# differential geometry – Is the tetrad formulation of General Relativity equivalent to a map from general spacetime to Minkowskian space

For those not aquainted, in General Relativity we consider spacetime as a 4-dimensional pseudo-Riemannian manifold with metric tensor components $$g_{munu}$$. There is an equivalent tetrad formulation of general relativity, wherein one utilizes (an often orthonormal) basis. In this case the metric tensor is written as:

$$g_{munu}=e_{mu}^{a}e_{nu}^{b}eta_{ab}$$

Where $$eta_{ab}$$ are components of the flat Minkowskian metric. In this case, the Greek and Latin indices refer to coordinate and orthonormal (generally non-coordinate) basis respectivly. One case use the tetrads to take quantities from coordinate to orthonormal indices. This formulation is extremely important, for example when using spinor fields in General relativity.

From a purely mathematical perspective, I perceive the tetrad formulation as a map from a spacetime manifold $$M$$ to Minkowskian spacetime $$eta$$.

$$Mrightarroweta$$

In which case it is rather natural that the spacetime metric tensor becomes the pullback of the metric from $$eta$$. Am I right in viewing it this way? The main arguments against this viewpoint on the physics stackexhange were:

“The tetrad formalism is not related to the pullback operation. The
similarity you point out has more to do with the statement that both
$$g_{munu}=frac{partial X^{a}}{partial x^{mu}}frac{partial > X^{b}}{partial x^{nu}}g_{ab}$$ and
$$g_{munu}=e_{mu}^{a}e_{nu}^{b}eta_{ab}$$ transform covariantly
under diffeomorphisms (which is a pullback of the spacetime to itself
via some diffeomorphism, if you prefer that language).”

However; in reading about the pullback it would seem that this argument only applies to the pullback by a diffeomorphism (see section in link). The pullback is a much more robust operation than this.

So I’m asking the mathematicians now: Can I regard the tetrad formulation as a map from a spacetime manifold to Minkowskian space which implies the spacetime metric tensor is the pullback? I do understand there are issues with pulling back contravariant tensor fields, but we can just stick with differential form versions if preferred.

Also I’ll add that in a great book on General relativity “Formulations of General Relativity Gravitry, Spinors and differential forms” by Krasnov. Section 3.1.3 says:

“given a tetrad, the metric on M is defined to be the pullback of the
metric on E $$g_{munu}=e_{mu}^{a}e_{nu}^{b}eta_{ab}$$

Where Krasnov is referring to the base and fiber metric, by M and E respectively.