Gravity as an emergent phenomenon

[nomadreid]

The concepts in "On the Origin of Gravity and the Laws of Newton" by Prof. Erik Verlinde, found at http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.0785v1.pdf, are intriguing. There are two points which I do not understand, and would be grateful for any enlightenment. First, why is the holographic principle necessary for his derivation? Secondly, I do not understand how he gets from the classical holographic principle operating on the border of the observable universe to the idea of the two-dimensional screens being able to be anywhere in space.

 

[atyy]

1. The holographic principle comes from considering general relativity and thermodynamics. GR predicts black holes which have an event horizon, which prevent you from seeing what's going on inside. For entropy not to disappear, it must somehow be preserved on the event horizon. Verlinde attempts to run the argument in the other direction to get gravity by assuming the holographic principle and thermodynamics.

2. An accelerated observer in special relativity has a horizon (Rindler horizon) has some of the properties of the event horizon of a black hole. In particular, it has a temperature called the Unruh temperature. Since we can have accelerated observers anywhere, we can have these horizons anywhere.

3. Verlinde's work is new and not generally accepted. However, the ideas he uses are "accepted speculation". The necessity for black hole event horizons to have entropy comes from Bekenstein. Classical black holes have no entropy, but Hawking made a very suggestive calculation that quantum black holes have entropy. Bekenstein and Hawking's work forms the basis of the holographic principle of 't Hooft and Susskind. See sections 2 and 3 of http://arxiv.org/abs/gr-qc/9508064. Derivations of the Einstein field equations of general relativity from the temperature of Rindler horizons were given first by Ted Jacobson http://arxiv.org/abs/gr-qc/9504004 and further expounded on by Thanu Padmanabhan http://www.springerlink.com/content/l403540q87606463/, http://arxiv.org/abs/0911.1403.

 

[nomadreid]

Thank you very much, atyy, for the excellent answer and the helpful article links. I look forward to reading them.

 

[Naty1]

I do not understand how he gets from the classical holographic principle operating on the border of the observable universe to the idea of the two-dimensional screens being able to be anywhere in space.

You can find a good discussion of the holographic principle in Leonard Susskind's THE BLACK HOLE WAR. Likely Gerard d'hooft, another originator of holographic theory and a collabarator of Susskinds, also has written material...

The idea is this: Think of some process in a bounded volume of space, and it's exact information representation on a two dimensional boundary, say an enclosing spherical surface...Then take a larger surface enclosing the prior boundary a larger concentric spherical surface perhaps...all the original information is still enclosed, still on the new boundary surface as it was before plus some new information...repeat until you get anywhere you'd like to observe from.

Horizons also arise from accelerated observers as in Unruh radiation/temperature...so horizons are relative here,too, similar to their relative nature at black holes, depending upon accleration. We just don't realize the "elativity" of things in our day to day lives.

 

[nomadreid]

Thank you, Naty1. Sorry for the delay in replying. I would be grateful for a little more clarification. Whereas your third paragraph corresponds to atty's answer and appears satisfactory, I am not sure whether the second paragraph is what Verlinde had in mind. Given that Verlinde is considering gravity, which is equivalent to considering accelerated frames, the Unruh concepts seem more germane. Can one subsume the second "onion skins" concept, which doesn't need acceleration, to the first one, the one connected to Unruh temperature?

 

[yuiop]

As I understand the holographic principle, the maximum amount of information (states) that can be contained in a given volume is proportional to the surface area. A black hole is the limiting case of maximal information density with the event horizon area being proportional to the contained information. Now it seems there is an easy counter example to this. Consider a cube of very strong material that has a mass density that is 1/4 of its Schwarzschild density. Twenty six similar cubes are brought into contact with the original cube making one large cube with 27 times the volume of the original cube and only 9 times the surface area of the original cube. Now the total information contained in the large cube is exactly 27 times that of the original single cube, but the maximum information that can be contained in the volume of the large cube according to the surface area principle is 9 times that of the single cube. It would appear that at this point, the information density of the large cube exceeds that of a black hole.

