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Holography

Black Hole Entropy

Holographic Universe

Holographic Space-time

light waves reflected from the object. The wave amplitudes are readily encoded on an ordinary photographic film. The phases are recorded as interference patterns produced by the reflected light and a reference coherent light (from the same laser). Each point on the hologram received light reflected from every part of the illuminated object and, therefore, contains the complete visual record of the object as a whole. When the hologram obtained from the development of a film exposed in this way is placed in a beam of coherent light, two sets of strong diffracted waves are produced - each an exact replica of the original signal bearing waves that impinged on the plate when the hologram was made. One set of diffracted waves produces a virtual image, which can be seen by looking through the hologram. It appears in a complete three-dimensional form with highly realistic perspective effects. In fact, the reconstructed picture has all the visual properties of the original object. | |

## Figure 01 Holography |
Figure 01depicts the process of making a hologram: |

- A single laser is split into two separate beam (a) and (b).
- Beam (b) passes through a lens and bounces off the object at (c) onto a photographic film (d).
- The reference beam at (a) passes through a lens (e) and merges with the reflected light from (b) to produce interference pattern on the film.

- Theoretical physicists has suggested that entropy or information of matter-energy inside the black hole is encoded on the surface of the event horizon similar to holography, which stores information in 3-dimensional space on a 2-D flat sheet. The derivation is surprisingly simply as shown in the following.
- The derivation begins by throwing a photon with wavelength into the black hole with mass m and radius R. In order to approach the black hole in an un-defined location, the wavelength should be about the size of the event horizon, namely, ~ R.
- The energy carries by such photon is E = h = hc/ = 2c/R.
- The corresponding increase in mass would be m = E/c
^{2}= 2/Rc (via the most celebrated formula E = mc^{2}). - The increase in surface area at the event horizon can be calculated from the definition R = 2Gm/c
^{2}, and yield:

RR = 4 (G/c^{3}) = 4 (L_{Pl})^{2},

where G/c^{3}= (L_{Pl})^{2}= (1.62x10^{-33}cm)^{2}is the Planck area - the smallest area (the most basic unit) conceivable before the full impact of quantum gravity set in. - Thus the number of basic unit on the event horizon surface area A
_{Sh}is:

n = A_{Sh}/RR = A_{Sh}/[4(L_{Pl})^{2}]. - Since the basic unit is either present or absent at any point on the surface, the number of arrangements is N = 2
^{n}. - According to definition, the absolute value of information |
**I**| = log_{2}(N) = n = (R_{Sh})^{2}/(L_{Pl})^{2}= 1.2x10^{66}(R_{Sh})^{2}bits, where we have used the formula A_{Sh}= 4(R_{Sh})^{2}. Note that the entropy S = (k_{B}ln2) |**I**|, where k_{B}is the Boltzmann constant.

If originally a roughly spherical distribution of matter is contained inside an area A', the matter is induced to collapse to form a black hole as shown in diagram (a), Figure 02. Then the black hole's area A must be smaller than A', so its entropy must be less than (c^{3}/G)(A'/4). Because entropy cannot decreases, so the original entropy must be less than that as well. This is the holographic bound as shown in diagram (c), Figure 02.The "universal entropy bound" is an even more stringent limit on the amount of entropy or information that can be contained within a volume of space. This time we throw some macroscopic mass m with radius R = d/2 into the black hole. It is derived by imaging that a capsule of matter is engulfed by a black hole about the same size (see diagram (b), Figure 02). The small increase in the black hole's size places a limit on how much entropy the capsule could have contained. If we put water in the capsule, then m = _{water}(4/3)R^{3}.
Thus the increment in event horizon surface area is A = 8RR = 8(2G/c^{2})_{water}(4/3)R^{4}.
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## Figure 02 Black Hole En-tropy [view large image] |
Following the prescription in the previous derivation, the increment in information is: | I| = (c^{3}/G)(A/4) 2x10^{39} R^{4} bits. |

1.6x10^{-33} cm (on the surface of the event horizon). It turns out to be around 3x10^{-13} cm as calculated from the size of the universe at about 10^{28} cm and the number of information n at the surface of the event horizon derived above (in the section for "Black Hole Entropy"). Or, to put it another way, a holographic universe is blurry, but the grainy structure is much easier to detect. Quantum effects will cause the space-time quanta to convulse wildly resulting in the noise picked up by GEO600. If this interpretation is proven to be correct, it will be ranked at the same level of achievement as the discovery of the CMBR, which also appeared as noise in the microwave detector.
| |

## Figure 03a GEO600 Detector [view large image] |

According to an article in the February 2012 issue of Scientific American, an experiment is being built in an abandoned shed at the now defunct Fermilab to probe the jitter of quantum space, which may be much larger than the Planck size as explained above. The setup is similar to the GE0600 experiment but with shorter arms at 40 meters (Figure 03b). In fact both of them are similar to the Michelson-Morley experiment in 1887 except that they are looking for something else this time round. Instead of going after the speed of light in the two arms, measurement of the jitter of space depends on the tiny movement of the beam splitter as the supporting space thrashing whichever way due to quantum fluctuation. It is similar to a boat pitching on the sea. The result could be a pattern of noise as shown in Figure 03b. A second interferometer is under the first one to make sure the measurement is repeatable. | |

## Figure 03b [view large image] Probing at Quantum Space |

Another example of the holographic principle at work involves the Euclidean Anti-de Sitter (AdS) space, which expands and possesses a boundary at infinity - much like our universe. The metric of such space has the form: ds ^{2} = dr^{2} + sinh^{2}(r) d_{4}^{2}where d _{4}^{2} is the (3+1)-D hypersphere at the boundary of the (4+1)-D Euclidean AdS space, where the (n+1)-D denotes n spatial and 1 temporal dimensions.
| |

## Figure 04 Holographic Space-time [view large image] |

A cosmological model based on the collapse of a four dimensional star was unveiled in 2014 (Figure 05). It shows that the three dimensional Holographic projection of this event corresponds to the Big Bang in our universe. The expanding shell of the 4-D supernova maps to our 3-D expansion. Its novel property includes an event horizon, which shields the singularity bothering so many cosmologists. The smoothness and flatness of our universe is just a reflection of the 4-D bulk, which had been in existence for a much longer time. The study suggests that the CMBR may reveal the spin and the presence of 4-D matter around the 4-D black hole. | |

## Figure 05 4-D Supernova |

cos(A) = g

where g

On a flat surface the metric ds

For the case of 2 unit vectors at the x and y axis respectively on a flat surface, u = (1,0), v = (0,1); cos(A) = 0 or A = 90^{o}. Now consider ds^{2} = g_{11}dx^{2} + g_{22}dy^{2}, if we make a conformal transformation x = ax', y = ay', where "a" is a constant. It follows that the new metric tensors become g'_{11} = g'_{22} = a^{2}. This kind of transformation will distort the distance between points, but the angle "A" remains unchanged (Figure 06). This is similar to the comoving frame in cosmic expansion. Figure 07 shows some more fanciful conformal mappings.
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## Figure 06 Conformal Transformation [view large image] |
## Figure 07 Conformal Mappings |