7.10 Aspects of general relativity

7.10.1 Introduction to general relativity

The general theory of relativity is one of the most beautiful and successful theories in classical physics. There is no disagreement between the theory and astrophysical and cosmological observations such as solar system tests, gravitational radiation from pulsars, gravitational red shifts, the recently discovered gravitational lens effect and so on. Readers not very familiar with general relativity may consult Berry (1989) or the primer by Price (1982).

Einstein proposed the following principles to construct the general theory of relativity

(I) Principle of General Relativity: All laws in physics take the same forms in any coordinate system. (II) Principle of Equivalence: There exists a coordinate system in which the effect of a gravitational field vanishes locally. (An observer in a freely falling lift does not feel gravity until it crashes.)

Any theory of gravity must reduce to Newton’s theory of gravity in the weak-field limit. In Newton’s theory, the gravitational potential $\Phi$ satisfies the Poisson equation

$$\Delta \Phi =4\pi G\rho \text{\hspace{1em}(7.200)}$$

where $\rho$ is the mass density. The Einstein equation generalizes this classical result so that the principle of general relativity is satisfied.

In general relativity, the gravitational potential is replaced by the components of the metric tensor. Then, instead of the LHS of (7.200), we have the Einstein tensor defined by

$$G_{\mu v}\equiv Ri{c}_{\mu v}-\frac{1}{2}{g}_{\mu v}\mathcal{R}.\text{\hspace{1em}(7.201)}$$

Similarly, the mass density is replaced by a more general object called the energy-momentum tensor $T_{\mu \nu }$ . The Einstein equation takes a very similar form to (7.200):

$$G_{\mu \nu }=8\pi G{T}_{\mu \nu }.\text{\hspace{1em}(7.202)}$$

The constant $8\pi G$ is chosen so that (7.202) reproduces the Newtonian result in the weak-field limit. The tensor $T_{\mu \nu }$ is obtained from the matter action by the variational principle. From Noether’s theorem, $T_{\mu \nu }$ must satisfy a conservation equation of the form ${\nabla }_{\mu }{T}^{\mu \nu }=0$ . A similar conservation law holds for $G_{\mu \nu }$ (but not for $Ri{c}_{\mu \nu }$ ). We shall see in the next subsection that the LHS of (7.202) is also obtained from the variational principle.

Exercise 7.24

Consider a metric

$$g_{00}=-1-\frac{2\Phi }{c^2}{g}_{0i}=0g_{ij}={\delta }_{ij}1\le i,j\le 3$$

and $T_{\mu \nu }$ given by $T_{00}=\rho c^2$ , $T_{0i}=T_{ij}=0$ which corresponds to dust at rest. Show that (7.202) reduces to the Poisson equation in the weak-field limit $(\Phi /c^2\ll 1)$ .

7.10.2 Einstein-Hilbert action

This and the next example are taken from Weinberg (1972). The general theory of relativity describes the dynamics of the geometry, that is, the dynamics of $g_{\mu \nu }$ . What is the action principle for this theory? As usual, we require that the relevant action should be a scalar. Moreover, it should contain the derivatives of $g_{\mu \nu }$ : $\int \sqrt{|g|}{\mathrm{d}}^{m}x$ cannot describe the dynamics of the metric. The simplest guess will be $S_{\mathrm{EH}}\propto \int \mathcal{R}\sqrt{|g|}{\mathrm{d}}^{m}x$ . Since $\mathcal{R}$ is a scalar and $\sqrt{|g|}\mathrm{d}{x}^1\mathrm{d}{x}^2\dots \mathrm{d}{x}^m$ is the invariant volume element, $S_{\mathrm{EH}}$ is a scalar. In the following, we show that $S_{\mathrm{EH}}$ indeed yields the Einstein equation under the variation with respect to the metric. Our connection is restricted to the Levi-Civita connection. We first prove a technical proposition.

