Spin connection explained

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.

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 ^{a} _{\mu} be the local Lorentz frame fields or vierbein (also known as a tetrad), which is a set of orthonormal space time vector fields that diagonalize the metric tensor$$g\_\; =\; e\_\backslash mu^\; e\_\backslash nu^\; \backslash eta\_,$$where g _{\mu} is the spacetime metric and η _{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} , when written in terms of the basis e ^{a} _{\mu} , is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e. g ^{\mu} or g _{\mu} . The Latin or "Lorentzian" vierbein indices can be raised or lowered by η ^{ab} or η _{ab} respectively. For example, e ^{\mu} = g ^{\mu\nu} e ^{a} _{\nu} and e _{\nu} =η _{ab} e ^{b} _{\nu}

The torsion-free spin connection is given by$$\backslash omega\_^=e\_\backslash nu^\; \backslash Gamma^\backslash nu\_e^\; +\; e\_\backslash nu^\; \backslash partial\_\backslash mu\; e^\; =\; e\_\backslash nu^\; \backslash Gamma^\backslash nu\_e^\; -\; e^\; \backslash partial\_\backslash mu\; e\_\backslash nu\; ^,$$where

\Gamma ^{\sigma} _{\mu\nu} 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 ^{ab} _{\mu} = e ^{a} _{\nu} \partial _{;\mu} e ^{\nu} = e ^{a} _{\nu} ( \partial _{\mu} e ^{\nu} + \Gamma ^{\nu} _{\sigma\mu} e ^{\sigma} ) using the gravitational covariant derivative \partial _{;\mu} e ^{\nu} of the contravariant vector e ^{\nu} . The spin connection may be written purely in terms of the vierbein field as^[1] $$\backslash omega\_^\; =\; \backslash tfrac\; e^\; (\backslash partial\_\backslash mu\; e\_\backslash nu^-\backslash partial\_\backslash nu\; e\_\backslash mu^)\; -\; \backslash tfrac\; e^(\backslash partial\_\backslash mu\; e\_\backslash nu^-\backslash partial\_\backslash nu\; e\_\backslash mu^)\; -\; \backslash tfrac\; e^e^(\backslash partial\_\backslash rho\; e\_-\backslash partial\_\backslash sigma\; e\_)e\_\backslash mu^,$$which by definition is anti-symmetric in its internal indices a, b .

The spin connection

\omega ^{ab} _{\mu} defines a covariant derivative D _{\mu} on generalized tensors. For example, its action on V ^{a} _{\nu} is$$D\_\backslash mu\; V\_\backslash nu^\; =\; \backslash partial\_\backslash mu\; V\_\backslash nu^\; +\; \_b\; V\_\backslash nu^\; -\; \backslash Gamma^\backslash sigma\_\; V\_\backslash sigma^$$

Cartan's structure equations

In the Cartan formalism, the spin connection is used to define both torsion and curvature. These are easiest to read by working with differential forms, as this hides some of the profusion of indexes. The equations presented here are effectively a restatement of those that can be found in the article on the connection form and the curvature form. The primary difference is that these retain the indexes on the vierbein, instead of completely hiding them. More narrowly, the Cartan formalism is to be interpreted in its historical setting, as a generalization of the idea of an affine connection to a homogeneous space; it is not yet as general as the idea of a principal connection on a fiber bundle. It serves as a suitable half-way point between the narrower setting in Riemannian geometry and the fully abstract fiber bundle setting, thus emphasizing the similarity to gauge theory. Note that Cartan's structure equations, as expressed here, have a direct analog: the Maurer–Cartan equations for Lie groups (that is, they are the same equations, but in a different setting and notation).