The question is does the holographic principle allow the maximum information density to be exceeded, even temporarily (the large cube would obviously collapse to a black hole soon after) or is it just a case that no material would have to the strength to remain as a cube at a density of 1/4 the Schwarzschild density, rather than forming a sphere at such densities and therefore preventing such a situation occurring? Does this mean that the maximum strength of the strongest material can be predicted by the holographic principle? Also, it should be noted that I used 27 cubes, but I could use more cubes, further increasing the volume to surface area of the combined super cube and could in principle start with an initial cube of much lower density. An additional question is whether the holographic surface area principle is restricted to spherical surfaces only, as that would negate the above arguments?

[EDIT]Some texts hint that the holographic principle may only apply to the surface area of event horizons. Can anyone clarify that issue? If that is true, it would resolve the above "paradox".

 

[yoron]

Thanks for the links, really sweet ones :)
And yes, it would be interesting if it was solution limited to event horizons.
But I don't think so myself, somehow it seems to fit so well in with the idea of Lorentz contraction and the way that one will be the only one available to you observing 'first hand scenario'. After all, as I understands it, a 'time dilation' can only be confirmed after a 'journey' whereas the Lorenz contraction will be with you the whole time journeying?

Which to my eyes gives you a more immediate confirmation of your velocity.
==

Not the holographic principle, I was referring to the idea I think it build on. The Bekenstein bound.

 

[yoron]

Or maybe I'm wrong?

There seems to be some limitations to the Bekenstein bound.

"Sad to say though, it has its limitations. In Sec.2.1, the conditions Bekenstein specified for the validity of his bound was not stated explicitly. The system under consideration must be of constant, finite size, have limited self-gravity and no matter components with negative energy density can be available. A system which satisfies these conditions will be referred to as a Bekenstein system."

http://www.teorfys.uu.se/files/Daniel_Domert_holographic.pdf from Uppsala university in Sweden.

?

 

[Q-reeus]

As I understand the holographic principle, the maximum amount of information (states) that can be contained in a given volume is proportional to the surface area. A black hole is the limiting case of maximal information density with the event horizon area being proportional to the contained information. Now it seems there is an easy counter example to this. Consider a cube of very strong material that has a mass density that is 1/4 of its Schwarzschild density. Twenty six similar cubes are brought into contact with the original cube making one large cube with 27 times the volume of the original cube and only 9 times the surface area of the original cube. Now the total information contained in the large cube is exactly 27 times that of the original single cube, but the maximum information that can be contained in the volume of the large cube according to the surface area principle is 9 times that of the single cube. It would appear that at this point, the information density of the large cube exceeds that of a black hole.

The question is does the holographic principle allow the maximum information density to be exceeded, even temporarily (the large cube would obviously collapse to a black hole soon after) or is it just a case that no material would have to the strength to remain as a cube at a density of 1/4 the Schwarzschild density, rather than forming a sphere at such densities and therefore preventing such a situation occurring? Does this mean that the maximum strength of the strongest material can be predicted by the holographic principle? Also, it should be noted that I used 27 cubes, but I could use more cubes, further increasing the volume to surface area of the combined super cube and could in principle start with an initial cube of much lower density...

Just an observation as a GR rank outsider: As you start to stack those cubes together, won't the (Schwarzschild?) metric severely warp and shrink, and your assumption about volume-to-surface-area ratios cannot be maintained? Hence even 'Born rigid' cubes distort and merge/collapse into a BH no matter how carefully one tries to avoid it.
Having said that, I'm also skeptical about whether the accepted BH entropy/surface area relation has any real validity. For starters only an exactly placed and pointed photon can hover at the horizon (non-rotating BH, and horizon radius as determined from a distant observer), but from that perspective the EH area has blown up to infinite size. So BH 'area' becomes a highly observer dependent quantity. Maybe that's all taken care of but it raises a question mark for me. Another thing I have trouble with is 'unitary evolution' of QM - information is always conserved no matter what. Supposing our universe began as a Planck Volume sized 'point' ala standard Big Bang/Inflationary scenario, can it really be that the information needed to construct say everything in the Louvre museum, let alone the entire universe's history, was all there in that 'primordial point'? Seems to me information must somewhere along the way be severely non-conserved! Hawking has conceded re the 'Information Paradox' issue in favor of Unitarity holding for BH's, but maybe he shouldn't have?