Proposition 7.2

Let $(M,g)$ be a (pseudo-)Riemannian manifold. Under the variation $g_{\mu \nu }\to g_{\mu \nu }+\delta g_{\mu \nu },g^{\mu \nu },g$ and $Ri{c}_{\mu \nu }$ change as

(a) $\delta g^{\mu \nu }=-g^{\mu \kappa }{g}^{\lambda \nu }\delta g_{\kappa \lambda }$ (7.203)
(b) $\delta g=g{g}^{\mu \nu }\delta g_{\mu \nu },$ 8√g|g=ggμv (7.204)
(c) $\delta Ri{c}_{\mu \nu }={\nabla }_{\kappa }\delta {\Gamma }_{\nu \mu }^{\kappa }-{\nabla }_{\nu }\delta {\Gamma }_{\kappa \mu }^{\kappa }$ (Palatini identity). (7.205)

Proof

(a) From $g_{\kappa \lambda }{g}^{\lambda \nu }={\delta }_{\kappa }^{\nu }$ , it follows that

$$0=\delta \left(g_{\kappa \lambda }{g}^{\lambda \nu }\right)=\delta g_{\kappa \lambda }{g}^{\lambda \nu }+g_{\kappa \lambda }\delta g^{\lambda \nu }.$$

Multiplying by $g^{\mu \kappa }$ we find that $\delta g^{\mu \nu }=-g^{\mu \kappa }{g}^{\lambda \nu }\delta g_{\kappa \lambda }$ .

(b) We first note the matrix identity $\ln (\det g_{\mu \nu })=\mathrm{tr}(\ln g_{\mu \nu })$ . This can be proved by diagonalizing $g_{\mu \nu }$ . Under the variation $\delta g_{\mu \nu }$ , the LHS becomes $\delta g\cdot g^{-1}$ while the RHS yields $g^{\mu \nu }\cdot \delta g_{\mu \nu }$ , hence $\delta g=g{g}^{\mu \nu }\delta g_{\mu \nu }$ . The rest of (7.204) is easily derived from this.
(c) Let $\Gamma$ and $\tilde{\Gamma }$ be two connections. The difference $\delta \Gamma \equiv \tilde{\Gamma }-\Gamma$ is a tensor of type (1, 2), see exercise 7.5. In the present case, we take $\tilde{\Gamma }$ to be a connection associated with $g+\delta g$ and $\Gamma$ with $g$ . We will work in the normal coordinate system in which $\Gamma \equiv 0$ (of course $\partial \Gamma \ne 0$ in general); see section 7.4. We find

$$\delta Ri{c}_{\mu \nu }={\partial }_{\kappa }\delta {\Gamma }_{\nu \mu }^{\kappa }-{\partial }_{\nu }\delta {\Gamma }_{\kappa \mu }^{\kappa }={\nabla }_{\kappa }\delta {\Gamma }_{\nu \mu }^{\kappa }-{\nabla }_{\nu }{\Gamma }_{\kappa \mu }^{\kappa }.$$

[The reader should verify the second equality.] Since both sides are tensors, this is valid in any coordinate system.

We define the Einstein-Hilbert action by

$$S_{\mathrm{EH}}\equiv \frac{1}{16\pi G}\int \mathcal{R}\sqrt{-g}{\mathrm{d}}^{4}x.\text{\hspace{1em}(7.206)}$$

The constant factor $1/16\pi G$ is introduced to reproduce the Newtonian limit when matter is added; see (7.214). We prove that $\delta S_{\mathrm{EH}}=0$ leads to the vacuum Einstein equation. Under the variation $g\to g+\delta g$ such that $\delta g\to 0$ as $|x|\to 0$ , the integrand changes as

$$\begin{array}{rl}& \delta (\mathcal{R}\sqrt{-g})=\delta (g^{\mu \nu }Ri{c}_{\mu \nu }\sqrt{-g})\\ & =\delta g^{\mu v}Ri{c}_{\mu v}\sqrt{-g}+g^{\mu v}\delta Ri{c}_{\mu v}\sqrt{-g}+\mathcal{R}\delta (\sqrt{-g})\\ & =-g^{\mu \kappa }{g}^{\lambda \nu }\delta g_{\kappa \lambda }Ri{c}_{\mu \nu }\sqrt{-g}\\ & +g^{\mu \nu }\left({\nabla }_{\kappa }\delta {\Gamma }_{\nu \mu }^{\kappa }-{\nabla }_{\nu }{\Gamma }_{\kappa \mu }^{\kappa }\right)\sqrt{-g}+\frac{1}{2}\mathcal{R}\sqrt{-g}{g}^{\mu \nu }\delta g_{\mu \nu }.\end{array}$$

