Spin connection

In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations.

Definition

Let $e_{\mu }^{\ a}$ be the local Lorentz frame fields or vierbein (also known as a tetrad), which is a set of orthogonal space time vector fields that diagonalize the metric tensor

g_{\mu \nu }=e_{\mu }^{\ a}e_{\nu }^{\ b}\eta _{ab},

where $g_{\mu \nu }$ is the spacetime metric and $\eta _{ab}$ is the Minkowski metric. Here, Latin letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices. This simply expresses that $g_{\mu \nu }$ , when written in terms of the basis $e_{\mu }^{\ a}$ , is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e. $g^{\mu \nu }$ or $g_{\mu \nu }$ . The Latin or "Lorentzian" vierbein indices can be raised or lowered by $\eta ^{ab}$ or $\eta _{ab}$ respectively. For example, $e^{\mu a}=g^{\mu \nu }e_{\nu }^{\ a}$ and $e_{\nu a}=\eta _{ab}e_{\nu }^{\ b}$

The spin connection is given by

\omega _{\mu }^{\ ab}=e_{\nu }^{\ a}\Gamma _{\ \sigma \mu }^{\nu }e^{\sigma b}+e_{\nu }^{\ a}\partial _{\mu }e^{\nu b}=e_{\nu }^{\ a}\Gamma _{\ \sigma \mu }^{\nu }e^{\sigma b}-e^{\nu b}\partial _{\mu }e_{\nu }^{\ a},

where $\Gamma _{\mu \nu }^{\sigma }$ are the Christoffel symbols. (Note that $\omega _{\mu }^{\ ab}=e_{\nu }^{\ a}\partial _{;\mu }e^{\nu b}=e_{\nu }^{\ a}(\partial _{\mu }e^{\nu b}+\Gamma _{\ \sigma \mu }^{\nu }e^{\sigma b})$ using the gravitational covariant derivative $\partial _{;\mu }e^{\nu b}$ of the contravariant vector $e^{\nu b}$ .) Or it may be written purely in terms of the vierbein field as^[1]

\omega _{\mu }^{\ ab}={\frac {1}{2}}e^{\nu a}(\partial _{\mu }e_{\nu }^{\ b}-\partial _{\nu }e_{\mu }^{\ b})-{\frac {1}{2}}e^{\nu b}(\partial _{\mu }e_{\nu }^{\ a}-\partial _{\nu }e_{\mu }^{\ a})-{\frac {1}{2}}e^{\rho a}e^{\sigma b}(\partial _{\rho }e_{\sigma c}-\partial _{\sigma }e_{\rho c})e_{\mu }^{\ c},

which by definition is anti-symmetric in its internal indices $a,b$ .

The spin connection $\omega _{\mu }^{\ ab}$ defines a covariant derivative $D_{\mu }$ on generalized tensors. For example its action on $V_{\nu }^{\ a}$ is

D_{\mu }V_{\nu }^{\ a}=\partial _{\mu }V_{\nu }^{\ a}+{{\omega _{\mu }}^{a}}_{b}V_{\nu }^{\ b}-\Gamma _{\ \nu \mu }^{\sigma }V_{\sigma }^{\ a}

Derivation

By the tetrad postulate

It is easy to deduce by raising and lowering indexes as needed that the frame fields defined by $g_{\mu \nu }={e_{\mu }}^{a}{e_{\nu }}^{b}\eta _{ab}$ will also satisfy ${e_{\mu }}^{a}{e^{\mu }}_{b}=\delta _{b}^{a}$ and ${e_{\mu }}^{b}{e^{\nu }}_{b}=\delta _{\mu }^{\nu }$ . We expect that $D_{\mu }$ will also annihilate the Minkowski metric $\eta _{ab}$ ,

D_{\mu }\eta _{ab}=\partial _{\mu }\eta _{ab}-{\omega _{\mu a}}^{c}\eta _{cb}-{\omega _{\mu b}}^{c}\eta _{ac}=0.

This implies that the connection is anti-symmetric in its internal indices, ${\omega _{\mu }}^{ab}=-{\omega _{\mu }}^{ba}.$ This is also deduced by taking the gravitational covariant derivative $\partial _{;\beta }({e_{\mu }}^{a}{e^{\mu }}_{b})=0$ which implies that $\partial _{;\beta }{e_{\mu }}^{a}{e^{\mu }}_{b}=-{e_{\mu }}^{a}\partial _{;\beta }{e^{\mu }}_{b}$ thus ultimately, ${\omega _{\beta }}^{ab}=-{\omega _{\beta }}^{ba}$ .

By substituting the formula for the Christoffel symbols ${\Gamma ^{\nu }}_{\sigma \mu }={1 \over 2}g^{\nu \delta }(\partial _{\sigma }g_{\delta \mu }+\partial _{\mu }g_{\sigma \delta }-\partial _{\delta }g_{\sigma \mu })$ written in terms of the ${e_{\mu }}^{a}$ , the spin connection can be written entirely in terms of the ${e_{\mu }}^{a}$ ,

{\omega _{\mu }}^{ab}=e^{\nu [a}({{e_{\nu }}^{b]}}_{,\mu }-{{e_{\mu }}^{b]}}_{,\nu }+e^{\sigma |b]}{e_{\mu }}^{c}e_{\nu c,\sigma })

where antisymmetrization of indices has an implicit factor of 1/2.

