rev Dodgson said:

diddly-squat said:

The Rev Dodgson said:

OK, but why would we assume that GR is applicable at scales and densities where there is excellent grounds to believe that it is not applicable?

I don’t think any actually believes that the singularity is truly ‘point like’ IRL, we just don’t have a quantum gravity model is all

But if that is the case why we would we take seriously a result that is entirely based upon the true point like nature of the mass within a black hole?

because, as one of my Undermanagers used to say, you can only f&$k with the c*%k you’ve got…

All we have is the GR interpretation, so until someone stumps up with a better model all we can do is assume that the singularity is ‘point like’

All we have is the GR interpretation, so until someone stumps up with a better model all we can do is assume that the singularity is ‘point like’

yes, we can only go on current observation and current theory. even while knowing they are not as good as probable future ones will be.

ChrispenEvan said:

All we have is the GR interpretation, so until someone stumps up with a better model all we can do is assume that the singularity is ‘point like’

yes, we can only go on current observation and current theory. even while knowing they are not as good as probable future ones will be.

I get the impression that for the most part people seem happy with the idea that the singularity is probably most likely to be small with very large density.

http://www.youtube.com/results?search_query=stanfordcosmology+susskind

looks a good series of lectures by Susskind. the first one is understandable to me.

This is just BC’s argument about “near infinite” worded differently! :P

diddly-squat said:

All we have is the GR interpretation, so until someone stumps up with a better model all we can do is assume that the singularity is ‘point like’

Well no, you can say we just don’t know how it works at that level. Pretending that a model that is almost certainly not applicable, is applicable, is almost certainly going to be misleading.

Riff-in-Thyme said:

This is just BC’s argument about “near infinite” worded differently! :P

It’s an example of why treating “very large” as being infinite can be misleading.

The Rev Dodgson said:

diddly-squat said:

All we have is the GR interpretation, so until someone stumps up with a better model all we can do is assume that the singularity is ‘point like’

Well no, you can say we just don’t know how it works at that level. Pretending that a model that is almost certainly not applicable, is applicable, is almost certainly going to be misleading.

but we have to use something to make our predictions… the fact is we don’t have anything that fundamentally disproves the GR interpretation

The Rev Dodgson said:

Riff-in-Thyme said:

This is just BC’s argument about “near infinite” worded differently! :P

It’s an example of why treating “very large” as being infinite can be misleading.

+1

the two are very, very different

diddly-squat said:

The Rev Dodgson said:

diddly-squat said:

All we have is the GR interpretation, so until someone stumps up with a better model all we can do is assume that the singularity is ‘point like’

Well no, you can say we just don’t know how it works at that level. Pretending that a model that is almost certainly not applicable, is applicable, is almost certainly going to be misleading.

but we have to use something to make our predictions… the fact is we don’t have anything that fundamentally disproves the GR interpretation

Not when we don’t have any observations to test it one way or the other we don’t.

But now I have to go and make some predictions based on inadequate information and simplistic theories.

The Rev Dodgson said:

diddly-squat said:

The Rev Dodgson said:

Well no, you can say we just don’t know how it works at that level. Pretending that a model that is almost certainly not applicable, is applicable, is almost certainly going to be misleading.

but we have to use something to make our predictions… the fact is we don’t have anything that fundamentally disproves the GR interpretation

Not when we don’t have any observations to test it one way or the other we don’t.

But now I have to go and make some predictions based on inadequate information and simplistic theories.

I’m sure the results will be close enough… ;)

diddly-squat said:

The Rev Dodgson said:

Riff-in-Thyme said:

This is just BC’s argument about “near infinite” worded differently! :P

It’s an example of why treating “very large” as being infinite can be misleading.

+1

the two are very, very different

I can only agree. I’d say that the term density is a description that may lose relevance in this application. However, the term ‘mass’ if applied as a verb may be more appropriate. Massing is the fundamental process that defines density in our FoR. We do not know if particles remain intact or are absorbed into the processes within the BH in another form. At the very least it seems that if the repulsive forces between particles are obsolete in this scenario, then density is also.

The Rev Dodgson said:

If Brian Greene has it right (and he’s not a cosmologist, so maybe he hasn’t), the maximum information content of a black hole is proportional to the surface area of the event horizon, and this same limit applies to all space, not just black holes.

Almost. The entropy / information content of a black hole is always maximum: it saturates the Bekenstein bound . If you add entropy to it, you get a bigger black hole.