V ^{a} _{b} of degree p is defined by$$DV^a\_=\; dV^a\_\; +\; \backslash omega^a\_\; \backslash wedge\; V^c\_\; -\; (-1)^p\; V^a\_\; \backslash wedge\; \backslash omega^c\_.$$Bianchi's second identity then becomes$$DR^a\_=0.$$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} = {e ^{a} _{\mu}} {e ^{b} _{\nu}} η _{ab} will also satisfy {e ^{a} _{\mu}} {e ^{\mu}} _{b} = \delta ^{a} _{b} and {e ^{b} _{\mu}} {e ^{\nu}} _{b} = \delta ^{\nu} _{\mu} . We expect that D _{\mu} will also annihilate the Minkowski metric η _{ab} ,$$D\_\backslash mu\; \backslash eta\_\; =\; \backslash partial\_\backslash mu\; \backslash eta\_\; -\; ^\; \backslash eta\_\; -\; ^\; \backslash eta\_\; =\; 0.$$This implies that the connection is anti-symmetric in its internal indices, {\omega _{\mu} }^ = - ^. This is also deduced by taking the gravitational covariant derivative \partial _{;\beta} ({e ^{a} _{\mu}} {e ^{\mu}} _{b)} = 0 which implies that \partial _{;\beta} {e ^{a} _{\mu}} {e ^{\mu}} _{b} = - {e ^{a} _{\mu}} \partial _{;\beta} {e ^{\mu}} _{b} thus ultimately, {\omega _{\beta} }^ = -^. 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} \left(\partial _{\sigma} g _{\delta} + \partial _{\mu} g _{\sigma} - \partial _{\delta} g _{\sigma} \right) written in terms of the {e ^{a} _{\mu}} , the spin connection can be written entirely in terms of the {e ^{a} _{\mu}} ,$$^\; =\; e^\; ^c\; e\_)$$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 ^{ab} _{\mu}} , one can use the same trick that was used to solve \nabla _{\rho} g _{\alpha} = 0 for the Christoffel symbols {\Gamma ^{\gamma}} _{\alpha \beta} . First contract the compatibility condition to give$$\_b\; \_c\; (\backslash partial\_\; +\; ^\; \backslash ;e\_)\; =\; 0.$$

Then, do a cyclic permutation of the free indices

a,b, and c , and add and subtract the three resulting equations:$$\backslash Omega\_\; +\; \backslash Omega\_\; -\; \backslash Omega\_\; +\; 2\; \_b\; \backslash omega\_\; =\; 0$$where we have used the definition \Omega _{bca} : = {e ^{\alpha}} _{b} {e ^{\beta}} _{c} \partial _{[\alpha} e _{\beta} . The solution for the spin connection is$$\backslash omega\_\; =\; \backslash tfrac\; ^b\; (\backslash Omega\_\; +\; \backslash Omega\_\; -\; \backslash Omega\_).$$

From this we obtain the same formula as before.

Applications

\gamma ^{a} are contracted onto vierbiens,$$\backslash gamma^a\; \_a\; (x)\; =\; \backslash gamma^\backslash mu\; (x).$$

We wish to construct a generally covariant Dirac equation. Under a flat tangent space Lorentz transformation the spinor transforms as$$\backslash psi\; \backslash mapsto\; e^\; \backslash psi$$

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 ^{ab} _{\mu}} that allows us to gauge the Lorentz group. The covariant derivative defined with the spin connection is,$$\backslash nabla\_\backslash mu\; \backslash psi\; =\; \backslash left(\backslash partial\_\backslash mu\; -\; \backslash tfrac\; ^\; \backslash sigma\_\backslash right)\; \backslash psi=\; \backslash left(\backslash partial\_\backslash mu\; -\; \backslash tfrac\; e^\; \backslash partial\_\; ^b\; \backslash sigma\_\backslash right)\; \backslash psi,$$and is a genuine tensor and Dirac's equation is rewritten as$$(i\; \backslash gamma^\backslash mu\; \backslash nabla\_\backslash mu\; -\; m)\; \backslash psi\; =\; 0.$$

The generally covariant fermion action couples fermions to gravity when added to the first order tetradic Palatini action,$$\backslash mathcal\; =\; -\; e\backslash ,\; \_a\; \_b\; ^\; [\backslash omega]\; +\; e\; \backslash overline\; (i\; \backslash gamma^\backslash mu\; \backslash nabla\_\backslash mu\; -\; m)\; \backslash psi$$where $e\; :=\; \backslash det\; ^a\; =\; \backslash sqrt$ and

{\Omega _{\mu} }^ 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 ^{i} _{a} (three-dimensional, spatial version of the tetrad). Here we extend the metric compatibility condition D _{a} q _{bc} = 0 to e ^{i} _{a} , that is, D _{a} e ^{i} _{b} = 0 and we obtain a formula similar to the one given above but for the spatial spin connection \Gamma ^{ij} _{a} .

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

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

See also

• Ashtekar variables
• Dirac operator
• Cartan connection
• Levi-Civita connection
• Ricci calculus
• Supergravity
• Torsion tensor
• Contorsion tensor
• Dirac equation in curved spacetime

References

• Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. (1976), "General relativity with spin and torsion: Foundations and prospects", Rev. Mod. Phys. 48, 393.
• Kibble, T.W.B. (1961), "Lorentz invariance and the gravitational field", J. Math. Phys. 2, 212.
• Poplawski, N.J. (2009), "Spacetime and fields", arXiv:0911.0334
• Sciama, D.W. (1964), "The physical structure of general relativity", Rev. Mod. Phys. 36, 463.

Notes and References

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