A tetrad at a point P of spacetime is a four-leg consisting of four vectors e_a (a=0,1,2,3) in the tangential space T_P of the manifold at P that are orthonormal wrt the metric (g_μν) at P, i.e. by representing the tetrad vectors e_a = e_a^μ ∂_μ by the basis vectors ∂_μ (μ=0,1,2,3) of the local coordinate basis we have the orthonormality relations:

g_μν e_a^μe_b^ν = η_ab                                                                 [C1;(3.118)]

where (η_ab) = diag(−1, 1, 1, 1) is the Minkowski matrix. Cf. [C1;(3.118)] and [C2;(J.5)].

Every local Lorentz transform (LLT) Λ=(Λ_a'^a) changes the tetrad vectors but leaving the canonical form of the metric unaltered:

e_a → e_a' = Λ_a'^a e_a                                                                 [C1;(3.125)]

Therefore we have

Λ_a'^a Λ_b'^b η_ab = η_a'b' .                                                                 [C1;(3.126)]

to obtain

η_a'b' e_μ^a'e_ν^b' = Λ_a'^a Λ_b'^b η_ab e_μ^a'e_ν^b' = η_ab (Λ_a'^a e_μ^a') (Λ_b'^b e_ν^b') = η_ab e_μ^a e_ν^b = g_μν .

Result:
The metric (g_μν) (and hence the geometry of the manifold) is not afflicted by a change of the (local) tetrad due to an arbitrary LLT.

[C2] Sean M. Carroll, Spacetime and Geometry,
        Addison & Wesley, ISBN 0-8053-8732-3
 

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