## Thursday, September 27, 2007

I forgot that there is a missing dimensionless variable factor x -corrected below where x is the ratio of two lengths, the position of the test particle to the intrinsic scale r of the geometrodynamic field that is related to the source distortion M. What is r? Here I take it as the radius of curvature at the surface r* of the source as the closest analog to the cosmological parameter in the de Sitter vacuum solution of constant space curvature. x < 1 is inside the source x > 1 is outside the source that may actually be exotic vacuum zero point energy.

The source mass density is M/r*^3 the radius of curvature at r* is r ~ (r*3/rs)^1/2 where rs = 2GM/c^2.

Therefore N below is computed using this "r".

Dvali's et-al work on extra space dimensions arrives at Abdus Salam's 1973 f-gravity that in the static Newtonian limit is simply for the gravity potential energy per unit test particle

V(r) = -(GM/r)(1 + a*e^-r/b*)

In my world hologram tetrad model, the Einstein-Cartan gravity tetrad 1-form field is conjectured to be

e^a = I^a + N^-1/3A^a

ds^2(1916GR) = e^ae^a = guvdx^udx^v

I^aIa is ds^2(1905SR) in which only global inertial frame (GIF) transformations are allowed.

As soon as one allows global non-inertial frames, A^a =/= 0 in such a GNIF. A^a = 0 in a GIF.

For a GNIF R^a^b = 0 (vanishing curvature 2-form) and T^a = 0 (vanishing torsion 2-form)

In 1916 GR localize rigid T4 to elastic T4(x) and now in general R^a^b =/= 0 but T^a = 0 still.

In the Jack Ng et-al world hologram conjecture for simple SSS vacuum model

4pir^2 = NLp^2/4

r = Schwarzschild radial coordinate for static LNIF observers when r > 2GM/c^2, M = source mass energy

Lp^2 = hG/c^3

N = number of Bekenstein c-BITS

r = N^1/2Lp/16pi

The size of the quantum gravity foam bubble is

&r ~ (Lp^2r)^1/3 ~ N^1/6Lp/16pi

Therefore r^3/&r^3 ~ N^3/2/N^1/2 ~ N

Therefore, there is a 1-1 hologram correspondance between each area hologram quantum and it's projected "volume without volume" hologram image quantum.

Conjecture

a* = N^1/3

b* = &r

Therefore

V(N,x) ~ -(GM/Lpx)(1/N^1/2)(1 + N^1/3e^-N^1/2/N^1/6) = -(GM/Lpx)(1/N^1/2)(1 + N^1/3e^-xN^1/3)