# I The Holographic Principle

1. Apr 25, 2017

### Buzz Bloom

I have just come across the following on the Astronomy Picture of the Day.
The Holographic Principle, yet unproven, states that there is a maximum amount of information content held by regions adjacent to any surface. Therefore, counter-intuitively, the information content inside a room depends not on the volume of the room but on the area of the bounding walls.​
I looked at the link for "Holographic Principle" and It gave a more formal definition of this concept, but not much about the reasoning behind it.

Can anyone explain this a bit further, or possibly post a link to a good discussion. The Wikipedia article was not particularly helpful to me, except for the following:
The holographic principle is a principle of string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional https://www.physicsforums.com/javascript:void(0) [Broken] to the region—preferably a light-like boundary like a gravitational horizon.​
In particular, can anyone estimate for me how likely this concept will turn out to be confirmed by observation?

Last edited by a moderator: May 8, 2017
2. Apr 25, 2017

### Grinkle

As I understand it, the maximum information in bits that any volume can contain before the particles inside that volume become a singularity is the number of plank areas on the sphere of the event horizon that would form for a singularity inside that volume.

The information content in any room you or I will ever observe is well below this threshold and not at all bottlenecked by the area of the room, since we observe densities very very far away from this limit.

3. Apr 25, 2017

### DennisN

I can not, my precognitive skills are not very good . Also, this belongs in the "Beyond the Standard Model" forum part, so I'll ask a mentor for a move of the thread to over there.

4. May 2, 2017

### Denis

My objection to the holographic principle is that our universe violates it. Our universe is described as homogeneous and flat, and flat means infinite, thus, you can use an arbitrary large radius R and compute what is inside. Given that it is homogeneous, the amount of information will increase like $R^3$. Its surface only like $R^2$. So, whatever the factor used in the holographic principle, and the information contained in a piece of the homogeneous universe, with sufficient big R the result will be fatal for the holographic principle.

As far as I have understood the idea, if there is more information inside a radius, the whole thing will collapse. No problem, and no contradiction. Our universe can be seen simply as a collapse solution reverted in time. So, at best the holographic principle can tell us something about GR solutions which can survive without singularity in the past as well as the future. But once we nicely live with a solution with a singularity in the past, what would be the point? As a fundamental principle of QG, that would be useless anyway.

5. May 2, 2017

### Grinkle

I don't follow that. Draw a radius of whatever size you please, then let that radius expand and add the extra bits of information in the expanded radius, and you will not find any region where the information has increased by order R^3. Even if that new (or old, I guess) radius contains an event horizon, the information inside the event horizon will be what is on the surface of the horizon, not what is in the volume. And all of the other space not containing an event horizon will have an information density of less than what is in the event horizon, which follows from saying that there is nothing dense enough to create an event horizon.

This is why I don't see how the information can increase faster than order R^2.

6. May 2, 2017

### Denis

Hm. We have a homogeneous universe. I think it is natural to propose that the information contained in a given volume is proportional to the number of particles inside the volume. Once we have constant density in a homogeneous universe, we have, then, also constant information density. Thus, the whole information in the sphere with radius R will be proportional to the volume, $R^3$. Not?
????? A large enough piece of matter can have quite small density but create an event horizon.

The Schwarzschild radius is proportional M, $r_S=2GM$. Take any density $\rho$, and a homogeneous piece with that density and radius R, then the mass will be $M=\rho R^3$. So that its Schwarzschild radius is $r_S=2G\rho R^3$. So, whatever $\rho$, you can chose an R so that $r_S > R$.

The, ok, a little correction: In this case, it will collapse if initially at rest. If it already explodes, like a white hole, it may not collapse. In this case, it will have a singularity in the past, the white hole singularity. So what, this is what we have in the BB solution too, a singularity in the past.

7. May 2, 2017

### Grinkle

I agree. The flaw in my reasoning is that the maximum possible information density increases order R^2. In our observable universe, the densities are much lower than this, and I agree with your reasoning that the actual information content will increase order R^3. This does not contradict the holographic principle, which is making a statement about maximum information density. The maximum possible density increases order R^2. If one believes in the existence or potential existence of black holes and event horizons, then I think its a matter of mathematics, not a matter of opinion.

I just mean that in the thought experiment one can either decide to include objects with event horizons or not.

8. May 3, 2017

### Denis

It does, because for a large enough R the $R^3$-proportional expression becomes greater than the $R^2$-proportional one. But there is nothing which changes the rule how to count the information contained in that volume. Whatever the proportionality factor for the information per volume, it remains the same, because there is nothing which says that the information contained on Earth is somehow smaller if Earth is considered as a part of some greater part of the universe. And the expanding universe solution remains a valid solution even for arbitrary large R, because the flat homogeneous universe is a solution even for infinite R, so that any large part of it is a valid solution too.
Correct, it is a matter of mathematics. We have the solution of the expanding flat homogeneous universe. It is the one which we use to describe our own universe, so not some artificial nonsense solution. In this solution, the holographic principle does not hold.

9. May 3, 2017

### Staff: Mentor

This might be a silly question, but doesn't your argument depend on the constant information density? How is this justified?

