Natural Science Forum / Physics / Relativity / January 2006

math typo - final equation on previous message on emergent gravity
should have been

R/\e + Lambda e/\e/\e = 0  vacuum equation

I. Plain Vanilla 1915 Einstein gravity  "The Cake"

The vacuum equation is the 1-form equation

*[R/\e + Lambda e/\e/\e] = 0

The 1-form matter source equation is

*[R/\e + Lambda e/\e/\e] = *J

*J is the stress-energy source current density 1-form, so that J is a
3-form.

The torsion 2-form is

S = De

First consider only 1915 GR i.e.

S = De = 0

e = 1 + B

D = d + W/\

Therefore the torsion-free spin connection 1-form is

W = -*[dB/\(1 - B)]

Note the subspace nonlinearity ~ dB/\B not found in electromagnetism.

Look at Maxwell's E&M gauge force theory

The U(1) internal symmetry gauge invariant exterior derivative on source
fields is

D = d + A/\

Compare this to the Diff(4) (AKA locally gauged T4) "background
independent" space-time covariant exterior derivative

D = d + W/\

So that A is analogous to W not to B. B lives in the antisymmetric
"square root" tetrad subspace of the symmetric geometrodynamic space.
The boson elements of space-time are pairs of the tetrads. That's one
way of thinking of the equivalence principle, AKA EEP.

In abelian U(1) electrodynamics

F = dA

is the 2-form EM field tensor

dF = 0 is the "Bianchi identity"

i.e. Faraday's law & no magnetic monopoles law

from d^2 = 0 locally ignoring global EMERGENT "More is different"
topological defects for now.

* is Hodge dual, i.e. 4D antisymmetric tensor depends on vacuum (or
material) structure.

d*F = *J   this is a 3-form since *F is a 2-form in 4D space-time

(no extra dimensions here)

i.e., Ampere's law with displacement (transverse radiation) current) +
Gauss's law

Again from d^2 = 0 locally NOT generically GLOBALLY in DeRham integration

d^2*F = d*J = 0

i.e. local conservation of electric current source density.

One can generalize this to non-Abelian SU(2) & SU(3) Yang-Mills gauge
force theories using

D = d + A/\

where

F = DA = dA + A/\A

A/\A = 0 in U(1) EM

A/\A =/= 0 in SU(2) weak force and in SU(3) strong force because of
non-trivial Lie algebra

[A^a,A^b] = fc^a^bA^c

Note that

A/\A means A^a^c/\Ac^b = (A/\A)^a^b

DF = 0 Bianchi identities

D*F = *J (source equations)

D^2 = 0 locally

D*J = 0

local conservation of Yang-Mills source current densities for weak and
strong internal gauge forces of the standard model without gravity.

What about gravity?

Obviously, the geodesic deviation tidal curvature 2-form is

R = DW = dW + W/\W

W = -*[dB/\(1 - B)]

Note Einstein's local invariant is the symmetric bilinear expression
(indices suppressed)

ds^2(curved) = guvdx^udx^v = (1 + B)(Minkowski)(1 + B)

= 1(Minkowski)1 + B(Minkowski)1 + 1(Minkowski)B + B(Minkowski)B

Note the nonlinear PLASTIC FRACTURE term B(Minkowski)B that gives "geon"
spontaneous self-organization to the fabric of space-time even in the
absence of "sources".

B = 0 globally means NO REAL GRAVITY FIELDS in sense of Landau &
Lifshitz "Classical Theory of Fields".

Rovelli shows that the Einstein-Hilbert action gives the field vacuum
equation

R/\e + (Lambda)e/\e/\e = 0

This is a 3-form equation

Note that

DR = 0 and De = 0

imply that in VACUUM

D(Lambda) = d(Lambda) ---> 4D gradient of 0-form Lambda = 0

i.e. Lambda is a uniform constant in this limiting case, though not in
general. This is so in the FRW cosmology of "dark energy" from Type 1a
supernovae anomalous redshifts.

The source equation is

R/\e + (Lambda)e/\e/\e = J

Still if no torsion

d(Lambda)e/\e/\e = DJ =/= 0

That is the locally conserved stress-energy density current is

J' = J - (Lambda)e/\e/\e

DJ' = 0

d(Lambda) =/= 0 locally coincident with DJ =/= 0

* This is a sloshing back and forth between off-mass-shell virtual zero
point vacuum stress-energy current densities and on-mass-shell real
matter-EM field stress-energy current densities.

II. Add the Russian Torsion Field from local gauging of O(1,3). This is
a poor man's string theory with a 10D manifold. The Whipped Cream with
the Cherry on Top.

S = De =/= 0

In vacuum

R/\e + Lambda e/\e/\e = 0

Therefore,

DR = 0 still true, but

R/\De + d(Lambda)e/\e/\e + (Lambda)d(e/\e/\e) = 0

Note the curvature-torsion coupling

R/\De = R/\S =/= 0

and the pure zero point energy torsion propellantless propulsion term

(Lambda)d(e/\e/\e) =/= 0

as well as the WWD term

d(Lambda)e/\e/\e =/= 0

One gets additional effect when J =/= 0

"Nuff said"

On Jan 26, 2006, at 9:29 PM, Jack Sarfatti wrote:

Let (hG/c^3) = 1 for now.

I have derived at least Einstein's classical GR vacuum equation as an
emergent Goldstone phase modulation.

I assume a Planck-scale emergence where the degenerate vacuum manifold

G/H ~ S2

So there are 2 independent local Goldstone phase coherent fields theta & phi

I define the invariant 1-form

B = (dtheta)/\(phi) - (theta)/\(dphi)

so that

dB = 2(dtheta)/\(dphi) =/= 0

d^2B = 0 locally

The invariant Einstein-Cartan 1-form is

e = 1 + B

where

e = eudx^u

eu = eu^a&a

Einstein's symmetric metric tensor is by the equivalence principle

guv = eu^a(Minkowski)abev^b

Note the nonlinear terms ~ B^2.

The spin connection is the 1-form W^a^b = Wu^a^bdx^u

The exterior covariant derivative is

D = d + W/\

Zero torsion means

De = 0

Therefore

W = -*[dB/\(1 - B)]

* is the Hodge dual

The curvature 2-form is

R = DW = dW + W/\W

i.e.

R^a^b = DW^a^b + Wa^c/\Wc^b

The Bianchi identities are

DR = 0

Define

The vacuum equation is the 1-form equation

*[R/\e + Lambda e/\e/\e] = 0