PatrickUrania said:
Why involve a vielbein at all? To me this looks like giving some special status to the linear term in a Taylor series.
People worked hard and proved theorems which allow us to throw away the rest of “Taylor series” whatever that means. In this context, the famous and decisive theorem is that of Geroch (1968):
A non-compact spacetime M^{4} has a spin structure
if and only if there exist 4 continuous vector fields on M^{4} which constitute a Minkowski tetrad, e^{a}{}_{\mu}(x), in the tangent space at each point of M^{4}. [*]
The point is this:
without spin structure, the very concept of a spinor field
does not exist [See Penrose & Rindler, Vol 1, “Spinor Calculus and Relativistic Fields”].
Your questions in the first post were answered mathematically in my first post. So, instead of hanging on to a dead fish (which does not buy you anything and nobody buy it of you), spend some of your time learning about Lie groups and their representation theory. Not just it is a beautiful subject, in fact, if undergraduate students can’t do without calculus, theoretical physicists can not do without group theory.
[*] By definition, a 4-dimensional space-time is a pair ( M , g ) consisting of a
connected, 4-dimentional,
Hausdroff C^{\infty} manifold M, together with a
Lorentz metric g on M: It can be shown that a manifold admits Lorentzian metric
if and only if there exists a globally defined (timelike) vector field X : M \to T (M), non-vanishing at each point of the manifold, i.e. M must be
time-orientable [Any non-compact manifold admits a Lorentzian metric. For a compact orientable manifold, existence of a Lorentzian metric is equivalent to the fact that the manifold has zero Euler characteristic]. Thus, two Lorentzian metrics on M are considered equivalent if they are related by the diffeomorphism group of M. Thus, our spacetime is modeled mathematically by
equivalence classes of pairs ( M , g ). When a space-time metric has been introduced, one can define (in addition to the diffeomorphism group) an action on M of the so-called local Lorentz group as follow. Let \{ e_{a} (p) = e_{a}{}^{\mu} ( x ) \partial_{\mu}\}, a = 0 , 1 , 2 , 3 be a set of 4 smooth vector fields on M, forming an orthonormal frame at each point p \in M, \langle e_{a} | e_{b} \rangle_{p} \equiv e_{a}{}^{\mu} (x) e_{b}{}^{\nu} (x) g_{\mu \nu} (x) = \eta_{a b} , where \eta is the Minkowski metric. The set \{ e_{a} (p) \} is called “veibein”, frame field or Minkowski tetrad in the tangent space at each point of M. The components e^{a}{}_{\mu} of the inverse veibein are related to the metric tensor g_{\mu \nu} by g_{\mu \nu} ( x ) = e^{a}{}_{\mu} ( x ) e^{b}{}_{\nu} ( x ) \eta_{a b} . Since e^{a}{}_{\mu} has n^{2} = 16 independent components while g_{\mu \nu} has n ( n + 1 ) / 2 = 10, the veibein contains n ( n – 1 ) / 2 = 6 extra degrees of freedom. They are nothing but the freedom to carry out (local)
Lorentz transformations in the tangent spaces.
