The vielbein formalism

This is the second post of a series of four posts concerned with the canonical quantization of general relativity.

 

The vielbein formalism

First the vielbein field is essentially the square root of the metric. Physically, it gives the local orientations $\xi^m$ of freely falling frames in the gravitational field associated with the metric $g_{\mu\nu}$, i.e.  $\xi^m$  are orientations of the local inertial frames with respect to the  coordinate axes  $x^{\mu}$ of the curved spacetime manifold, viz
\begin{eqnarray}
e_{\mu}^m=\frac{\partial \xi^m}{\partial x^{\mu}}.
\end{eqnarray}
The metric is the square toot of the vielbein which means that we have ($\eta$ being the flat metric with signature $-1$, $+1$, $+1$, $+1$)
\begin{eqnarray}
g_{\mu\nu}=e_{\mu}^me_{\nu}^n\eta_{mn}.\label{fundamental}
\end{eqnarray}
The inverse of $e_{\mu}^m$ is denoted by $e_m^{\mu}$, viz $e_m^{\mu}e_{\mu}^n=\eta_m^n$ and $e_{\mu}^me_m^{\nu}=\eta_{\mu}^{\nu}$. Thus, we also have $g^{\mu\nu}=e^{\mu}_m e^{\nu}_n\eta^{mn}$.

The fundamental equation  (\ref{fundamental}) can be derived in a straightforward way from the Clifford algebra of the Dirac matrices in the curved spacetime manifold which is of the usual form but only with the replacement $\eta\longrightarrow g$, i.e.
\begin{eqnarray}
\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}. \label{clifford}
 \end{eqnarray}
Indeed, equation (\ref{fundamental}) is obtained by replacing in the Clifford algebra (\ref{clifford}) the ansatz (where $\gamma^m$ are the flat spacetime  Dirac matrices)
\begin{eqnarray}
\gamma^{\mu}=\gamma^m e_m^{\mu}.
\end{eqnarray}
Two solutions $e_{\mu}^m$ and $f_{\mu}^m$ of (\ref{fundamental}) are related by a local Lorentz transformation $\Lambda\in SO(1,3)$ (where $SO(1,3)$ is the restricted Lorentz group with antisymmetric generators) as follows
\begin{eqnarray}
f_{\mu}^m=\Lambda_k^me_{\mu}^k~,~\Lambda_k^m\eta_{mn}(\Lambda^T)_l^n=\eta_{kl}.
\end{eqnarray}
Obviously, the index $\mu$ refers to the curved coordinates $x^{\mu}$ of the manifold whereas the index $m$ refers to the flat coordinates $\xi^m$. A vector field $v$ will then have components $v^{\mu}$  in the system $x^{\mu}$ and components $v^m$ in the system $\xi^m$. They are related by means of the vielbein, i.e. $v^{\mu}=v^me_m^{\mu}$ and $v^m=v^{\mu}e_{\mu}^m$.

The covariant derivative with respect to the curved index $\mu$ is defined by means of the parallel transport with respect of the affine connection $\Gamma$ given by the Christoffel symbols, viz
\begin{eqnarray}
\tilde{v}^{\mu}(x+\Delta x)=v^{\mu}(x)-\Delta x^{\rho}\Gamma_{\rho\sigma}^{\mu}v^{\sigma}\iff \tilde{v}^{\mu}(x)+\Delta x^{\rho}\nabla_{\rho}\tilde{v}^{\mu}(x)=v^{\mu}(x).
\end{eqnarray}
Similarly, the covariant derivative with respect to the flat index $m$ is defined by means of the parallel transport with respect of the spin connection $\omega$, viz
\begin{eqnarray}
\tilde{v}^{m}(x+\Delta x)=v^{m}(x)-\Delta x^{\rho}\omega_{\rho n }^{m}v^{n}\iff \tilde{v}^{m}(x)+\Delta x^{\rho}\nabla_{\rho}\tilde{v}^{m}(x)=v^{m}(x).
\end{eqnarray}
Since we are dealing with the parallel transport of the same vector we must have the property that the parallel transport of the curved index from $x$ to $x+\Delta x$ followed by its the projection to a flat index equal to the projection at $x$ of the curved index to a flat index followed by the parallel transport of the flat index from $x$ to $x+\Delta x$, viz
\begin{eqnarray}
\tilde{v}^{\mu}(x+\Delta x)e_{\mu}^m(x+\Delta x)=\tilde{v}^{m}(x+\Delta x).
\end{eqnarray}
This gives immediately the so-called vielbein postulate which states that the vielbein $e_{\mu}^m$ is covariantly constant with respect to both connections affine $\Gamma$ (which acts on the curved index $\mu$) and spin $\omega$ (which acts on the flat index $m$), i.e.
\begin{eqnarray}
{\cal D}_{\rho}e_{\mu}^m&\equiv &\partial_{\rho}e_{\mu}^m-\Gamma_{\rho\mu}^{\nu}e_{\nu}^m+\omega_{\rho n}^me_{\mu}^n\nonumber\\
&=&0.
\end{eqnarray}
This equation gives us the spin connection $\omega$ (which defines the spinor bundle) in terms of the affine connection  $\Gamma$ (which defines the tangent/co-tangent bundle) or vice versa. Clearly, the spin connection encodes the local Lorentz (special coordinate transformations) invariance of the theory whereas the affine connection encodes diffeomorphism (general coordinate transformations) invariance.

