Holographic World

[khan]

Do we live in a holographic world with random noise at the very foundations?

http://en.wikipedia.org/wiki/Holographic_principle

In a larger and more speculative sense, the theory suggests that the entire universe can be seen as a two-dimensional information structure "painted" on the cosmological horizon, such that the three dimensions we observe are only an effective description at macroscopic scales and at low energies. Cosmological holography has not been made mathematically precise, partly because the cosmological horizon has a finite area and grows with time.[4][5]

The holographic principle was inspired by black hole thermodynamics, which implies that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected.

...

$$\hbar = \frac{h}{2 \pi}$$



$$l_p = \sqrt{\frac{\hbar G}{c^3}}$$



$$\frac{l_p}{2} = \frac{\sqrt{\frac{\hbar G}{c^3}}}{2}$$



$$(\frac{l_p}{2})^2 = \frac{\hbar G}{4c^3}$$



$$\pi (\frac{l_p}{2})^2 = \frac{h G}{6c^3}$$



$$e$$ = base of the natural logarithm




$$\frac{h G}{6c^3} \approx e \times 10^{-70}$$


Circles with pi*[(Planck length)/2]^2 would be the most fundamental building blocks - the informational "bits" of physical existence

 

[khan]

There is an error in my calculation

:shrug:

What a mistake

 

[arfa brane]

How would you relate:

$$ S_B\; =\; klog(V) $$​

to:

$$ S_{BH} \;=\;\frac {kc^3(A)} {4G \hbar} $$​

 

[khan]

How would you relate:

$$ S_B\; =\; klog(V) $$​

to:

$$ S_{BH} \;=\;\frac {kc^3(A)} {4G \hbar} $$​

$$k log W$$

$$S_{BH} = \frac{A}{4(l_P)^2}$$


Black hole entropy?

:shrug:

 

[arfa brane]

The formulas I posted are descriptions of 'entropy'. The first should be recognisable as Boltzmann entropy, but expressed in terms of a volume. The second is the Bekenstein-Hawking entropy of a black hole expressed in terms of area.

But how are they related? What's a good "expression" that relates area to volume, and volume of what?

 

[Emil]

Holometer experiment to test if the universe is a hologram :runaway:

 

[khan]

The formulas I posted are descriptions of 'entropy'. The first should be recognisable as Boltzmann entropy, but expressed in terms of a volume. The second is the Bekenstein-Hawking entropy of a black hole expressed in terms of area.

But how are they related? What's a good "expression" that relates area to volume, and volume of what?

$$A(x^2) = V(x^3)$$

I am guessing that an N dimensional Ricci flow can be equated to an N-1 dimensiopnal holographic encoding :shrug:

http://arxiv.org/ftp/arxiv/papers/0710/0710.2556.pdf

http://homepage.mac.com/photomorphose/documents/qpdf.pdf

http://www.cosmolearning.com/courses/edward-witten-lecture-2-1-dimensional-gravity-revisited/

http://en.wikipedia.org/wiki/Holographic_principle#Energy.2C_matter.2C_and_information_equivalence

Energy, matter, and information equivalence

Shannon's efforts to find a way to quantify the information contained in, for example, an e-mail message, led him unexpectedly to a formula with the same form as Boltzmann's. Bekenstein summarizes that "Thermodynamic entropy and Shannon entropy are conceptually equivalent: the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information one would need to implement any particular arrangement..." of matter and energy. The only salient difference between the thermodynamic entropy of physics and the Shannon's entropy of information is in the units of measure; the former is expressed in units of energy divided by temperature, the latter in essentially dimensionless "bits" of information, and so the difference is merely a matter of convention.[citation needed]

The holographic principle states that the entropy of ordinary mass (not just black holes) is also proportional to surface area and not volume; that volume itself is illusory and the universe is really a hologram which is isomorphic to the information "inscribed" on the surface of its boundary.[9]

Holometer experiment to test if the universe is a hologram :runaway:

The holographic scenario is strange and also interesting, yes.

http://www.physorg.com/news/2010-10-holometer-universe-hologram.html

The idea is that space on the ultra-small Planck scale is two-dimensional, and the third dimension is inextricably linked with time.

 

[khan]

My new equation following from the OP :m:

$$(\frac{\pi e}{\pi+1}) \times 10^{-70} \approx \frac{hG}{8c^3}$$

A diagram of the possible two dimensional structure of the Planck length scales of space-tme of a holographic world...

