How is entropy defined by boundary area in Holographic Principle?

In summary, the Holographic Principle states that the description of a volume of space can be encoded on its boundary, such as a gravitational horizon. This means that for a black hole, all the information about objects that fall in is contained in the surface fluctuations of the event horizon. Leonard Susskind's book "The Black Hole War" discusses the concept in relation to quantum gravity, where the maximum amount of information that can be stored in a volume is equal to that which can be stored on its boundary. The mathematics behind this principle is still being explored.
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Reference: http://en.wikipedia.org/wiki/Holographic_Principle

The principle states that the description of a volume of space should be thought of as encoded on a boundary to the region, preferably a light-like boundary like a gravitational horizon. For a black hole, the principle states that the description of all the objects which will ever fall in is entirely contained in surface fluctuations of the event horizon.

Leonard Susskind, in his book "The Black Hole War", reflects on quantum gravity whereby the total amount of information (bits) that can be stored within a spactial volume is no more than that which can reside on its spatial boundary. This would be calculated in terms of plank length.

Surface area of box, A = 2*(L.l*L.w + L.w*L.h + L.l*L.h))
Internal volume of box, V = L.l*L.w*L.h

In Euclidean space, V>A except for L.l, L.w, L.h < 6 and other trivial proportions.

Can anyone illustrate (or direct me to) the mathematics that validates the Holographic Principle?
 
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Still curious ... Bump.
 
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The Holographic Principle is a concept in theoretical physics that suggests that the information and properties of a three-dimensional volume of space can be described by a two-dimensional surface, similar to how a hologram contains all the information of a three-dimensional object on a two-dimensional surface. In this context, entropy is defined as the measure of the disorder or randomness in a system.

According to the Holographic Principle, the entropy of a system is directly related to the surface area of its boundary. This means that as the surface area of the boundary increases, the entropy of the system also increases. This concept is supported by the fact that the entropy of a black hole is proportional to its surface area, as described by the famous Hawking radiation.

The mathematics behind the Holographic Principle is based on the principles of quantum mechanics and general relativity. In particular, it is derived from the AdS/CFT (Anti-de Sitter/Conformal Field Theory) correspondence, which is a duality between a theory of gravity in a higher-dimensional space and a lower-dimensional quantum field theory. This duality allows for the calculation of the entropy of a black hole using only the properties of its boundary, without needing to know the details of what is inside the black hole.

The holographic principle has been extensively studied and has been shown to hold true in various contexts, including black holes, cosmology, and condensed matter systems. While there is still ongoing research and debate on the exact nature and implications of the Holographic Principle, it has provided valuable insights into the fundamental nature of space, time, and information in the universe.
 

1. What is the Holographic Principle?

The Holographic Principle is a theory that suggests that all the information contained within a three-dimensional space can be represented on a two-dimensional surface. It was proposed by physicist Leonard Susskind in the late 1990s and is based on the idea that the universe is a hologram.

2. How is entropy defined in the context of the Holographic Principle?

In the Holographic Principle, entropy is defined as the amount of disorder or randomness within a system. It is believed that the amount of entropy in a given three-dimensional space is equivalent to the amount of entropy on its two-dimensional boundary. This means that the boundary area serves as a measure of the system's entropy.

3. Why is the boundary area used to define entropy in the Holographic Principle?

The boundary area is used to define entropy in the Holographic Principle because it is believed to be a more fundamental quantity than the volume of the space. This is because the information contained within a space can be fully described by its boundary area, rather than its volume. Therefore, using the boundary area to define entropy allows for a simpler and more accurate understanding of the system.

4. How does the Holographic Principle relate to other theories, such as black hole thermodynamics?

The Holographic Principle is closely related to other theories, such as black hole thermodynamics. In fact, the Holographic Principle was initially inspired by black hole thermodynamics. Both theories suggest that the information contained within a space can be represented on its boundary, and that the entropy of the system is related to the boundary area.

5. Is there evidence to support the Holographic Principle?

While there is no direct evidence to support the Holographic Principle, there are several theoretical arguments and calculations that lend support to the idea. For example, the Holographic Principle has been used to explain the entropy of black holes and the behavior of certain quantum systems. Additionally, some theories of quantum gravity, such as string theory, are consistent with the Holographic Principle.

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