Gravitational Fluctuations from Non-Commutative Algebra in Einsteins Equations
The Einstein field equations are written in the following form:
Rμv - 1/2 Rgμv = 8πG[Tμv]/c^4
To get the curvature scalar, with the desired inverse length^2, we have to contract the above equation on both sides and I calculate this as
R = -(8πG/c^4) T
alternatively
R = -(8πG/c^2) T(matter)
In which T is now the matter scalar [[Density]]. In this form, it does have less information in it than the Ricci curvature tensor because that particular tensor depends on Bianchi identities.
With this minimal information though, we know that R has dimensions of 1/l^2 so we can apply the string uncertainty to this now - And in fact there is a bit more to all this, if you plug the second equation into the first, you get back the standard Poisson analogy of Einsteins equations taking the form:
Rμv = 8πG/c^4[Tμv - 1/2Tgμv]
If we correct the Einstein curvature equation for the Planck length in a cosmological background, we have
1/ℓ(P)² = 8πG[Tμv]/c^4
(which is equivalent to a quantum length corrected curvature tensor G_μv). An uncertainty principle can be directly plugged in, by taking a relationship from string theory by Yoneya (1987, 1989, 1997), given as
Δℓ cΔt ~ ℓ(P)²
This gives us an uncertainty in G_μv for the curvature tensor corrected to the Planck scale as,
1/(Δℓ cΔt) = 8πG[Tμv]/c^4
Taking the reciprocal of the equation, we find the Planck scale uncertainty implies the relationship
Δℓ cΔt ≤ c^4/8πG[Tμv] = Għ/c³
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
1/δL² ≥ 8πGδT/c^4
as
δT = ħc/δL^4
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):
ħc/δL^4 ~ Gm²/δL^4
Plugging this in we have
1/δL² ≥ k Gm²/δL^4
k = 8πG/c^4
Could this be taken that a change in the 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
N/δL² ≥ k Gm²N/δL^4 = k Gm²/δL n
Where N is the number of particles in the enclosed area of uncertainty 1/δL² and n = N/V 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 Gm²/δL as
<φ> =
Knowing this, we then take the operator value for φ which gives the creation and annihilation operators
Black hole entropy is proportional to the area of the horizon, as
S ~ k(B) Gm²/ħc ~ k(B)c³A/Għ
The holographic bound implies, using C Sivaram, is
S ≥ k(B)c³A/4Għ
In which Sivaram also stated, it can be given in the following form:
S ≥ k(B) A/L²
Where the length in his example, is taken to the Planck length and A is the area and k(B) is Boltzmann's constant.
The scaling of S is performed with area, instead of volume, which itself was the manifestation of the holographic principle.
Conveniently, the fundamental length equation is scaled by some Planck area, so we can get the dimensionless form of S from the following equation for entropy:
δS = NA/δL² ≥ k Gm²NA/δL^4 = k Gm/δL nmA
This isn't just about the entropy in terms of G, ħ or c. It has something to do, not only with the potential Gm/δL but in this is has probably the only important dynamical feature: the number of particles in the system N.
If we were considering the horizon of a black hole, then the following quantization of the scalar field, produces fluctuations of the order of the temperature, which can be as I have shown before, be recorded in terms of the Boltzmann factor which is given altogether now as
δS ≥ k(B) Gm/δL nmA
→ δS ≥ k N/Z (m/V A exp{-E/kT})
(we will drop the Boltzmann constant so that no confusing arises using Einstein's k)
where 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.
n = N/Z 1/V exp{-E/kT}
The only interesting physics I can recover for this in terms of the holography of the black hole is from deriving the usual: if temperature T is very high, then this will correspond to a relatively small black hole with large fluctuations in
<φ> = = [af + a^†f]
Where a and a^† are the creation and annihilation operators, which are calculated in the usual the way ~
a|n> = (√n_k + 1|) n_k + 1>
a^†|n> = (√n_k + 1|) n_k - 1>
The expectation of N, in
δS ≥ k N/Z (m/V A exp{-E/kT})
Will yield the number of particles in a given slice of time
S ≥ k /Z (m/V A exp{-E/kT}) = k(B) Σ_k aa^†/Z (m/V A exp{-E/kT}) (12)
It is this specific set of coefficients
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
[a,a^†] = aa^† - a^†a = 1
so that
aa^† = a^†a + 1
Which is pretty neat. So this factor
can be understood in the following way based on almost only creation and annihilation operators:
[af + a^†f] Σ_k a^†a + 1
