Explaining the Holographic Principle - Physics & Math

[Mononoke]

Can someone please explain the holographic principle to me?

I'm decent in my physics and math, so you don't need to dumb things down too much

 

[Physics Monkey]

Hi Mononoke,

The origin of the holographic principle is in the physics of black holes. In ordinary statistical mechanics and thermodynamics, the entropy $$S$$ of a system of linear size $$R$$ in $$d$$ spatial dimensions is proportional to $$R^{d}$$ i.e. it is extensive. The energy is also extensive.

Now imagine increasing the temperature and energy $$E$$ of the system while keeping $$R$$ fixed. In a world without gravity there is no obvious limit to this process. However, in a world with gravity, the Schwarzschild radius $$R_s = 2 G E$$ also increases with the energy (G is Newton's constant with c and hbar set to one). Once $$R_s > R$$ the system is consumed by a black hole!

Now black holes have a temperature and an entropy, but this entropy is proportional to the area of the black hole event horizon. In $$d$$ spatial dimensions, the black hole's entropy is proportional to $$R^{d-1}_s$$. Note the different scaling of entropy with "system size" $$R_s$$ as compared with the ordinary thermal system we considered above.

This chain of ideas is the most basic manifestation of the holographic principle. The holographic principle states roughly that the maximum entropy and hence the total number of degrees of freedom should ultimately grow as the area rather than as the volume of some region. Essentially, this is because black holes exist and are the most entropic objects available. There are some subtleties, caveats, etc, for example, its not quite clear what the meaning of $$R$$ is in a curved space-time, but I can address these further if you're interested.

The most sophisticated modern realization of holography is provided by AdS/CFT. Briefly, quantum gravity in a particular space-time called anti de Sitter space is described by a quantum field theory living, in some sense, on the boundary of the space-time. The technical statement is that these two theories are dual to each other. Heuristically, what it means is that the gravitational theory is ultimately describable in terms of a quantum system of lower dimension i.e. the gravitational theory is holographic.

I'm happy to say more if you like. Also, the more background you give about your knowledge, the better I can tailor my answers to you.

Hope this helps.

 

[Mononoke]

Thanks Physics Monkey,

So in essence what it says is that the quantum mechanical description of gravity of a particular space time can be described as a function of the surface area of that space. Am I right, or did I muck it up.

Also, the more background you give about your knowledge, the better I can tailor my answers to you.

My background is that I have major in Maths & Physics. I've also taken a couple of graduate level classes in physics one in dynamic systems and the other in QM. I don't think either of those will help me much here.

I'm interested in this question because a friend of mine forwarded me this paper by Erik Verlinde. Where Verlinde uses the holographic principle to show gravity as emergent from entropy.

You can find the paper http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.0785v1.pdf"

 

[cristo]

You can find the paper http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.0785v1.pdf"

There is already a thread open on that paper: https://www.physicsforums.com/showthread.php?t=368785.

 

[Naty1]

"So in essence what it says is that the quantum mechanical description of gravity of a particular space time can be described as a function of the surface area of that space. Am I right, or did I muck it up..."

if you said instead "..a particular volume of spacetime..." I'd say "ok"...that's the basic idea.

Whereas as quantum mechanics says where a bit of information, for example, is located is slightly uncertain, the holographic principle says as steadily move the boundary surface area out to the very "edge" of the universe...and each time you do the "location" appears on successively larger boundaries...in the case of black holes, we call these boundaries or surface areas..."event horizons.

Leonard Susskind in THE BLACK HOLE War describes his work in the field, with Gerard d'Hooft and others and has lengthy insights into the implications...no fancy math, written for the general public...

 

[atyy]

There is a very nice exposition of the holographic principle in sections 2 & 3, "The Bekenstein argument" and "Consequences of Bekenstein’s bound", of Smolin's http://arxiv.org/abs/gr-qc/9508064 .

 

[Physics Monkey]

So in essence what it says is that the quantum mechanical description of gravity of a particular space time can be described as a function of the surface area of that space. Am I right, or did I muck it up.

Something like that. The trouble with our understanding right now is that we don't know how to make this statement generally. For example, when the boundary of the space-time is of a particular form, as in anti de Sitter space, then we know what to do. But we don't yet know in general what the right degrees of freedom are.

Your QM background may actually be quite useful. There is a nice little book by Susskind and Lindesay called "An Introduction To Black Holes, Information And The String Theory Revolution: The Holographic Universe" which discusses many of these concepts at level beyond Susskind's pop sci books. You may enjoy reading and learning about some general relativity while exploring these holographic ideas.

 

[p-brane]

Now black holes have a temperature and an entropy...

A system with entropy must have a temperature. Is the reverse true?

The holographic principle states roughly that the maximum entropy and hence the total number of degrees of freedom should ultimately grow as the area rather than as the volume of some region.

I`m sure you meant the area bounding the region.

...what it means is that the gravitational theory is ultimately describable in terms of a quantum system of lower dimension

What`s really interesting is that the dual system describing gravity involves only nongravitational interactions.

 

[Physics Monkey]

Hi p-brane,

A system with entropy must have a temperature. Is the reverse true?

You can imagine all kinds of strange and interesting things. Systems with finite entropy at zero temperature, systems with negative temperature, systems far out of equilibrium, and on and on. In my opinion, these were needless complications for answering the original question. I'm happy to talk with you about them if you like.

