Spin connection

In

rotations

.

The spin connection occurs in two common forms: the Levi-Civita spin connection, when it is derived from the Levi-Civita connection, and the affine spin connection, when it is obtained from the affine connection. The difference between the two of these is that the Levi-Civita connection is by definition the unique torsion-free connection, whereas the affine connection (and so the affine spin connection) may contain torsion.

Definition

Let $e_{\mu }^{\;\,a}$ be the local Lorentz

vierbein

(also known as a tetrad), which is a set of orthonormal 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 torsion-free 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. This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.

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 $e^{\nu b}$ . The spin connection may be written purely in terms of the vierbein field as^[1]

\omega _{\mu }^{\ ab}={\tfrac {1}{2}}e^{\nu a}(\partial _{\mu }e_{\nu }^{\ b}-\partial _{\nu }e_{\mu }^{\ b})-{\tfrac {1}{2}}e^{\nu b}(\partial _{\mu }e_{\nu }^{\ a}-\partial _{\nu }e_{\mu }^{\ a})-{\tfrac {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}

Cartan's structure equations

In the

Maurer–Cartan equations for Lie groups

(that is, they are the same equations, but in a different setting and notation).

Writing the vierbeins as differential forms

e^{a}=e_{\mu }^{\;\,a}dx^{\mu }

for the orthonormal coordinates on the cotangent bundle, the affine spin connection one-form is

\omega ^{ab}=\omega _{\mu }^{\;\;ab}dx^{\mu }

The torsion 2-form is given by

\Theta ^{a}=de^{a}+\omega _{\;b}^{a}\wedge e^{b}

while the curvature 2-form is

R_{\;\,b}^{a}=d\omega _{\;\,b}^{a}+\omega _{\;c}^{a}\wedge \omega _{\;\,b}^{c}={\tfrac {1}{2}}R_{\;\,bcd}^{a}e^{c}\wedge e^{d}

These two equations, taken together are called Cartan's structure equations.^[2] Consistency requires that the

Bianchi identities

be obeyed. The first Bianchi identity is obtained by taking the exterior derivative of the torsion:

d\Theta ^{a}+\omega _{\;b}^{a}\wedge \Theta ^{b}=R_{\;\,b}^{a}\wedge e^{b}

while the second by differentiating the curvature:

dR_{\;\,b}^{a}+\omega _{\;c}^{a}\wedge R_{\;\,b}^{c}-R_{\;\,c}^{a}\wedge \omega _{\;b}^{c}=0.

The covariant derivative for a generic differential form

V_{\;\,b}^{a}

of degree p is defined by

DV_{\;\,b}^{a}=dV_{\;\,b}^{a}+\omega _{\;c}^{a}\wedge V_{\;\,b}^{c}-(-1)^{p}V_{\;\,c}^{a}\wedge \omega _{\;b}^{c}.

Bianchi's second identity then becomes

DR_{\;\,b}^{a}=0.

The difference between a connection with torsion, and the unique torsionless connection is given by the contorsion tensor. Connections with torsion are commonly found in theories of teleparallelism, Einstein–Cartan theory, gauge theory gravity and supergravity.

Derivation

Metricity

It is easy to deduce by raising and lowering indices 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}

. This is sometimes called the metricity condition;^[2] it is analogous to the more commonly stated metricity condition that

g_{\mu \nu ;\alpha }=0.

Note that this condition holds only for the Levi-Civita spin connection, and not for the affine spin connection in general.

By substituting the formula for the Christoffel symbols ${\Gamma ^{\nu }}_{\sigma \mu }={\tfrac {1}{2}}g^{\nu \delta }\left(\partial _{\sigma }g_{\delta \mu }+\partial _{\mu }g_{\sigma \delta }-\partial _{\delta }g_{\sigma \mu }\right)$ 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 in 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}={\tfrac {1}{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 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 generated by the $\sigma _{ab}$ 's, such that $\epsilon _{ab}$ 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 =\left(\partial _{\mu }-{\tfrac {i}{4}}{\omega _{\mu }}^{ab}\sigma _{ab}\right)\psi =\left(\partial _{\mu }-{\tfrac {i}{4}}e^{\nu a}\partial _{;\mu }{e_{\nu }}^{b}\sigma _{ab}\right)\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

{\textstyle 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}=\left|\det(e)\right|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.

References

^ M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory", Vol. 2.
^ ^a ^b Tohru Eguchi, Peter B. Gilkey and Andrew J. Hanson, "Gravitation, Gauge Theories and Differential Geometry", Physics Reports 66 (1980) pp 213-393.