Now by requiring that the length of the vector $v$ to be invariant under the parallel transport we get by using the flat and the curved indices respectively the two equivalent results that the spin connection is antisymmetric (by using the components $v^m$) and the metric is covariantly constant (by using the components $v^{\mu}$), viz
\begin{eqnarray}
\omega_{\mu}^{mn}=-\omega_{\mu}^{nm}~,~\omega_{\mu}^{mn}=\omega_{\mu k}^m\eta^{kn}.
\end{eqnarray}
\begin{eqnarray}
{\nabla}_{\rho}g_{\mu\nu}\equiv \partial_{\rho}g_{\mu\nu}-\Gamma_{\rho\mu \nu}-\Gamma_{\rho\nu \mu}~,~\Gamma_{\rho\mu \nu}=\Gamma_{\rho\mu}^{\lambda}g_{\lambda\nu}.
\end{eqnarray}
The spin connection will play the role of a local $SO(1,3)$ gauge field representing local Lorentz invariance and hence this connection must be antisymmetric by construction and as a consequence the invariance of the length and the covariant constancy of the metric follow naturally from a local symmetry principle.

The Riemann curvature tensor can be computed from the commutator of two covariant derivatives. The action of this commutator on the vielbein field should vanish identically. We have then
\begin{eqnarray}
[{\cal D}_{\rho},{\cal D}_{\sigma}]e_{\mu}^m=0.
\end{eqnarray}
From this equation we obtain the result that the Riemann curvature tensor in terms of $\Gamma$ is equal to the Riemann curvature tensor in terms of $\omega$, viz
\begin{eqnarray}
R_{\rho\sigma\mu}^{\nu}(\Gamma)e_{\nu}^{m}=R_{\rho\sigma n}^m(\omega)e_{\mu}^n.
\end{eqnarray}
Where
\begin{eqnarray}
R_{\rho\sigma\mu}^{\nu}(\Gamma)=\partial_{\rho}\Gamma_{\sigma\mu}^{\nu}+\Gamma_{\rho\tau}^{\nu}\Gamma_{\sigma\mu}^{\tau}-(\rho\leftrightarrow\sigma).
\end{eqnarray}
\begin{eqnarray}
R_{\rho\sigma n}^{m}(\omega)=\partial_{\rho}\omega_{\sigma n}^{m}+\omega_{\rho k}^{m}\omega_{\sigma n}^{k}-(\rho\leftrightarrow\sigma).
\end{eqnarray}Hence we get
\begin{eqnarray}
R_{\rho\sigma \mu\tau}(\Gamma)=R_{\rho\sigma \mu\tau}(\omega)~,~R_{\rho\sigma \mu\tau}(\Gamma)=-R_{\rho\sigma \tau\mu}(\Gamma).
\end{eqnarray}
The action of general relativity written in terms of the metric $g$  (the Hilbert-Einstein action) is therefore the same as the action written in terms of the vielbein field $e$ and the spin connection $\omega$ (the Palatini action).

Another important property satisfied by the affine connection $\Gamma$ in general relativity is torsionless (the Christoffel symbol $\Gamma_{\mu\nu}^{\rho}$ is symmetric in its two lower indices $\mu$ and $\nu$). In general torsion is generated from the vielbein field ($T=de+\omega\wedge e$) in the same way that curvature is generated from the spin connection ($R=d\omega+\omega\wedge\omega$).

Also, similarly to the fact that the curvature tensor measures the gap if one parallel transports a vector parallel to itself along a closed curved the torsion tensor measures the gap if one parallel transports one vector along another one minus the other way around.

As we have said the affine Levi-Civita connection defines the tangent bundle which is the associated vector bundle corresponding to the $O(1,3)$ bundle of orthonormal frames. In fact the affine connection is induced from the  connection  on the $O(1,3)$ bundle of orthonormal frames and since the spacetime manifold is orientable this connection can be restricted to the $SO(1,3)$ bundle of orthonormal frames and then lifted to a connection on the corresponding spinor bundle.  Thus, the spin connection is indeed more fundamental than the affine connection in every respect.

Finally, we note that spinors provide a representation of the Lorentz group $SO(1,3)$. Thus, they transform covariantly under Lorentz transformations with the covariant derivative given explicitly in terms of the spin connection $\omega$  by (with $\gamma_{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]/2$)
\begin{eqnarray}
D_{\mu}\psi=\partial_{\mu}\psi+\frac{1}{4}\omega_{\mu}^{mn}\gamma_{mn}\psi.
\end{eqnarray}
The covariant Dirac action in a curved spacetime manifold (discovered by Wigner in $1929$) is therefore given by (with $e=\sqrt{-{\rm det} (g_{\mu\nu})}={\rm det}(e_{\mu}^m)$)
\begin{eqnarray}
{\cal L}_D=-\frac{e}{2}\bar{\psi}\gamma^{\mu}D_{\mu}\psi.
\end{eqnarray}