# Bekenstein bound on cosmological scales

1. Jun 24, 2015

### tzimie

Except the extreme case of Black Holes, all other objects in the Universe (say, Galaxies) are very far from the Bekenstein bound (BB). Any object saturating the BB tends to be BH. But we can try to saturate the BB increasing the radius of a sphere. In flat static Universe, sooner or later we saturate the BB and we get a BH. This is one of the reasons why such Universe is not possible in GR.

As we know, big volumes in our Universe don't become BHs because of the expansion. My questions is, what's about the BB on the cosmological scales? If we take a sphere bigger than Cosmological Horizon, then BB is probably not applicable to it (because oppisite regions are not causally connected)? What's about the simpler case of the Universe without Dark Energy - Universe without Cosmological Horizons, where sooner or later all regions mutually cross their 'future' light cones?

2. Jun 25, 2015

### marcus

I like your questions even though I cannot answer and am not the right person to try to answer. I hope someone else here will respond. I do not see a natural way to define the Bekenstein Bound in the case of expanding geometry. You must specify a radius R. Is this in terms of comoving distance or proper distance?
Comoving distance is convenient but seems unphysical because it does not experience expansion. Proper distance is defined only for the specific moment of Universe time (Friedmann time) and you must imagine somehow pausing the expansion process long enough to make enough time to measure it.
Operationally speaking what could be the meaning of the Bek. Bound on such a large scale? Everything in physics presumably comes down to measurements, whatever meaning it has boils down to relations between measurements---operations that in principle could be done. I am having difficulty making sense of the question in operational terms.
But the Bek. Bound is extremely interesting! I will quote, for people unfamiliar with it:
==Wikipedia on Bek. Bound==
The universal form of the bound was originally found by Jacob Bekenstein as the inequality[1][2][3]

where S is the entropy, k is Boltzmann's constant, R is the radius of a sphere that can enclose the given system, E is the total mass–energy including any rest masses, ħ is the reduced Planck constant, and c is the speed of light. Note that while gravity plays a significant role in its enforcement, the expression for the bound does not contain Newton's Constant G.

In informational terms, the bound is given by

where I is the information expressed in number of bits contained in the quantum states in the sphere. The ln 2 factor comes from defining the information as the logarithm to the base 2 of the number of quantum states.[4] Using mass energy equivalence, the informational limit may be reformulated as

where is the mass of the system in kilograms, and the radius is expressed in meters.
==endquote==
You know in GR there is no energy conservation law and one might not be sure how to define R when it is very large.
But suppose the "E" is all radiation and the "R" is proper distance. Then RE could be constant because as R increases the energy inside the volume declines just enough to compensate. the wavelength of each photon, so to speak, grows proportional to R and so the expansion effect cancels.
(this is just a crude oversimplification but it suggests that the bound might extend in some fashion to this more general case)

Maybe this has already been realized and done long ago.

3. Jun 25, 2015

### ShayanJ

It seems to me this paper may help.

Last edited: Jun 25, 2015
4. Jun 26, 2015

### tzimie

Thank you for the article! Interesting.
But I have one another question.
Is BB fundamental in TOE (Loop gravity or superstrings) or is it merely an emergent property on larger (non-Planks) scales?

5. Jun 26, 2015

### ShayanJ

This question seems strange. Thermodynamics is about the collective behaviour of a large number of degrees of freedom. How can you think of it as fundamental in anything, whether air conditioning or quantum gravity?!
And the BB is an inequality involving entropy which is, like other concepts in thermodynamics, about a large number of degrees of freedom.
So the answer to your question is, I think, the BB may arise from the ultimate quantum theory of gravity or may not. Its just more probable that it will. But even if it arises, it will be about an emergent property, because entropy itself is emergent.

6. Jun 26, 2015

### tzimie

Correct, you can look at thermodynamics this way, so it makes no sense to talk about thermodynamics of a single electron. However, some aspects of thermodynamics are more fundamental - for example, "no information loss" (quantum determinism)

7. Jun 26, 2015

### ShayanJ

That's about the unitary evolution of the quantum state of a system. Its an aspect of quantum mechanics, not thermodynamics.

8. Jun 27, 2015