 

[arfa brane]

If you have the formula

$$ S_{BH} \;=\;\frac {kc^3(A)} {4G \hbar} $$​

and set all the constants to 1, you have $$ S_{BH} \;=\;\frac {A} {4} $$

So the entropy should scale with a fourfold symmetry--each unit of entropy should be four 'Planck units' of area. You've drawn something different.

 

[khan]

If you have the formula

$$ S_{BH} \;=\;\frac {kc^3(A)} {4G \hbar} $$​

and set all the constants to 1, you have $$ S_{BH} \;=\;\frac {A} {4} $$

So the entropy should scale with a fourfold symmetry--each unit of entropy should be four 'Planck units' of area. You've drawn something different.

It is difficult to draw overlapping circles where the pattern is 4 fold symmetry.

Lorentz invariance may not hold for the Planck scale so the pattern might not be circles... :shrug:

 

[khan]

Gravity could be the thermodynamic limit of statistical arrangements of the Planck-scale sized atoms of space-time.

http://www.imsc.res.in/~iagrg/IagrgSite/Activities/IagrgMeetings/25th_Iagrg/VRtalk.pdf

http://www.science20.com/hammock_physicist/it_bit_entropic_gravity_pedestrians-66244

A rough idea of an equation


[Maxwell's equations] + [Thermodynamics] $$\Rightarrow$$ [Unification of Gravity and Electromagnetism]


My depth perception tells me that I am experiencing 3 dimensions of space - including perceptions of the manifold changes in that 3-D world. Perceived changes may be defined as a sequence of events, AKA time.

If space is 2 dimensional holographic encodings and "time" is a form of depth perception, how would the equations change? :idea:

 

[AlphaNumeric]

If you have the formula

$$ S_{BH} \;=\;\frac {kc^3(A)} {4G \hbar} $$​

and set all the constants to 1, you have $$ S_{BH} \;=\;\frac {A} {4} $$

So the entropy should scale with a fourfold symmetry--each unit of entropy should be four 'Planck units' of area. You've drawn something different.

The various factors are not to do with a 4 fold symmetry, they are due to black hole surface gravity, areas and the volume of spheres.

 

[arfa brane]

What I was getting at is the way Planck units are drawn as equilateral triangles, for instance in the diagram below. Four such triangles can make another equilateral triangle, which is the 'symmetry'. I don't see how this could work with circles.

 

[khan]

I am assuming randomness at the quantum Planck length scales and the fundamental entropy bits are entangled with the others. A wave interference pattern? Circles overlapping is my guess and approximation of the random entangled nature of the geometric 2-D shapes. Circles are also Lorentz invariant shapes. I am also guessing that Lorentz invariance might be true, or approximately true for Planck length scales, getting truer and truer at larger scales :shrug:

http://www.maa.org/devlin/devlin_3_99.html

With such a variety of behavior, it's not obvious that such sequences follow the nice kind of growth pattern of the Fibonacci sequence.

But they do. Last fall, Viswanath, who recently finished a Ph.D. in computer science at Cornell University in New York, showed that the absolute value of the Nth number in any random Fibonacci sequence generated as described is approximately equal to the Nth power of the number 1.13198824 . . . .

Actually, that's not quite accurate. Because the sequences are generated randomly, there are infinitely many possibilities. Some of them will not have the 1.13198824 property. For example, the sequence that cycles endlessly through 1, 1, 0 does not have the property, nor does the original Fibonacci sequence. But those are special cases. What Viswanath showed is that if you actually start to generate such a sequence, then with probability 1 the sequence you get will have the 1.13198824 property. In other words, you can safely bet your life on the fact that for your sequence, the bigger N is, the closer the absolute value of the Nth number gets to the Nth power of 1.13198824 . . .

http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm

http://www.sciencedaily.com/releases/2010/01/100107143909.htm

Golden Ratio Discovered in Quantum World: Hidden Symmetry Observed for the First Time in Solid State Matter

[...]


For these interactions we found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture." Radu Coldea is convinced that this is no coincidence. "It reflects a beautiful property of the quantum system -- a hidden symmetry. Actually quite a special one called E8 by mathematicians, and this is its first observation in a material," he explains.

:shrug: