175. Holographic principle for dummies
Save This Post as PDFSince today is Sunday, nobody should be allowed to overload your brain with too technical discussion of a new paper in ArXiv (there are no new papers till Monday, anyway!). But does it mean that I will devote part of this Sunday to posting something about financial crisis instead of science? No way! 
Since, as generally accepted, Sunday should be devoted to fun, let me have some physics related fun.
Namely, let me try to explain holographic principle in less than 700 words using less than 5 formulae.
So, what is holographic principle in a few words?
Holographic principle is a fundamental concept that is believed to introduce a compatibility between quantum mechanics and general relativity. Ok, saying this, I did not explain anything at all, so let me try to start again Holographic principle has to do with information. In quantum mechanics, information is measured in qubits. For example, if you know that the spin projection of a given (fermion) particle on the z-axis is equal to 1/2 (or -1/2), you have one bit of information. Generally, a two-level system can carry at most one qubit of information. As seen from this example, information is carried by some kind of physical carriers – electrons, photons, etc. The more you know about quantum states of the carriers, the more qubits of information you have.
How much information can be contained in the volume 
?
Well, you can squeeze rather large number of photons in a given volume. However, any of these photons carry energy (every photon carries energy proportional to its frequency or inversely proportional to its wavelength), and as we know from general relativity, this energy curves spacetime. If the number of photons within a given volume becomes very large (so does the energy stored in the given volume), spacetime may become so strongly curved, that a single photon will be unable to leave the volume , and the information stored within this volume will never be accessible to us.
The maximal amount of energy that can be jammed into a sphere of the size 
coincides with the mass of a black hole with radius :
. (1)
As a result, there is a maximum amount of information that can be crammed inside a sphere of the radius , and this amount is proportional to
, (2)
where 
is the surface area of a sphere with radius and 
is a very tiny area called the Planck area.
This is actually a teapot
Volume scaling and area scaling
What is the most surprising about the formula (2) is its extreme counter-intuitiveness. Naively, one would expect that the maximal amount of information contained in a volume should be proportional to the volume itself! Indeed, that’s what our everyday experience teaches us: take a glass and start filling it with grains of sand. The total number of grains that the full glass will contain is clearly proportional to the volume of the glass.
No so, if gravity is taken into account As we see from (2), the maximum amount of information contained in the volume is proportional to 
.
Holographic principle, finally
This observation allows many people to think that maybe relevant degrees of freedom in physical problems involving gravitation actually live on a surface rather than in a volume – and that is where the term “holographic principle” comes from. Relevant degrees of freedom live on a surface, interact with each other there, and the 3d world we see is in a sense fiction – reflection of this dynamics on a surface.
Update: Richard Epp and Robert McNees from the Perimeter Institute do a great job explaining what is holographic principle in less than 50 slides (and less than 40 minutes) – at least, they do it much better than I just did. Just in case you decided to watch the presentation – do you also have an impression that kids in the auditory are scared to death by the holographic principle?
Update 2: Do you see a teapot on the picture above? If you don’t or if you want to learn how stereograms work, kidly let me PR the Lubos Motl’s blog. Honestly, my explanation of physics behind the stereogram would never be better than the one offered by him.
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The stereogram is a fun picture for this topic. When we were undergrads in Prague, we had so much fun with it. My friends saw it at an exhibition. I couldn’t really see it well, with my fuzzy vision, but after some time, I cracked how it worked and wrote some elementary programs that converted a function z(x,y) into this colorful picture.
http://www.kolej.mff.cuni.cz/~.....node2.html
In fact, I should recreate it now for Mathematica, even though it might be slower than a compiled program in Turbo Pascal. We’ll see. It should be fast today even if interpreted.
Hoho, I can have some fun with your program
Would you want to translate the page to English and put it on your blog? I believe, you would have some serious boost of traffic.
Cheers,
Dmitry.
Hi Dmitry, I still maintain some kind of Turbo Pascal on my desktop PC but I don’t think that it is really an up-to-date software. So it would be better to rewrite the program into something more modern, including Mathematica, despite a possible slowdown.
I can write an English version of the explanation why the “dinograms”, as I called it, work, but your indication that it would boost the traffic sounds unlikely, unless you count roughly 3.1415926 visits added as clicks from your blog!
Eventually, I will sink you in traffic, Motl
Incidentally, in advance, the technical “Wikipedia” name for these particular stereographs are
http://en.wikipedia.org/wiki/Autostereogram
Hi Dmitry, here is an TRF article with a Mathematica notebook
http://motls.blogspot.com/2009.....grams.html
Nice! Updated the post to link to yours. But you did not explain why stereograms are good as example of holographic principle
Cheers,
Dmitry.
Priv?t Dmitrij,
it’s actually only one sentence later in the text – saying that the additional, radial, holographic dimension is encoded in the scale of the stereogram patterns (the periodicity in the x-direction), much like the information about the r-position in AdS is encoded in the scale in quantum gravity, too.
Best wishes
Lubos
What is the radius of a black hole? Is that the radius of its event horizon? Does a black hole have a radius? I thought that a black hole is pointlike object with infinite density? Emphasis on ‘pointlike’.
Interesting food for thought.
Thanks.