 

[yoron]

Maybe it has to do with the definitions?

If you expect 'discrete events' to be the states defining information, and also define 'SpaceTime' as a closed universe, where no information can be lost then you have a paradox accepting the 'arrow of time' and the Big Bang.

If you instead define all 'discrete events' and states as 'relations' created out of something we can't know, maybe under Planck size, then you could see what we have as an enclosed bubble in one way, in another just as an organized 'information space' containing its own logic and causality chain(s) inside a different one, where we are perfectly adapted to take in our 'information' and from it observe a 'SpaceTime'.

If it would be so then the 'information' we observe, bits, states whatever, could be contained in that first point of the BB too as what creates SpaceTime is the part of the larger infinity's 'information' we use, their relations creating us. Sort of as we were contained inside another 'reality', a small 'infinity' contained in a 'larger infinity'.

That would make SpaceTime into 'relations' instead of 'forces' and ''discrete events' though, as I see it.

Weird?
==

It would crave principles to exist though, maybe like the Feigenbaum's constant that, to me, speaks of a linear function 'surrounding' a non-linear universe. :)

 

[yuiop]

As I understand the holographic principle, the maximum amount of information (states) that can be contained in a given volume is proportional to the surface area. A black hole is the limiting case of maximal information density with the event horizon area being proportional to the contained information. Now it seems there is an easy counter example to this. Consider a cube of very strong material that has a mass density that is 1/4 of its Schwarzschild density. Twenty six similar cubes are brought into contact with the original cube making one large cube with 27 times the volume of the original cube and only 9 times the surface area of the original cube. Now the total information contained in the large cube is exactly 27 times that of the original single cube, but the maximum information that can be contained in the volume of the large cube according to the surface area principle is 9 times that of the single cube. It would appear that at this point, the information density of the large cube exceeds that of a black hole.

The question is does the holographic principle allow the maximum information density to be exceeded, even temporarily (the large cube would obviously collapse to a black hole soon after) or is it just a case that no material would have to the strength to remain as a cube at a density of 1/4 the Schwarzschild density, rather than forming a sphere at such densities and therefore preventing such a situation occurring? Does this mean that the maximum strength of the strongest material can be predicted by the holographic principle? Also, it should be noted that I used 27 cubes, but I could use more cubes, further increasing the volume to surface area of the combined super cube and could in principle start with an initial cube of much lower density. An additional question is whether the holographic surface area principle is restricted to spherical surfaces only, as that would negate the above arguments?

I wonder why no one noticed the flaw in the above argument? What I had not taken into account was that the required density to form a black hole gets less as the mass increases. For example let us say initially the radius was such that 2GM/(rc^2) =1/4 (am I approximating a cube as a sphere as a first order back of the envelope approximation). When the mass increases 27 fold, by stacking the cubes in a 3X3X3 formation, the radius will have increased by 3. This means that we now have 2GM/(rc^2)*27/3 = 9/4 and the mass/radius ratio of the stacked cube is now more than twice that required to be a black hole even before collapsing.

 

[Passionflower]

I wonder why no one noticed the flaw in the above argument? What I had not taken into account was that the required density to form a black hole gets less as the mass increases. For example let us say initially the radius was such that 2GM/(rc^2) =1/4 (am I approximating a cube as a sphere as a first order back of the envelope approximation) and the mass is. When the mass increases 27 fold by stacking the cubes in a 3X3X3 formation, the radius will have increased by 3. This means that we now have 2GM/(rc^2)*27/3 = 9/4 and the mass/radius ratio of the stacked cube is now more than twice that required to be a black hole even before collapsing.

Isn't the issue here the usage of r?

If we leave out r all together then apparently mass is equivalent to an area, this mass can be arranged inside an area not smaller than 4 times otherwise the mass will become a black hole, this is irrespective of the size of the mass.

E.g.:

<br /> {A_{occupied} \over A_{mass} } &lt; 4<br />

 

[Q-reeus]

yoron; information theory seems to be something you are into in a big way! Afraid to say it's something else I'm a rank outsider at, so my comments may be completely skew but here goes:

...If you expect 'discrete events' to be the states defining information, and also define 'SpaceTime' as a closed universe, where no information can be lost then you have a paradox accepting the 'arrow of time' and the Big Bang.

Isn't that part 'just' a standard thermodynamic-arrow-of-time thing, a matter of 'fine-grained' vs 'coarse-grained' perspective? Recently read through Sean Carroll's interesting article here http://arxiv.org/abs/0811.3772 that goes into that.
Edit: A better link may be to part 4 of http://arxiv.org/abs/hep-th/0410270" . Actually the earlier sections 3.2 - 3.3 perhaps cover what you mean in general and supersede my comments below. Too obscure and speculative for me so bailing out of this one!

If you instead define all 'discrete events' and states as 'relations' created out of something we can't know, maybe under Planck size, then you could see what we have as an enclosed bubble in one way, in another just as an organized 'information space' containing its own logic and causality chain(s) inside a different one, where we are perfectly adapted to take in our 'information' and from it observe a 'SpaceTime'.

Hmm - let me guess. Much more potentially accessible information than a single bit at the Planck scale - let's go sub Planckian a la Gerard't Hooft and co? If so, highly speculative wouldn't you say?

If it would be so then the 'information' we observe, bits, states whatever, could be contained in that first point of the BB too as what creates SpaceTime is the part of the larger infinity's 'information' we use, their relations creating us. Sort of as we were contained inside another 'reality', a small 'infinity' contained in a 'larger infinity'.

Tentative translation - maybe not sub-Planckian here but a Multiverse thingy - our initial pre BB 'bubble universe' sucked info from it's Mother Universe via some umbilical cord so to speak (Linde, Vilenkin et al)? Wasn't the cord broken very early on in the 'standard' picture, way before much info could have accumulated? What seems like death to all that sort of thing as explanation seems to be that Bekenstein Bound formula you linked to earlier:http://en.wikipedia.org/wiki/Bekenstein_bound. A tiny initial bubble just can't contain much info, period! All the scenarios seem to require a hugely unlikely but necessary low-entropy beginning, but that's very different IMHO from a large info beginning.

That would make SpaceTime into 'relations' instead of 'forces' and ''discrete events' though, as I see it. Weird?

You bet - we have Unity (not necessarily Unitarity) on that one! Seriously, as an admittedly unqualified amateur the whole notion of 'it from bit' ala J.A.Wheeler seems crazy to me - a kind of Platonic philosophy run riot. Things first, info as a derived construct in my view.

It would crave principles to exist though, maybe like the Feigenbaum's constant that, to me, speaks of a linear function 'surrounding' a non-linear universe. :)

Err - OK, I guess!?

 

[yuiop]

Isn't the issue here the usage of r?

If we leave out r all together then apparently mass is equivalent to an area, this mass can be arranged inside an area not smaller than 4 times otherwise the mass will become a black hole, this is irrespective of the size of the mass.

E.g.:

<br /> {A_{occupied} \over A_{mass} } &lt; 4<br />

I am not sure how you get that result and I do not think you have clearly expressed yourself. What is the the "area of a mass" if it is not the "area occupied by the mass"? A kilogram of Polystyrene has greater surface area than a kilogram of lead under normal circumstances, so there is no such thing as the intrinsic "area of a mass".