We note that the second term is a total divergence,

$$\begin{array}{rl}& {\nabla }_{\kappa }\left(g^{\mu \nu }\delta {\Gamma }_{\nu \mu }^{\kappa }\sqrt{-g}\right)-{\nabla }_{\nu }\left(g^{\mu \nu }\delta {\Gamma }_{\kappa \mu }^{\kappa }\sqrt{-g}\right)\\ & ={\partial }_{\kappa }\left(g^{\mu \nu }\delta {\Gamma }_{\mu \nu }^{\kappa }\sqrt{-g}\right)-{\partial }_{\nu }\left(g^{\mu \nu }\delta {\Gamma }_{\kappa \mu }^{\kappa }\sqrt{-g}\right)\end{array}$$

and hence does not contribute to the variation. From the remaining terms we have

$$\delta S_{\mathrm{EH}}=\frac{1}{16\pi G}\int \left(-Ri{c}^{\mu v}+\frac{1}{2}\mathcal{R}{g}^{\mu v}\right)\delta g_{\mu v}\sqrt{-g}{d}^{4}x.\text{\hspace{1em}(7.207)}$$

If we require that $\delta S_{\mathrm{EH}}=0$ under any variation $\delta g$ , we obtain the vacuum Einstein equation,

$$G_{\mu \nu }=Ri{c}_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }\mathcal{R}=0\text{\hspace{1em}(7.208)}$$

where the symmetric tensor $G$ is called the Einstein tensor.

So far we have considered the gravitational field only. Suppose there exists matter described by an action

$$S_{\mathrm{M}}\equiv \int \mathcal{L}(\varphi )\sqrt{-g}{\mathrm{d}}^{4}x\text{\hspace{1em}(7.209)}$$

where $\mathcal{L}(\varphi )$ is the Lagrangian density of the theory. Typical examples are the real scalar field and the Maxwell fields,

$$S_{\mathrm{S}}\equiv -\frac{1}{2}\int \left[g^{\mu v}{\partial }_{\mu }\varphi {\partial }_v\varphi +m^2{\varphi }^2\right]\sqrt{-g}{\mathrm{d}}^{4}x\text{\hspace{1em}(7.210a)}$$

$$S_{\mathrm{ED}}\equiv -\frac{1}{4}\int F_{\mu v}{F}^{\mu v}\sqrt{-g}{\mathrm{d}}^{4}x\text{\hspace{1em}(7.210b)}$$

where $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }={\nabla }_{\mu }{A}_{\nu }-{\nabla }_{\nu }{A}_{\mu }$ . If the matter action changes by $\delta S_{\mathrm{M}}$ under $\delta g$ , the energy-momentum tensor $T^{\mu \nu }$ is defined by

$$\delta S_{\mathrm{M}}=\frac{1}{2}\int T^{\mu v}\delta g_{\mu v}\sqrt{-g}{\mathrm{d}}^{4}x.\text{\hspace{1em}(7.211)}$$

Since $\delta g_{\mu \nu }$ is symmetric, $T^{\mu \nu }$ is also taken to be so. For example, $T_{\mu \nu }$ of a real scalar field is given by

$$\begin{array}{rl}{T}_{\mu \nu }(x)& =2\frac{1}{\sqrt{-g}}\frac{\delta }{\delta g^{\mu \nu }(x)}{S}_S\\ & ={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -\frac{1}{2}{g}_{\mu \nu }\left(g^{\kappa \lambda }{\partial }_{\kappa }\varphi {\partial }_{\lambda }\varphi +m^2{\varphi }^2\right).\end{array}\text{\hspace{1em}(7.212)}$$

Suppose we have a gravitational field coupled with a matter field whose action is $S_{\mathrm{M}}$ . Now our action principle is

$$\delta \left(S_{\mathrm{EH}}+S_{\mathrm{M}}\right)=0\text{\hspace{1em}(7.213)}$$

under $g\to g+\delta g$ . From (7.207) and (7.211), we obtain the Einstein equation

$$G_{\mu v}=8\pi G{T}_{\mu v}.\text{\hspace{1em}(7.214)}$$

Exercise 7.25

We may add an extra scalar to the scalar curvature without spoiling the invariance of the action. For example, we can add a constant called the cosmological constant $\Lambda$ ,

$${\tilde{S}}_{\mathrm{EH}}=\frac{1}{16\pi G}{\int }_M(\mathcal{R}+\Lambda )\sqrt{-g}{\mathrm{d}}^{4}x.\text{\hspace{1em}(7.215)}$$

Write down the vacuum Einstein equation. Other possible scalars may be such terms as ${\mathcal{R}}^2$ , $Ri{c}^{\mu \nu }Ri{c}_{\mu \nu }$ or $R_{\kappa \lambda \mu \nu }{R}^{\kappa \lambda \mu \nu }$ .

