Why do we need a quantum correction for black hole entropy?

[somok]

TL;DR Summary

It is now generally believed that information is preserved in black-hole evaporation.
why need a correction to bekenstein bound if it s proven valid by casini and quantum theory..??

Hey to all,...

It is now generally believed that information is preserved in black-hole evaporation.

This means that the predictions of quantum mechanics are correct whereas Hawking's original argument that relied on general relativity must be corrected.

However, views differ as to how precisely Hawking's calculation should be correctedwhy need a correction to bekenstein bound if it s proven by casini and quantum theory..??casini proves the thermodynamics interpretation in the form of bekenstein bound as valid..

we know following the work of casini in 2008 about the von neumann entropy and the bekenstein bound ,
that the proof of the Bekenstein bound is no more matters of debat

here is how...

Casini defines the right-hand side of the Bekenstein bound as the difference between the expectation value of the modular Hamiltonian in the excited state and the vacuum state,

K V = t r ( K ρ V ) − t r ( K ρ V 0 ) .

With these definitions, the bound reads S V ≤ K V , which can be rearranged to give t r ( ρ V log ⁡ ρ V ) − t r ( ρ V log ⁡ ρ V 0 ) ≥ 0.

This is simply the statement of positivity of quantum relative entropy, which proves the Bekenstein bound.
[Casini, Horacio (2008). "Relative entropy and the Bekenstein bound". Classical and Quantum Gravity. 25 (20): 205021. arXiv:0804.2182] in 2008 using quantum field theory.

Quantum information is entangled, all the information accumulated inside the black hole is conserved after the evaporation of the black hole...

Bekenstein bound is entropy of a black hole

The Bekenstein–Hawking entropy is a statement about the gravitational entropy of a system

S = A/4ℓ_p^2 for ℓ = √{Għ/c^3} and A = 4πr_s^2 and r_s the Schwarzschild radius r_s = 2GM/c^2.

 

[Demystifier]

First, you ask why do we need a correction to the Bekenstein bound, but you don't cite any reference which claims that we indeed need it.

Second, the Casini work rests on some assumptions that can be questioned. In particular, it is not obvious that simply subtracting the vacuum contribution gives the physically right values of entropy and energy. This is like saying that the (old version of the) cosmological constant problem is solved by subtracting the vacuum contribution.

Third, this forum offers you a possibility to write equations with LaTeX.

 

[somok]

hey to you Demystifier,...

""

First, you ask why do we need a correction to the Bekenstein bound, but you don't cite any reference which claims that we indeed need it.

Second, the Casini work rests on some assumptions that can be questioned.

"we discuss Hawking’s original formulation of the paradox, and how it can be easily resolved by the appropriate exponentially small corrections in Hawking radiation. "

from
"Lessons from the Information Paradox"
https://arxiv.org/abs/2012.05770
Suvrat RajuAbout Casini assumptions ::

from https://en.wikipedia.org/wiki/Bekenstein_bound

"the precise formulation of the bound was a matter of debate until Casini's work in 2008.[2][3][7][8][9][10][11][12][13][14][15]"

 

[Demystifier]

First, corrections to Hawking radiation are not the same thing as corrections to the Bekenstein bound. Second, the bound is still a matter of debate, wikipedia is not a completely reliable source.

 

[somok]

First, corrections to Hawking radiation are not the same thing as corrections to the Bekenstein bound. Second, the bound is still a matter of debate, wikipedia is not a completely reliable source.

i have never say that ... but it s all about black hole entropy as mentioned in the title

even if "wikipedia is not a completely reliable source.", i have provide bibliographic reference about casini work and bekenstein entropy proof... can you provide some from your side.?

 

[Demystifier]

even if "wikipedia is not a completely reliable source.", i have provide bibliographic reference about casini work and bekenstein entropy proof... can you provide some from your side.?

The references are OK, but only the wikepedia claims that "it is not a matter of debate". It is a matter of debate, that's all I claim. (After all, we are debating it right here.)

For example, Page says that Casini's proof corresponds to one particular definition of $S$ and $RE$ https://arxiv.org/abs/1804.10623 , but it's not clear whether this definition is the right one.

Rovelli argues that entropy can be much larger than the Bekenstein bound
https://arxiv.org/abs/1710.00218 . I myself made similar arguments in https://arxiv.org/abs/1507.00591 .

 

[somok]

Rovelli argues that entropy can be much larger than the Bekenstein bound
https://arxiv.org/abs/1710.00218 . I myself made similar arguments in https://arxiv.org/abs/1507.00591 .

from this article...
""
This conclusion is not in contrast with the the various arguments leading to identify Bekenstein-Hawking
entropy with a counting of states. "
""

For example, Page says that Casini's proof corresponds to one particular definition of S and RE https://arxiv.org/abs/1804.10623 , but it's not clear whether this definition is the right one.

In informational terms, the relation between thermodynamic entropy S
and Shannon entropy H is given by relation between S & H :

S=kHln(2)whence

H≤2πRE/ℏcln(2)where H

is the Shannon entropy expressed in number of bits contained in the quantum states in the sphere.

as the ln 2 factor comes from defining the information as the logarithm to the base 2 of the n

 

[somok]

casini proves the thermodynamics interpretation in the form of bekenstein bound as valid..
here is how...

Casini defines the right-hand side of the Bekenstein bound as the difference between the expectation value of the modular Hamiltonian in the excited state and the vacuum state,

K V = t r ( K ρ V ) − t r ( K ρ V 0 ) .

With these definitions, the bound reads S V ≤ K V , which can be rearranged to give t r ( ρ V log ⁡ ρ V ) − t r ( ρ V log ⁡ ρ V 0 ) ≥ 0.

This is simply the statement of positivity of quantum relative entropy, which proves the Bekenstein bound.

With S(ρ) : von Neumann entropy
S(ρ) is maximal and equal to ln N for a maximally mixed state, N being the dimension of the Hilbert space.

given a spatial region V , Casini defines the entropy on the left-hand side of the Bekenstein bound as
S V = S ( ρ V ) − S ( ρ V 0 ) = − t r ( ρ V log ⁡ ρ V ) + t r ( ρ V 0 log ⁡ ρ V 0 )

where S ( ρ V ) is the Von Neumann entropy of the reduced density matrix ρ V associated with V in the excited state ρ , and S ( ρ V 0 ) is the corresponding Von Neumann entropy for the vacuum state ρ 0 .

On the right-hand side of the Bekenstein bound, a difficult point is to give a rigorous interpretation of the quantity 2 π R E , where R is a characteristic length scale of the system and E is a characteristic energy.

This product has the same units as the generator of a Lorentz boost, and the natural analog of a boost in this situation is the modular Hamiltonian of the vacuum state K = − log ⁡ ρ V 0 .

 

[mitchell porter]

I think corrections come e.g. from Hawking radiation that gets out and falls back in... You could look at references in Ashoke Sen's paper, which is cited in Casini et al (page 30).

 

[somok]

I think corrections come e.g. from Hawking radiation that gets out and falls back in... You could look at references in Ashoke Sen's paper, which is cited in Casini et al (page 30).

from the Ashoke Sen's paper :

""In this case the dominant contribution to the entropy comes from the Bekenstein-Hawking
term, but it can receive subleading corrections proportional to the logarithm of the horizon area""...