Holographic Principle

String Theory

All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History	S-Matrix Theory Regge Trajectory
Bosonic String Theory	Worldsheet String Bosonic String Theory String Perturbation Theory Tachyon Condensation
Supersymmetric Revolution	Supersymmetry RNS Formalism GS Formalism BPS
Superstring Revolutions
First Superstring Revolution	GSO Projection Type II String Theory Type IIB String Theory Type IIA String Theory Type I String Theory Type H String Theory Type HO String Theory Type HE String Theory
Second Superstring Revolution	T-Duality D-Brane S-Duality Horava-Witten String Theory M-Theory Holographic Principle N=4 Super-Yang-Mills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory F-Theory String Field Theory Pure Spinor Formalism
After the Revolutions
Phenomenology	String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology

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

Motivating examples[]

Stokes-Navier Theorem[]

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' Theorem[]

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:

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

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 Theorem[]

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 Hole[]

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 Entropy[]

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 theorem[]

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 Physics[]

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 examples[]

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.