In order to be a black hole the following condition must be satisfied:

\frac{2Gm}{rc^2} &gt;= 1

Which can also be expressed as:

r &lt;= \frac{2GM}{c^2}

Assuming the area of a sphere is A = 4 \pi r^2 then the condition for a given mass to be a black hole in terms of its surface area can be expressed as:

\frac{A}{M^2} &lt;= \frac{16 \pi G^2}{c^4}

or:

M &gt;= \frac{1}{4} \, \frac{c^2}{G} \, \sqrt{\frac{A}{\pi}}I think you are thinking of the Area/Mass < 4 relationship for the entropy of a black hole.

 

[Passionflower]

I am not sure how you get that result and I do not think you have clearly expressed yourself. What is the the "area of a mass" if it is not the "area occupied by the mass"?

Notice that the r coordinate in a Schwarzschild solution is the 'reduced circumference', e.g. the area and circumference are primary and physical while the 'radius' is not a physical radius at all, in fact it is related to the Gaussian curvature.

The area equivalent to mass makes as much sense as the 'distance' equivalent to mass as expressed in geometric units in terms of r, but that is what we have to enter as a parameter in the Schwarzschild solution.

If you work out the function that gives the reduced circumference r based on a given area and you determine the area equivalent to a given mass in geometric units you get the above mentioned equation.

For instance the function to get the reduced circumference is:

<br /> 1/2\,{\frac {\sqrt {A}}{\sqrt {\pi }}}<br />

As you can see the r value is derived but not a physical distance while the area and circumference is primary in a Schwarzschild solution.

 

[yoron]

Well, if you find the pdf I linked fit your standards you can use that one as an start. Most of what we discuss is inside what I call the 'mind space' no matter if we can validate the theories mathematically, and experimentally. Time dilation may exist :) But your clock will always tick 'the same' to you, will it not? You might sit in a spaceship watching the universe 'die' but how do you prove that it was time dilation? Your 'history'? The theories you know about :)

I find the universe just as weird as my suggestions above shows :)
And they're only suggestions, not any proven facts.
We just have to wait a while.
==

And I'm not sure about 'information'. To me it has to do with 'definitions' again :) You can either assume that there exist 'discrete events' (e.g. bits on a hard disk, or defined 'shapes' against some background.) and then define 'information' from that idea. It makes perfect sense to do so, but if what you see as 'discrete events', to me, are 'relations' taking shape, then what you call 'information' is like 'condensed states' shaped 'somewhere else'. And that is also what the Planck scale is, the place where physics 'breaks down'. Looked at my way you would have a 'SpaceTime' created out of relations, with the relations becoming our 'discrete events' and macroscopic property's or 'emergences' as they use in chaos math.

And, as I think Einstein's universe shows us, in its 'room-time plasticity' everything do become 'relations' with the only 'true reality' becoming your own unchanging 'frame of reference', no matter where you are, at the EV or on Earth :) Your clock ticks 'the same as always', and nothing really happens. Except when using your 'history', for example comparing what you see outside that 'space ship', coming to you as 'information', with what you 'remember' of how it should look before that velocity was introduced.

But don't get too excited now :)
It's just me looking at it this way, there are several other ways to define a SpaceTime.
==

And thinking of it, No, my view of a 'information space' I've arrived to recently, before I felt like I guess you do, that it was too 'abstract' as a concept for 'reality'. It's just that when testing what I think I have learned against what I deem 'reality' they clashes terribly, just as my two grey do :

But now as I am there I find it quite okay, surprisingly enough. It just that I've invented this 'mind-space' helping me differ somewhat between them, otherwise I would be in a constant meltdown :) It's just like what we perceive, against what we find to be true when comparing conceptually.

ah well :)
==

Try Lee Smolin's 'Three roads to quantum gravity'. I became quite happy reading it as we seemed to think in similar ways. With exception taken to 'discrete events' where he seems to find them necessary, well, maybe not? It's a nice presentation of strings, the holographic idea, etc. Made me want to write :)