The Rev Dodgson said:

My understanding is that this result depends on the following two assumptions:
1) Matter inside a black hole is compressed to a region of infinite density
2) The 2nd law of thermodynamics is applicable under all situations, even inside black holes.

No, this doesn’t depend on the matter inside a black hole being compressed to infinite density, but it does depend on it having zero entropy. In other words, the matter at the core of a BH has no degrees of freedom, like a perfect crystal, so the description of its micro-structure is mathematically well-determined from its macro-structure.

As for the 2nd law of thermodynamics, I guess it could come in handy if black holes destroyed entropy, but without any evidence that they do that physicists are not keen to throw away the 2nd law.

I suspect that it wouldn’t be a calamity for the theory if there was some kind of 2nd law violation at the BH core, as long as the external behaviour of the BH conforms to the 2nd law.

The Rev Dodgson said:

What mechanism could prevent space having much more information than this? In particular, if you had a spherical lump of very dense matter at just pre-black hole stage, why would the maximum information content of this lump not be proportional to its volume, rather than its surface area, and why would this information content not be vastly greater than that of a black hole with an event horizon of the same size?

From Holographic principle

An object with entropy is microscopically random, like a hot gas. A known configuration of classical fields has zero entropy: there is nothing random about electric and magnetic fields, or gravitational waves. Since black holes are exact solutions of Einstein’s equations, they were thought not to have any entropy either.

But Jacob Bekenstein noted that this leads to a violation of the second law of thermodynamics. If one throws a hot gas with entropy into a black hole, once it crosses the event horizon, the entropy would disappear. The random properties of the gas would no longer be seen once the black hole had absorbed the gas and settled down. The second law can only be salvaged if black holes are in fact random objects, with an enormous entropy whose increase is greater than the entropy carried by the gas.

Bekenstein argued that black holes are maximum entropy objects—that they have more entropy than anything else in the same volume. In a sphere of radius R, the entropy in a relativistic gas increases as the energy increases. The only limit is gravitational; when there is too much energy the gas collapses into a black hole. Bekenstein used this to put an upper bound on the entropy in a region of space, and the bound was proportional to the area of the region. He concluded that the black hole entropy is directly proportional to the area of the event horizon.

Now, it might seem weird that the entropy of a BH is proportional to the square of the Schwarzschild radius (and is in fact one quarter of the area of the EH of a Schwarzschild BH when using natural units), rather than being proportional to the cube. But bear in mind that the Schwarzschild radius is proportional to the mass (r_s = 2GM / c²), so the entropy is in fact proportional to the square of the BH mass.

But having said all that, from the Bekenstein bound page:

Bekenstein derived the bound from heuristic arguments involving black holes. If a system exists that violates the bound, i.e. by having too much entropy, Bekenstein argued that it would be possible to violate the second law of thermodynamics by lowering it into a black hole. In 1995, Ted Jacobson demonstrated that the Einstein field equations (i.e., general relativity) can be derived by assuming that the Bekenstein bound and the laws of thermodynamics are true. However, while a number of arguments have been devised which show that some form of the bound must exist in order for the laws of thermodynamics and general relativity to be mutually consistent, the precise formulation of the bound has been a matter of debate.

PM 2Ring said:

However, while a number of arguments have been devised which show that some form of the bound must exist in order for the laws of thermodynamics and general relativity to be mutually consistent, the precise formulation of the bound has been a matter of debate.

One cannot take offence without taking the gates too?

The Rev Dodgson said:

If Brian Greene has it right (and he’s not a cosmologist, so maybe he hasn’t), the maximum information content of a black hole is proportional to the surface area of the event horizon, and this same limit applies to all space, not just black holes.

My understanding is that this result depends on the following two assumptions:
1) Matter inside a black hole is compressed to a region of infinite density
2) The 2nd law of thermodynamics is applicable under all situations, even inside black holes.

I can’t add much more than has already been said above. Yes, the holographic principle applies. No, this doesn’t assume that the centre of a black hole has infinite density; some string theories say that the matter within a black hole has a finite density and is not an infinite density singularity, but they still satisfy the holographic principle. Yes, the applicability of the 2nd law of thermodynamics to black holes is assumed and, although this looks likely, it isn’t yet proved.

Peak Warming Man said:

What ‘information’ are we talking about?

structure, order.. detail.. could be anything as simple as the spin of an electron that existed before it crossed the horizon.. any ‘data observation’ you can make is ‘information’

PM 2Ring said:

so the entropy is in fact proportional to the square of the BH mass.

