The Holographic Principle From Einsteins Equations

[Dubbelosix]

There are fluctuations that arise in spacetime spontaneously - these fluctuations are called virtual particles that exist for generally such a short time frame that we cannot actually detect them. They often appear in reality, much shorter than the distance of spacetime that we can probe. 

 

We do believe in their existences however; par a few. Those who deny the existence of virtual particle activity and zero point fields, just suggests to me they don't truly understand the quantum theory at hand. Quantum theory actually requires the existence of these short-lived fluctuations of spacetime!

 

The zero point field can be thought about in a number of ways, but one of the best, experimental ways to understand zero point fields is in our inability to completely freeze a system to zero Kelvin. It doesn't matter how much energy you put into the system, the system will refuse to freeze. The reason lies in the fact that even a ground state, like Zero Kelvin, is not actually the absence of residual motion, due to the motion of virtual particles in the vacuum. 

 

General Relativity suffers that it has no sensible model for vacuum fluctuations: A number of problems persist, but one of the main ones is that the source terms in the field equations cannot normally be taken for their respective operators, because of a string of problems, notwithstanding the divergence problems.

 

Nevertheless, it is only acceptable within quantum theory that vacuum fluctuations exist and somehow it needs to translate into the stress energy tensor, which was one of those pesky sources which is riddled with problems when quantized. Theoretically, the stress energy has to be related to the stress of a single particle confined within a region of uncertainty.

 

There is no heuristic derivation normally to allow this, but if somehow the contracted stress energy could be interpreted in an equation I derived straight from the Einstein field equations, using a specific string uncertainty relationship:

 

[math]\frac{1}{\delta L^2_P} \geq \frac{8 \pi G \delta T}{c^4}[/math] 

 

as

 

[math]T = \frac{\hbar c}{L^4_P}[/math]

 

Which is identical to an energy density on the RHS in terms of electromagnetic zero point fluctuations. A theoretical look into just a gravitational fluctuation would take the form from a Weyl invariance relationship (Motz 1963):

 

[math]\frac{\hbar c}{L^4_P} = \frac{Gm^2}{L^4_P}[/math]

 

Plugging this in we have, in our picture a gravitational charge confined within [math]L^4_P[/math]

 

[math]\frac{1}{\delta L^2_P} \geq k \frac{Gm^2}{\delta L^4_P}[/math] 

 

where 

 

[math]k = \frac{8 \pi G}{c^4}[/math]

 

 

Could this be taken that a change in the certainty in a length gives rise to the gravitational fluctations Gm²/L^4? Quantum theory predicts that particles are spontaneously being created at the Planck level, perhaps such uncertainty in the length does give rise to fluctations. If so, we may find something like the following

 

[math]\frac{N}{\delta L^2_P} \geq k \frac{Gm^2N}{\delta L^4_P} = k \frac{Gm^2}{\delta L_P} \mathbf{n}[/math]

 

Where [math]N[/math] is the number of particles in the enclosed inverse area of uncertainty [math]\frac{1}{\delta L^2_P}[/math] and [math]n = \frac{N}{V}[/math] is the number density of the particles presumably being created at the fundamental level. One thing may be extrapolated, in that being, if we can probe space to a degree of certainty in δL², then the larger the fluctuation of zero point energy in δE = Gm²/δL that will be produced. Density number, ok... but how do you actually describe the creation or annihilation of the particles? That part is naturally covered by the respective operator for the gravitational potential, which is encoded in the expression [math]\frac{Gm^2}{\delta L}[/math] as

 

[math]\phi = \frac{Gm}{L_P}[/math]

 

Knowing this, we then take the operator value for [math]\phi[/math] which gives the creation and annihilation operators - then we can rewrite the whole thing with the quantum operator

 

[math]\frac{N}{\delta L^2_P} \geq k <\frac{Gm}{\delta L}> \mathbf{n}m[/math]

 

This type of quantization created the scalar particles and  is the same as doing

 

[math] <\phi> = [a_k f_k + a^{\dagger}_{k} f_k][/math]

 

Where a and a^† are the creation and annihilation operators, which are calculated in the usual the way ~

 

[math]a|n> = \sqrt{n_k + 1}| n_k + 1>[/math]

 

[math]a^{\dagger}|n> = (\sqrt{n_k + 1}| n_k - 1>[/math]

 

The expectation of N, in

 

[math]\delta S \geq k <\frac{Gm}{\delta L}> (\frac{<N>}{Z} \frac{m}{L^3_P} A e^{-E/kT})[/math]

 

*(where does the exponential come from? See notes below. )

 

Will yield the number of particles in a given slice of time

 

[math]<N> = \sum_k aa^{\dagger}[/math]

 

[math]\delta S \geq  k <\frac{Gm}{\delta L_P}> \sum_k \frac{aa^{\dagger}}{Z} \rho A e^{-E/kT}[/math] 

 

It is this specific set of coefficients

 

[math]<\frac{Gm}{\delta L_P}> \sum_k aa^{\dagger}[/math]

 

that produces the fluctuations on the horizon, characterized by the area A, and even counts how many are produced. The creation and annihilation follow a commutation rule

 

[math][a, a^{\dagger}] = aa^{\dagger} - a^{\dagger}a = 1[/math]

 

[math]aa^{\dagger} = a^{\dagger}a + 1[/math]

 

Which is pretty neat. So this factor

 

[math]<\frac{Gm}{\delta L_P}> \sum_k aa^{\dagger}[/math]

 

can be understood in the following way based on almost only creation and annihilation operators:

 

[math][a_kf_k + a^{\dagger}_{k} f_k] \sum_k a^{\dagger}a + 1[/math]

 

In full expression of equation, we have something like the following

 

[math]\delta S \geq  k <\frac{Gm}{\delta L_P}> \sum_k \frac{aa^{\dagger}}{Z} \frac{m}{L^3_P} A e^{-E/kT}[/math] 

 

Black hole entropy is proportional to the area of the horizon, as

 

[math]S \propto k_B \frac{Gm^2}{\hbar c} \propto k_B \frac{A}{L^2_P}[/math]

 

The holographic bound implies, using C Sivaram, is

 

[math]S \geq \frac{k(B)c^3A}{G\hbar}[/math]

 

The scaling of S is performed with area, instead of volume, which itself was the manifestation of the holographic principle. 

 

Conventiently, the fundamental length equation (1) is scaled by some Planck area, so we can get the dimensionless form of S from the following equation for entropy:

 

[math]\delta S = \frac{NA}{\delta L^2} \geq k \frac{Gm^2 NA}{\delta L^4} = k \frac{Gm}{\delta L} \mathbf{n}mA[/math]

 

 

 

 

 

notes:

 

 

The number density is related to the Boltzmann factor, in which the temperature T dictates the behaviour of the quantum system in interesting ways I won't cover here. 

 

[math]\mathbf{n} = \frac{N}{Z} \frac{1}{L^3_P} e^{\frac{-E}{kT}}[/math]

 

and [math]Z[/math] is the partition function.

Edited April 6, 2017 by Dubbelosix