Thursday, December 30, 2004

Einstein's Cosmological Constant from The World Hologram

R(t)^2 = [1 - Lp^3|Vacuum Coherence|^2]^-1

Note when Vacuum Coherence = 0 PRE-INFLATION

R(Initial Singularity) = 1

i.e. 1 Bit thinking of universe as cosmic computer with David Finkelstein's chronons 10^-44 sec of erasures - resetting the register so to speak.

As the large-scale Vacuum Coherence limits to

|Vacuum Coherence|^2 -> 1/Lp^3

R(t) -> infinity

Vacuum Coherence can exceed this limit on smaller non-cosmological scales.

On Dec 31, 2004, at 12:43 PM, Jack Sarfatti wrote:

Inflation demands total stuff is at critical density for flat 3D space, i.e. k = 0 in large-scale FRW metric

Omega(random ZPF) + Omega(matter) + Omega(radiation) + Omega(vacuum coherence) = 1

Omega(on-mass-shell matter) ~ 1/R(t)^3, i.e. w = 0 in R(t)^-3(1 + w) for v/c << 1

Omega(on-mass-shell radiation) ~ 1/R(t)^4, i.e. w = +1/3

with Omega(random zpf) + Omega(vacuum coherence) ~ 1/R(t)^0 independent of R(t) since w = -1 for both random "normal fluid" zpf and condensate (vacuum coherence)

/\ is from random zpf causing both dark energy (negative pressure) and dark matter (positive pressure) exotic vacuum phases.

Clumps of w = -1 positive pressure, possibly like the Galactic Halo mimic w = 0 dark matter in their gravity lensing. Positive and negative pressure exotic vacua can each either universally attract or repel depending on their detailed distribution relative to the test particle. However localized negative pressure exotic vacuum clumps anti-gravitate for test particles outside their domain of support. Similarly, localized positive pressure clumps gravitate for test particles outside their support. The effective gravity strength G* of clumps of exotic vacuum regions can be larger than Newton's G.

Guv + /\zpfguv = 0

is the zero torsion field GR Poisson equation for exotic vacua.

In the Newtonian weak-field slow-speed limit, neglecting gravimagnetism, this is

Grad^2(Potential Energy per unit test mass of exotic vacuum field) ~ c^2/\zpf

Susskind's World Hologram Conjecture requires

Einstein's cosmological constant /\ ~ [LpR(t)]^-2

With critical energy density hc/Lp^4R(t)^2 = (mpc^2/Lp^3)R(t)^-2

Therefore Omega(/\) is independent of R(t) consistent with w = -1 Lorentz covariance and EEP of GR. Similarly for Omega(Vacuum Coherence), which is my new term.

OK trash what I wrote late last night on this when I was very tired. I woke up fresh with a start with the correct way to think about the problem - after taking a 600 mg ibuprofen for shoulder ache from over exercise at the health club.

Here is how Lenny Susskind's world hologram really does work and how it explains the dark energy!

1. Trash H. We don't need it.

2. The FRW space expansion factor R(t) is dimensionless. Everything is in units of Lp.

3. The Susskind-Hawking-Bekenstein-t'Hooft hologram entropy of the Universe is simply

S/kB = (1/4)R(t)^2

I have written this before of course.

R(now) = 10^61

This is a nice formula because it also explains the Arrow of Time since the entropy of the universe is 0 at the initial singularity here, i.e. R(initial singularity) = 0 or, if you prefer Finkelstein's chronons it is 1-bit at t = 0.

4. OK roughly model everything as photons.

The photon thermal distribution is

hf/(e^hf/kBT - 1) + hf/2

But the dark energy is, to first approximation virtual photons hf/2 whose mean value is hc/LpR(t).

5. Therefore we need virtual energy hf/2 to "erase" each bit every time the cosmic quantum computer clears its register to step forward another chronon to compute the history of the universe

Therefore, the dark energy density the vacuum fabric of curved spacetime needs to compute itself is

(R(t)^2/4)(hc/LpR(t))(1/Lp^3R(t))^3 = (hc/Lp^4)1/4R(t)^2 = (hc/4Lp^2)/
Einstein's cosmological constant /\ = 1/Lp^2R(t)^2 = 10^-56 cm^-2 NOW

hc/Lp^2 = c^4/G

6. So this says that Einstein's cosmological constant /\ is getting smaller as the Universe expands. On the other hand, w = pressure/(energy density) = -1 for random micro-quantum zero point energy.

The energy density of cosmic stuff scales as R(t)^-3(1 + w)

For example w = 0 for ordinary matter so that Omega(matter) ~ R(t)^-3

w = +1/3 for cosmic black body real photons so that Omega(CMB) ~ R(t)^-4

w = -1 for RANDOM micro-quantum vacuum zero point fluctuations (from covariance) so that

Omega(ZPF) ~ R(t)^-0 = constant.

7. However in my macro-quantum theory of the world hologram

/\ = (1/Lp^2)[1 - Lp^3|Vacuum Coherence|^2]

Equating this with the world hologram formula in the large-scale FRW metric limit

1/Lp^2R(t)^2 = (1/Lp^2)[1 - Lp^3|Vacuum Coherence|^2]

1/R(t)^2 = [1 - Lp^3|Vacuum Coherence|^2]

R(t)^2 = [1 - Lp^3|Vacuum Coherence|^2]^-1

Where from inflation

Omega(random ZPF) + Omega(matter) + Omega(radiation) + Omega(vacuum coherence) = 1

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