Which implies that when two blackholes merge, the entropy of the merged blackhole will always be greater than the sum of the entropies of the original blackholes, consistent with the 2nd law of thermodynamics. That is, for a collection of blackholes, the area when the blackholes are merged into a single blackhole will always be greater that the sum of the areas of the original balckholes.

It should be noted that the volume of a blackhole is not well-defined. The Schwarzschild radius actually isn’t a measure of the blackhole’s radius, as the radius is also not well-defined. But in spite of this, the area of a blackhole is well-defined, and ideally, the characteristics of a blackhole should be specified in terms of its area.

The Rev Dodgson said:

and this same limit applies to all space, not just black holes.

Possibly relevant to this is the analogous relationship between Hawking radiation (which applies to blackhole event horizons) and Unruh radiation (which applies to any observer horizons, such as that associated with an accelerated observer).

As I understand it, Hawking derived the quantum fields in the curved spacetime surrounding a blackhole and concluded that this was thermal in character, thus giving the blackhole a temperature. By considering the addition of heat energy to the blackhole of a given temperature, the entropy of the blackhole could be derived by standard thermodynamic principles (for example, dS = dq/T).

KJW said:

As I understand it, Hawking derived the quantum fields in the curved spacetime surrounding a blackhole and concluded that this was thermal in character, thus giving the blackhole a temperature. By considering the addition of heat energy to the blackhole of a given temperature, the entropy of the blackhole could be derived by standard thermodynamic principles (for example, dS = dq/T).

However, the pure thermal character of Hawking radiation created a problem known as the black hole information paradox where blackholes emitting Hawking radiation apparently violated the unitary evolution of the quantum wavefunction. However, it seems to be accepted that the paradox is resolved by the radiation being not purely thermal and that the information of the infalling matter is retained within the perturbations of the event horizon, thus justifying the holographic principle.

A fundamental problem with unifying general relativity with the second law of thermodynamics is that general relativity, as a classical theory, operates in a continuum, and the informational definition of entropy becomes infinite in the continuum limit. To calculate the entropy in statistical mechanics requires that the Heisenberg uncertainty principle be invoked to keep the subdivision of the phase space finite.

KJW said:

A fundamental problem with unifying general relativity with the second law of thermodynamics is that general relativity, as a classical theory, operates in a continuum, and the informational definition of entropy becomes infinite in the continuum limit. To calculate the entropy in statistical mechanics requires that the Heisenberg uncertainty principle be invoked to keep the subdivision of the phase space finite.

What this means is that statistical mechanics (or statistical thermodynamics) is only <I>semi</I>-classical. That is, the statistics is classical, but the system it is applied to must be quantum mechanical.

Black holes might have firewalls:

http://www.huffingtonpost.com/2013/04/08/black-hole-firewall-theory-paradox-einstein-equivalence_n_3036733.html

Bubblecar said:

http://www.huffingtonpost.com/2013/04/08/black-hole-firewall-theory-paradox-einstein-equivalence_n_3036733.html

Interesting. However, from the article:

“To escape this paradox, Polchinski and his co-workers realized, one of the entanglement relationships had to be severed. Reluctant to abandon the one required to encode information in the Hawking radiation, they decided to snip the link binding an escaping Hawking particle to its infalling twin. But there was a cost. “It’s a violent process, like breaking the bonds of a molecule, and it releases energy,” says Polchinski. The energy generated by severing lots of twins would be enormous.”

Since when is entanglement like the bonds of a molecule that releases energy when broken?

Actually, if I understand correctly, for an isolated blackhole, the multi-particle quantum state of the entire radiation external to the event horizon must remain entangled with the multi-particle quantum state of the corresponding partner radiation inside the event horizon. In other words, even if the quantum state of the individual particles become disentangled from their partners inside the blackhole due to interaction with the other particles of the radiation, the total radiation must remain entangled as a whole. But interaction of the radiation with an external object would break this entanglement.

The Rev Dodgson said:

My understanding is that this result depends on the following two assumptions:
1) Matter inside a black hole is compressed to a region of infinite density
2) The 2nd law of thermodynamics is applicable under all situations, even inside black holes.

I have several questions on this:

- Has Brian Greene got it right?

- Is this hypothesis widely accepted?

- Is my understanding in points 1) and 2) correct?

