An orthonormal frame separates local measurements from coordinate labels

Write the orthonormal basis vectors as eₐ=eₐμ∂μ and the dual one-forms as θᵃ=eᵃμdxμ, with separate frame and coordinate indices. The two component matrices are inverses. Orthonormality means their metric contraction is ηab=diag(−1,1,1,1). A vector’s measured frame components are obtained by applying the coframe matrix, not by multiplying by whichever tetrad array happens to have convenient-looking indices. This is the same dual-basis logic that distinguished vectors from covectors earlier.

The observer’s timelike frame vector selects a local rest frame; spatial frame vectors specify its axes. A different observer can choose a locally boosted frame at the same event. This changes measured energy and momentum components without changing the event or requiring a coordinate change. Coordinate transformations act on the Greek indices, while local Lorentz transformations act on the frame indices. Both can occur together. The word local refers to this independent choice at each point, not to a requirement that curvature vanish in the neighborhood.

For the flat polar spatial metric choose θʳ=dr and θφ=r dφ. The dual vectors are ∂r and r⁻¹∂φ. Their spatial metric is the identity, but the vectors do not commute: their commutator is −r⁻²∂φ. They therefore cannot both be coordinate partial derivatives on a patch. A connection expressed in such a frame includes this anholonomy. The simple coordinate-basis statement that torsion-free coefficients are symmetric in the lower indices cannot be applied without the frame commutator correction.

Tetrads are often called square roots of the metric because contracting two coframe factors with η reconstructs g. They are not unique: a position-dependent Lorentz transformation preserves the same metric. The associated spin connection accounts for the varying frame and is essential for differentiating frame components or coupling spinors. In curved geometry it is not generally possible to make all connection information vanish throughout an open region by choosing orthonormal axes. At one event a freely falling construction can simplify it, while curvature still controls the next-order variation.

A local tetrad exists for a regular Lorentzian metric, but global frame choices can face topological or orientation restrictions. Overlaps relate local frames through the appropriate Lorentz transformations. A tetrad is therefore neither a proof of one global inertial coordinate system nor an elimination of gravitational tides. Physical frame components should also be tied to a specified observer and transport convention when comparisons are made along a trajectory; rotating an observer’s axes can create inertial terms even in flat spacetime.

The vector lab illustrates the spatial rotation subgroup. Reset and set both controls to angle 30 degrees and magnitude two, predicting components (√3,1). Reset and set angle 60 degrees and magnitude two, predicting (1,√3). Both have squared Euclidean spatial magnitude four. This compares two spatial component orientations, not full spacetime boosts or a computed tetrad field. The lab has no spin connection or coordinate metric, so its preserved vector length cannot be used to infer that a spacetime is flat.