Thursday, January 26, 2006

Simple derivation of Einstein's gravity as an emergent collective effect
I think 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


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


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

* is the Hodge dual

The curvature 2-form is

R = DW = dW + W/\W


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

The Bianchi identities are

DR = 0


The vacuum equation is the 1-form equation

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

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