- Why is it assumed that matter is compressed to infinite density inside a Black Hole, when there is no observational evidence for this, and it is incompatible with many hypotheses about the behaviour of matter at that scale?

- What mechanism could prevent space having much more information than this? In particular, if you had a spherical lump of very dense matter at just pre-black hole stage, why would the maximum information content of this lump not be proportional to its volume, rather than its surface area, and why would this information content not be vastly greater than that of a black hole with an event horizon of the same size?

I haven’t seen it specifically mentioned in this thread, or maybe it was and I just missed it…

Its my understanding that when physicists talk about “black hole singularities” and “big bang singularities” they are talking about mathematical singularities rather than physical ones.

A mathematical singularity is simply a function which provides a null result when applied mathematically.

Although, from the OP, Brian Greene seems to disagree… There is no reason to think that the mathematics of general relativity describes the physics inside the event horizon of a black hole.

GR is not used to model the centre of black holes. Quite specifically not used. GR models the universe in general, but when it comes to things like black holes the entire point of describing it as a singularity is to indicate that the GR maths does not and can not model what is physically happening in those volumes.

The Rev Dodgson said:

diddly-squat said:

The Rev Dodgson said:

- Why is it assumed that matter is compressed to infinite density inside a Black Hole, when there is no observational evidence for this, and it is incompatible with many hypotheses about the behaviour of matter at that scale?

This is a direct result of GR

OK, but why would we assume that GR is applicable at scales and densities where there is excellent grounds to believe that it is not applicable?

We don’t. GR is defined as a singularity at these scales specifically because it is not applicable. Treating a Black Hole as being a singularity at its centre s akin to calculating orbits based on centers of gravity.

Defining the centre of gravity of an object doesn’t tell you much about that object other than its COG. But by treating massive objects as if they ave a COG we an make useful calculations around those objects.

The Rev Dodgson said:

diddly-squat said:

All we have is the GR interpretation, so until someone stumps up with a better model all we can do is assume that the singularity is ‘point like’

Well no, you can say we just don’t know how it works at that level. Pretending that a model that is almost certainly not applicable, is applicable, is almost certainly going to be misleading.

If Greene has led you to believe that GR s applicable to black hole singularities, and describes some kind of physical effect then he has misled you and is wrong.

As I said above, the entire point of describing the centre f a black hole as a singularity is to show that the mathematical models we have which describe the universe break down and do not describe the centre of a black hole.

Most simply put, GR math at a black hole singularity = 1/0

The Rev Dodgson said:

… why we would we take seriously a result that is entirely based upon the true point like nature of the mass within a black hole?

We shouldn’t. We don’t. Treating a black hole as having an infinitely dense centre works in the same way as treating the Earth as if its gravity is centered on one point.

It’s useful for external calculations, but useless on its own for describing things like the size, shape or density of the planet.

esselte said:

Its my understanding that when physicists talk about “black hole singularities” and “big bang singularities” they are talking about mathematical singularities rather than physical ones.

For the Schwarzschild solution, there are two singularities: the event horizon and the central singularity. The event horizon singularity is a coordinate singularity in that it is the result of the particular coordinate system used to describe the solution and can be removed by expressing the solution in a different coordinate system. As such, this singularity is non-physical. By contrast, the central singularity is a true singularity of the local curvature invariants and cannot be removed by changing the coordinate system. As such, this singularity is physical (to the extent that general relativity can validly describe this).

The Big Bang singularity is a little different to blackhole singularities in that the metric does not become infinite but a key component becomes zero instead, making the metric non-invertible as a matrix. While the scale-dependent form of the curvature does become infinite, the scale-independent form of the curvature does not, but the local curvature invariants do become infinite as these are scale-dependent.

esselte said:

There is no reason to think that the mathematics of general relativity describes the physics inside the event horizon of a black hole.

There is: What happens to an object after it crosses the event horizon from the point of view of the object? Note that because the event horizon is only a coordinate singularity in the Schwarzschild metric which can be removed by changing to the Kruskal-Szekeres coordinate system, to the object, the spacetime inside the event horizon is not much different to the spacetime outside the event horizon. The issue is topology, the way the spacetime continues inside the event horizon from the outside. Actually, some time ago (on the old forum), I derived a wormhole solution that pushes the entire interior of the blackhole into the complex number domain. Nevertheless, in the complex number domain, the singularity still exists and indeed must exist for every physically reasonable spacetime.