By the metric compatibility

This formula can be derived another way. To directly solve the compatibility condition for the spin connection ${\omega _{\mu }}^{ab}$ , one can use the same trick that was used to solve $\nabla _{\rho }g_{\alpha \beta }=0$ for the Christoffel symbols ${\Gamma ^{\gamma }}_{\alpha \beta }$ . First contract the compatibility condition to give

{e^{\alpha }}_{b}{e^{\beta }}_{c}(\partial _{[\alpha }e_{\beta ]a}+{\omega _{[\alpha a}}^{d}\;e_{\beta ]d})=0

.

Then, do a cyclic permutation of the free indices $a,b,$ and $c$ , and add and subtract the three resulting equations:

\Omega _{bca}+\Omega _{abc}-\Omega _{cab}+2{e^{\alpha }}_{b}\omega _{\alpha ac}=0

where we have used the definition $\Omega _{bca}:={e^{\alpha }}_{b}{e^{\beta }}_{c}\partial _{[\alpha }e_{\beta ]a}$ . The solution for the spin connection is

\omega _{\alpha ca}={1 \over 2}{e_{\alpha }}^{b}(\Omega _{bca}+\Omega _{abc}-\Omega _{cab})

.

From this we obtain the same formula as before.

Applications

The spin connection arises in the Dirac equation when expressed in the language of curved spacetime, see Dirac equation in curved spacetime. Specifically there are problems coupling gravity to spinor fields: there are no finite-dimensional spinor representations of the general covariance group. However, there are of course spinorial representations of the Lorentz group. This fact is utilized by employing tetrad fields describing a flat tangent space at every point of spacetime. The Dirac matrices $\gamma ^{a}$ are contracted onto vierbiens,

\gamma ^{a}{e^{\mu }}_{a}(x)=\gamma ^{\mu }(x)

.

We wish to construct a generally covariant Dirac equation. Under a flat tangent space Lorentz transformation the spinor transforms as

\psi \mapsto e^{i\epsilon ^{ab}(x)\sigma _{ab}}\psi

We have introduced local Lorentz transformations on flat tangent space, so $\epsilon _{ab}$ is a function of space-time. This means that the partial derivative of a spinor is no longer a genuine tensor. As usual, one introduces a connection field ${\omega _{\mu }}^{ab}$ that allows us to gauge the Lorentz group. The covariant derivative defined with the spin connection is,

\nabla _{\mu }\psi =(\partial _{\mu }-{i \over 4}{\omega _{\mu }}^{ab}\sigma _{ab})\psi =(\partial _{\mu }-{i \over 4}e^{\nu a}\partial _{;\mu }{e_{\nu }}^{b}\sigma _{ab})\psi

,

and is a genuine tensor and Dirac's equation is rewritten as

(i\gamma ^{\mu }\nabla _{\mu }-m)\psi =0

.

The generally covariant fermion action couples fermions to gravity when added to the first order tetradic Palatini action,

{\mathcal {L}}=-{1 \over 2\kappa ^{2}}e\,{e^{\mu }}_{a}{e^{\nu }}_{b}{\Omega _{\mu \nu }}^{ab}[\omega ]+e{\overline {\psi }}(i\gamma ^{\mu }\nabla _{\mu }-m)\psi

where $e:=\det {e_{\mu }}^{a}={\sqrt {-g}}$ and ${\Omega _{\mu \nu }}^{ab}$ is the curvature of the spin connection.

The tetradic Palatini formulation of general relativity which is a first order formulation of the Einstein–Hilbert action where the tetrad and the spin connection are the basic independent variables. In the 3+1 version of Palatini formulation, the information about the spatial metric, $q_{ab}(x)$ , is encoded in the triad $e_{a}^{i}$ (three-dimensional, spatial version of the tetrad). Here we extend the metric compatibility condition $D_{a}q_{bc}=0$ to $e_{a}^{i}$ , that is, $D_{a}e_{b}^{i}=0$ and we obtain a formula similar to the one given above but for the spatial spin connection $\Gamma _{a}^{ij}$ .

The spatial spin connection appears in the definition of Ashtekar-Barbero variables which allows 3+1 general relativity to be rewritten as a special type of $\mathrm {SU} (2)$ Yang–Mills gauge theory. One defines $\Gamma _{a}^{i}=\epsilon ^{ijk}\Gamma _{a}^{jk}$ . The Ashtekar-Barbero connection variable is then defined as $A_{a}^{i}=\Gamma _{a}^{i}+\beta c_{a}^{i}$ where $c_{a}^{i}=c_{ab}e^{bi}$ and $c_{ab}$ is the extrinsic curvature and $\beta$ is the Immirzi parameter. With $A_{a}^{i}$ as the configuration variable, the conjugate momentum is the densitized triad $E_{a}^{i}=|\det(e)|e_{a}^{i}$ . With 3+1 general relativity rewritten as a special type of $\mathrm {SU} (2)$ Yang–Mills gauge theory, it allows the importation of non-perturbative techniques used in Quantum chromodynamics to canonical quantum general relativity.