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Holographic Principle

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The Holographic Principle is a principle that states that all the information within a reigon is completely encoded onto it's boundary.

Motivating examplesEdit this section

Stokes-Navier TheoremEdit this section

The Stokes-Navier Theorem states that the work done along a path is exactly equivalent to the flux through the surface it bounds.

\oint\vec f\cdot\mbox{d}\vec r=\iint\left(\nabla\times\vec f\right)\cdot\hat n\mbox{d}S

In other words, the flux through the surface is equal to the work along the boundary. This means that something on the surface has an alternate description on its boundary.

Stokes' TheoremEdit this section

The Stokes-Navier Theorem is merely a special case of the more general, all-intuitive theorem, known as [[Stokes' Theorem]. It can be stated as:

   
    \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega.

Again, on the left - hand - side, the integral is taken along \partial\Omega, but, whereas, on the right - hand - side, it is taken along \Omega, which means that it relates a property on a surface or manifold to an alternate description of that on it's boundary.

Gauss's TheoremEdit this section

The Gauss's Theorem statesj thaty:

\int\!\!\!\!\int\!\!\!\!\int_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)dV=\oiint\scriptstyle S(\mathbf{F}\cdot\mathbf{n})\,dS .

This again means that some information about the region has an alternate description on the boundary.

A Schwarzschild Black HoleEdit this section

Consider an in - falling observer towards a Black Hole. For simplicity, let's say that it is a Schwarzschild Black Hole. Then, since the time dilation is given by:

       \frac{\mbox{d}t}{\mbox{d}\tau}= \frac{-c_0^2\mbox{d}t^2}{\mbox{d}s^2}

For a Schwarzschild Metric, this becomes 0 at the event horizon, so that this in - falling observer would appear to an external observer as stopping at the event horizon.

Therefore, all activities that happen to this in - falling observer would appear to the external observer as appearing at the event horizon; where - as the in - falling observer itself would perceive it as happening within the event horizon of the black hole.

Again, something in a reigon, has an alternate description on the boundary of this reigon.

Black Hole EntropyEdit this section

In the framework of Semi-Classical Gravity, Hawking Radiation is the radiation emitted from black holes due to Quantum Mechanicsal processes. The entropy associated with Hawking Radiation is given by (in a convinient system of natural units):

S=\frac{A}4

This thus means that the entropy of the (three-dimensional) black hole can be expressed in terms of an alternative description, on it's boundary; the area.

Statement of the theoremEdit this section

From these observations, we can thus state that:

The information in a region is can be completely be described by the information on its boundary.

This statement is known as the Holographic Principle.

Applications in PhysicsEdit this section

The Holographic Principle has many implications in Physics, such as AdS/CFT, and it's extensions, such as AdS/QCD, AdS/CMT, and so on.

A word of caution about the motivating examplesEdit this section

Most of the motivating examples mentioned, except for the one on black holes is not the same sort of the Holographic Principle as that which is used in Physics, but merely just "Holographic Statements", as they relate the information about a region to information on it's boundary.

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