10. May 3, 2017

### Denis

What else could be justified if we have a universe with (on the large scale) constant density of matter? The same number of particles in the same volume, add the same temperature if necessary, how could one justify something else as the result than constant information density? And what would be the alternative to simply adding the information from different parts, in a way similar to adding the masses?

11. May 3, 2017

### Staff: Mentor

I thought the density of "matter" would be decreasing in an expanding universe.

12. May 3, 2017

### Denis

It decreases with time. But we are not talking about evolution in time, but about a fixed moment of time.

13. May 3, 2017

### Grinkle

You are missing my point. The HP says nothing about empirically observed information per volume in a flat expanding universe with low average information density. The HP does not predict or contradict such a universe. The HP makes a statement about maximum possible information density in any given volume of spacetime.

Again, not the point. The information contained in the Earth's particles gets more dense if the Earth is shrunk. If the Earth is shrunk to its critical mass, an EH will form that will have a surface area such that exactly as many Plank areas on the EH exist as there are bits of information inside the EH. The amount of information it takes to describe the Earth has not changed, just the density of that information.

14. May 3, 2017

### mitchell porter

I too felt sure that Denis must be overlooking something. But not only is he right that this principle is violated by sufficiently large regions in a flat universe; Susskind (and Fischler) already knew this back in 1998! (See section 2.) You have to explicitly say that you are only talking about a volume of spacetime that doesn't cross the cosmological horizon.

15. May 3, 2017

### Denis

Thanks. The natural question which follows is what would be the point of such a particular observation of a relation between a cosmological horizon and its content. And, in particular, why such an observation should be relevant for quantum gravity.

16. May 3, 2017

### Grinkle

Thanks for that reference. This is a different description of the holographic principle than the one I thought myself to be arguing. @Denis, I concede that I didn't understand there is a cosmological version and that is the one your were talking about. One quibble I have remaining (and its minor) is that it doesn't seem that you disagree with the cosmological HP, you are pointing out that it does not hold unconditionally.

I am trying to get a conceptual understanding of where the described limit comes from.

From section 2, just prior to formula 2.2, "The entropy contained within a volume of coordinate size Rh should not exceed the area of the horizon in Planck units." Higher entropy means that more thermodynamically equivalent states exist for a given system at equilibrium. So, if the number of particles is the same, to specify which of the many possible states a volume is in requires more bits for higher entropy - is that a more or less correct way to think about it?

17. May 3, 2017

### Haelfix

Ok, a few things here.
The Holographic principle strictly speaking comes from the study of black hole physics. However, it was very quickly realized that it was impossible to have it remain a black hole only phenomenon, and it had to be a more general property of quantum mechanics coupled to gravity.

The very simple heuristic baby version can be justified with a thought experiment. Take a given spherical region of space time with mass M and radius R. Now outside of this radius much further away, collapse a spherically symmetric lightlike cloud of radiation (photons) where you pick the mass of the shell such that, coupled with the inner mass M it will form a black hole with Schwarschild radius exactly R.

Now, Hawking calculated the entropy of this quantity. The entropy of the system is given by S=A/4G. In other words explicit computation of the black hole entropy yields that it's proportional to the area of the enclosing horizon.

Now, if the second law of thermodynamics holds during this process, it must be that the initial uncollapsed region had S <= A/4G. Which is completely at odds with what you would naively expect. Normally the maximal entropy of a quantum field theory would scale with the volume, but here it's given by at most something proportional to the area. So the statement is that either Hawkings calculation is wrong, the second law is violated or you must consider a holographic bound to be a general property of quantum systems coupled to gravity

What this doesn't tell you of course, is how it applies exactly to cosmology and where to put your boundary conditions. For that you need to not just consider spacelike hypersurfaces like the above, but to generalize the whole thing to timelike surfaces and more general geometries. That was done, and in general these carry fancy names like the Bousso bound, the Fischer Susskind bound etc. You need more sophisticated tools like the focusing theorem in GR to prove this, but in some cases this has been done.

Of course the cleanest version occurs with Anti De Sitter spaces, where the holographic principle strikingly applies and is an integral part of the understanding of the system and to Ads/CFT duality

18. May 14, 2017

### Krill

Can i ask a side question ?

Our brains are supposed to construct a 3D experience of the world from information in the curved plane of our retinas.

Would that be analogous in any way to the holographic theory ?

(Granted that retinas aren't truly 2D).

19. May 15, 2017

### craigi

This is actually an incredibly difficult principle to understand fully. It would typically become apparent at the tail end of a postgrad theoretical physics course. Few physicists understand it in a meaningful way. Leonard Susskind has popularised the principle and I recommend searching for his lectures on the subject until you find one which is at the level you prefer. I would suggest that anything else beneath the levels which Susskind explains it at are nothing more than interpretations of Susskind's own attempts to make it available to a wider audience.

Regarding whether it's likely to be confirmed by observation, then we're talking about layers of mathematical abstraction. I have Susskind and Lindesay on my desk right here and even with that I can't tell you that the Holographic Principle is a property of Quantum Gravity or even String Theory. A tiny percentage of physicists might be able to divine that information.

Last edited: May 15, 2017
20. May 30, 2017

### Krill

Well, OK. Did you ever hear anything about Donald Hoffman, visual perception scientist ?