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
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
Thursday, January 26, 2006
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