I`m sure you meant the area bounding the region.

True, I did not include the phrase "bounding the region". I think the original poster understood my meaning, as I assume you did.

What`s really interesting is that the dual system describing gravity involves only nongravitational interactions.

I agree that this is one particularly interesting aspect of the duality, at least in the AdS cases we understand. However, it is not at all clear that this is the most general situation in holography.

Hope this helps.

 

[p-brane]

You can imagine all kinds of strange and interesting things. Systems with finite entropy at zero temperature...

Forget about other systems. I was referring to your remarks about black holes. Can black holes have zero entropy and finite temperature or finite temperature and zero entropy? I posted the question for your benefit, not mine.

I agree that this is one particularly interesting aspect of the duality, at least in the AdS cases we understand. However, it is not at all clear that this is the most general situation in holography.

What is "not at all clear" is what your point is. Are you saying that once we understand holography, we`ll see that AdS/CFT is somehow wrong? Whatever the origin of holography or the true meaning of the gauge/gravity correspondence, there do seem to be purely nongravitational descriptions of purely gravitational systems.

Hope this helps.

Right back at you dude.

 

[atyy]

You can imagine all kinds of strange and interesting things. Systems with finite entropy at zero temperature, systems with negative temperature, systems far out of equilibrium, and on and on.

Is a system with finite entropy at zero temp one with degenerate ground states?

I suppose a system with negative temp is thought not relevant since those have finite numbers of energy levels, and systems out of equilibrium not relevant because Hawking radiation is supposed to be thermal?

 

[Finbar]

A system with entropy must have a temperature.

An extremal charged black hole can have an entropy and zero temperature.

 

[Physics Monkey]

Hi p-brane,

Sorry to be blunt, but do you have a problem with what I've said?

I suspect you know very well that there exist extremal black holes and black branes with vanishing temperature and non-zero entropy. That doesn't change the fact that the generic black hole has both a non-zero entropy and a non-zero temperature.

I also imagine you know that there have been serious proposals that holography in de sitter, for example, may require gravity in the dual theory. My point is simply that we don't understand all situations yet, surely you agree?

Furthermore, I have never said that ads/cft will be found to be "somehow wrong". Personally, I find the duality amazing, and I've written papers about it, so I have nothing against ads/cft. If you look around this forum you'll find that I'm a pretty loud proponent of ads/cft.

My goal in this thread is simply to answer Mononoke's question and encourage everyone to think about holography.

 

[Physics Monkey]

Hi atyy,

Yes, finite entropy at zero temperature means ground state degeneracy. In some cases this degeneracy is protected by supersymmetry. This fits with the fact that extremal black holes are sometimes bps and preserve extra supersymmetry.

 

[Fra]

So in essence what it says is that the quantum mechanical description of gravity of a particular space time can be described as a function of the surface area of that space. Am I right, or did I muck it up.

Something like that. The trouble with our understanding right now is that we don't know how to make this statement generally. For example, when the boundary of the space-time is of a particular form, as in anti de Sitter space, then we know what to do. But we don't yet know in general what the right degrees of freedom are.

I personally see this problem as quite entangled with lacking depth of understanding also the conceptual foundations of QM, and a missing intrinsic information&representation theory such as.

What IS information and information capacity physically and generically? And thus identification of natural entropy measures.
What IS the meaning of action on this information?

What I find interesting is that the holographic principle, bekenstein bound etc can be interpreted also so that the inferrable information of system, relative to another system, is somehow bounded/constrained in terms of the communication channel between the systems and that also, right in line with intuition that the "communication channel" capacity somehow constrains the inferrable encoded information even given an infinite information sink on the other side.

I think the points needed to be improved in order to make a generalisation precise is our understanding of information, degrees of freedom etc without resorting to some kind of realism that is not inferrable.

If we usually think of the holographic principle in terms of SPACE, then a possible generalisation in terms of generic surfaces thoughr of as interaction interfaces or communication channels, might even be a more fundamental starting point to also help us understand the degrees of freedom of spacetime from a deeper perspective.

/Fredrik

 

[atyy]

Yes, finite entropy at zero temperature means ground state degeneracy. In some cases this degeneracy is protected by supersymmetry. This fits with the fact that extremal black holes are sometimes bps and preserve extra supersymmetry.

So even "fundamental" theories could have ground state degeneracy? That's interesting, I had thought it only occurred in condensed matter physics. So a black hole can be part of a ground state? I naively thought black holes were formed by high energy collisions, so they'd be an excited state.

 

[Finbar]

In terms of the black hole mechanics the zero temperature is dynamical.

$$dE= T dS - dW$$

Where E is the energy(mass of the black hole) and W is a work term due to eg. a non zero charge.

rearranging

$$T= \frac{dE}{dS}(1+\frac{dW}{dE})$$

So an extremal BH has

$$\frac{dW}{dE}=-1$$

Hence T=0. It also corresponds to Anti -de Sitter space at the horizon. So I guess if the true vacuum is de Sitter. The "false vacuum" is a dynamically induced one where in extreme situations where the metric is locally Anti-de Sitter.

Also a black hole is a system of the gravitational field interacting with matter fields so its not clear that its "fundamental" i.e T=0 for a charged BH is obtained solving the Einstein-Maxwell equations. But maybe there are solutions to just the Einstein equations that can give T=0 for a black hole e.g Kerr??