Black Hole Horizon Information

[Naty1]

Here's what for me was a fascinating (and straight forward) description of How Jacob Beckenstein concluded information is displayed on the horizon of the black hole. The formulas are so simple and the results so profound I wanted to post them here...I'd never seen this before...

This is excerpted from THE BLACK HOLE WAR 2008, by Leonard Susskind, most of which appears around pages 151 to 155...

The most difficult conceptual idea for me is the starting point:
Beckenstein assumed a one bit photon falls into a black hole. To make the photon location as "uncertain" as possible, minimizing its information content, following Heinsenberg's uncertainty principle, he assumed the wavelength to about equal the Schwarzchchild radius, the longest feasible wavelength for capture to spread the photon to be as uncertain as possible. Susskind says longer wavelength photons would bounce off the horizon.

Beckenstein wondered how the size of a black hole would change if a single bit of information were dropped in...today we take for granted it's one Planck area, but how did he calculate that? Here's the derivation:

To figure the increase in mass, let's figure the energy of the photon and then convert that to an equivalent mass.

Photon energy is E = hf and if wavelength is Rs/SUB], from v = f x wavelength, frequency (f) is c/Rs/SUB] so E = hf becomes hc/Rs/SUB]. (1)

From E =mc², dividing energy by c² gives mass, so the change in mass becomes h/R_sc. (2)

The Schwarzschild radius is given by R_s - 2MG/c² (3)

and substituting the above change in mass for the photon, substituting (2) in (3) increases the radius 2hG/R_sc³...


So we have a photon adding a bit (pun) of mass to a black hole. Plugging in all the numbers gives a radius increase of 10^-72 meters...

Finally the (spherical) horizon area of a stable black hole is given by
A= 4(pi) R_s² so if the radius increases by a power of -72, the area increases by a power of -70...

What a coincidence...that's Planck area! (the square of Planck length of 10^-35 meter) ... so adding one bit adds one Planck area to the horizon! It works for any size black hole.

Susskind concludes:

Somehow, hidden in the principles of quantum mechanics and the general theory of relativity there is a mysterious connection between indivisible bits of information and Planck sized bits (pieces) of area.

If anyone can explain a bit more about the logic underlying one bit per photon in this example, I'd appreciate it.

 

[MeJennifer]

What a coincidence...that's Planck area!

That is not a coincidence at all, it simply follows from the applied formula.

 

[Jon_Trevathan]

Naty1 asked:

If anyone can explain a bit more about the logic underlying one bit per photon in this example, I'd appreciate it.

In Black Hole Wars, it is my recollection that Dr. Susskind, within the context of string theory, described each photon as a bit of string. Elsewhere in his book, it is my recollection that Dr. Susskind seemed to associate one bit of information to each dimensional twist and turn that the string might potentially manifest. Using an example that I recall in his book was limited to three spatial dimensions, I believe Dr. Susskind expressly associated two bits of information to each string loop. Based on the forgoing, I would have thought each photon might have been associated with two bits of information, and not just one.
Obviously, I am missing something important.
Help?

I recognize that strings are believed to be one-dimensional lines oscillating in eleven spacetime dimensions. If so, it would seem that a string might also twist and turn in any of the ten spatial dimensions that string theory allows. Would string theory's additional spatial dimensions increase the bits of information potentially associated with each sting loop; and by implication the bits of information each photon might possess? I recognize that Dr. Susskind's book provided very simplified descriptions and examples for readers with limited scientific backgrounds. However, I again feel that I am missing something important.
Further explanations from the readers of this forum would be very helpful.

Thank you.

Jon

 

[Naty1]

Elsewhere in his book, it is my recollection that Dr. Susskind seemed to associate one bit of information to each dimensional twist and turn that the string might potentially manifest. Using an example that I recall in his book was limited to three spatial dimensions, I believe Dr. Susskind expressly associated two bits of information to each string loop. Based on the forgoing, I would have thought each photon might have been associated with two bits of information, and not just one.
Obviously, I am missing something important.
Help?

That is actually a two dimensional lattice example on page 373...each link is two bits. He goes on to relate this to a black hole event horizon, also two dimensional...

You bring up a good point I had not thought about...maybe a photon present is a "1"; absence of a photon a "0"...

 

[George Jones]

Two comments:

1) An Introduction to Black Holes, Information, and the String Theory Revolution: The Holographic Universe by Susskind and Lindesay,

https://www.amazon.com/dp/9812560831/?tag=pfamazon01-20,

is a (somewhat) short introduction to Susskind's ideas at about the senior undergrad/beginning grad student level;

2) the holographic principle is not universally accepted.

 

[Naty1]

Would string theory's additional spatial dimensions increase the bits of information potentially associated with each sting loop; and by implication the bits of information each photon might possess?

Great question. Stumps me...some thoughts...

Entropy: if considered as information and measured in bits which depends on the total degrees of freedom of matter/energy...this seems to confirm your thought.

Holographic Principle: everything inside a region of space is described by information bits restricted to the boundary...seems to contradict. But what is the "boundary" of say ten dimensional space...a surface of 2 dimensions... or 9 dimensions??

Additional dimensions: If compactified, I don't think strings would "fit"...quantum jitters would maybe go crazy due to confinement??...yet the extra dimensions supposedly provide strings with characteristic vibration modes...could be a source of additional bits...like different wrapping, for example...

 

[Jon_Trevathan]

Great question. Stumps me...some thoughts...

Thank you for your efforts.

But what is the "boundary" of say ten dimensional space...a surface of 2 dimensions... or 9 dimensions??

If we are dealing with 10 spatial dimensions plus time (10+1 space), I believe the holographic principle requires the boundary to have 9 dimensions.

Would string theory's additional spatial dimensions increase the bits of information potentially associated with each sting loop; and by implication the bits of information each photon might possess?

The more I have reflected on this question, the more I feel the answer must be Yes.
However, noting that in string theory each particle-type is manifested through a unique vibrational pattern within the string, it would seem possible that the strings associated with our observable photons vibrate such that only one or two bits of information are involved. I am not sure I like this explanation, but I think it could answer the question I posed.

Any ideas?

Additional dimensions: If compactified, I don't think strings would "fit"...quantum jitters would maybe go crazy due to confinement??...yet the extra dimensions supposedly provide strings with characteristic vibration modes...could be a source of additional bits...like different wrapping, for example...

I recognize that there are a number of black hole theories. However, your observation about quantum jitters introduces an important observation. The Heisenberg uncertainty principle would seem to preclude a black hole singularity of infinite density. Dr. Susskind, in fact, makes this observation in the Black Hole wars. I can only presume that he and others properly considered the compact dimensions that String Theory seems to require.

Does anyone know what the maximum density of a black hole singularity is predicted to be (when the quantum jitters and the compacted dimensions are appropriately considered)?
It would be fascinating if the result corresponded with the Planck mass.

Again, any ideas?