Last update: 9.02.2005 17:00

SOME NOTES ON METRIC COMPATIBILITY
AND THE TETRAD POSTULATE

M.W. Evans

(from the handwritten original in
http://www.atomicprecision.com/new/ametriccompatibilityfromtetradpostulate.pdf )

with a short remark on logics by

Gerhard W. Bruhn, Darmstadt University of Technology

The metric compatibility condition of Riemann geometry is well known to be

D_ρ g_μν = 0                                                                                 (1)

where D_ρ is the covariant derivation and g_μν is the symmetric metric. If it is assumed that the torsion tensor is zero:

T_μ^κ_ν = Γ_μ^κ_ν − Γ_ν^κ_μ = 0                                                                 (2)

we obtain the well known expression for the Christoffel connection of equ. (2):

Γ_ν^σ_μ = ½ g^σρ (∂_μg_νρ + ∂_νg_ρμ − ∂_ρg_μν)                                                 (3)

Einstein and Hilbert used eqns (1), (2) and (3) to develop the famous theory of general relativity (1915 - 1916).

In the Palatini variation of general relativity, developed mainly by Cartan, the tetrad q_μa is the fundamental field.

The symmetric metric is

g_μν = q_μa q_νb η_ab                                                                 (4)

where η_ab is the Minkowski metric of the tangent bundle. The tetrad postulate is:

D_σ q_μa = 0                                                                                 (5)

and is true for any correction. The tetrad postulate is not predicated on the assumption of metric compatibility or absense of torsion.

The metric compatibility of Riemann geometry is obtained from the tetrad postulate as follows. From equ. (4):

D_σ g_μν = D_σ (q_μa q_νb η_ab)                                                                                 (6)
= η_ab (q_νb D_σq_μa + q_μa D_σq_νb) = 0                                                            

where we have used equ.(5) and the Leibnitz theorem, together with:

D_σ η_ab = 0 .                                                                                 (7)

Therefore given equ. (4) we obtain equ. (1) from equ. (5). This is a powerful and simple demonstration that the tetrad postulate is more fundamental than the compatibility condition used by Einstein and Hilbert.

Remark

Logics is not everybody's cup of tea, but Einstein and Hilbert had some of it, I'm sure:
Evans has shown above that his (and S.M. Carroll's) tetrad postulate implies metric compatibility.