7.10.3 Spinors in curved spacetime

For concreteness, we consider a Dirac spinor $\psi$ in a four-dimensional Lorentz manifold $M$ . The vierbein $e_{\mu }^{\alpha }$ defined by

$$g_{\mu \nu }=e_{\mu }^{\alpha }{e}_{\nu }^{\beta }{\eta }_{\alpha \beta }\text{\hspace{1em}(7.216)}$$

defines an orthonormal frame $\{{\hat{\theta }}^{\alpha }=e_{\mu }^{\alpha }\mathrm{d}{x}^{\mu }\}$ at each point $p\in M$ . As noted before, $\alpha ,\beta ,\gamma ,\dots$ are the local orthonormal indices while $\mu ,\nu ,\lambda ,\dots$ are the coordinate indices. With respect to this frame, the Dirac matrices ${\gamma }^{\alpha }=e_{\mu }^{\alpha }{\gamma }^{\mu }$ satisfy $\{{\gamma }^{\alpha },{\gamma }^{\beta }\}=2{\eta }^{\alpha \beta }$ . Under a local Lorentz transformation ${\Lambda }_{\beta }^{\alpha }(p)$ , the Dirac spinor transforms as

$$\psi (p)\to \rho (\Lambda )\psi (p)\bar{\psi }(p)\to \bar{\psi }(p)\rho (\Lambda {)}^{-1}\text{\hspace{1em}(7.217)}$$

where $\bar{\psi }\equiv {\psi }^{†}{\gamma }^0$ and $\rho (\Lambda )$ is the spinor representation of $\Lambda$ . To construct an invariant action, we seek a covariant derivative ${\nabla }_{\alpha }\psi$ which is a local Lorentz vector and transforms as a spinor,

$${\nabla }_{\alpha }\psi \to \rho (\Lambda ){\Lambda }_{\alpha }^{\beta }{\nabla }_{\beta }\psi .\text{\hspace{1em}(7.218)}$$

If we find such a ${\nabla }_{\alpha }\psi$ , an invariant Lagrangian may be given by

$$\mathcal{L}=\bar{\psi }(\mathrm{i}{\gamma }^{\alpha }{\nabla }_{\alpha }+m)\psi \text{\hspace{1em}(7.219)}$$

$m$ being the mass of $\psi$ . We note that $e_{\alpha }^{\mu }{\partial }_{\mu }\psi$ transforms under $\Lambda (p)$ as

$$e_{\alpha }^{\mu }{\partial }_{\mu }\psi \to {\Lambda }_{\alpha }^{\beta }{e}_{\beta }^{\mu }{\partial }_{\mu }\rho (\Lambda )\psi ={\Lambda }_{\alpha }^{\eta }{e}_{\beta }^{\mu }[\rho (\Lambda ){\partial }_{\mu }\psi +{\partial }_{\mu }\rho (\Lambda )\psi ].\text{\hspace{1em}(7.220)}$$

Suppose ${\nabla }_{\alpha }$ is of the form

$${\nabla }_{\alpha }\psi =e_{\alpha }^{\mu }\left[{\partial }_{\mu }+{\Omega }_{\mu }\right]\psi .\text{\hspace{1em}(7.221)}$$

From (7.218) and (7.220), we find that ${\Omega }_{\mu }$ satisfies

$${\Omega }_{\mu }\to \rho (\Lambda ){\Omega }_{\mu }\rho (\Lambda {)}^{-1}-{\partial }_{\mu }\rho (\Lambda )\rho (\Lambda {)}^{-1}.\text{\hspace{1em}(7.222)}$$

To find the explicit form of ${\Omega }_{\mu }$ , we consider an infinitesimal local Lorentz transformation ${\Lambda }_{\alpha }^{\beta }(p)={\delta }_{\alpha }^{\beta }+{\epsilon }_{\alpha }^{\beta }(p)$ . The Dirac spinor transforms as

$$\psi \to \exp \left[\frac{1}{2}\mathrm{i}{\epsilon }^{\alpha \beta }{\Sigma }_{\alpha \beta }\right]\psi \simeq \left[1+\frac{1}{2}\mathrm{i}{\epsilon }^{\alpha \beta }{\Sigma }_{\alpha \beta }\right]\psi \text{\hspace{1em}(7.223)}$$

where ${\Sigma }_{\alpha \beta }\equiv \frac{1}{4}\mathrm{i}\left[{\gamma }_{\alpha },{\gamma }_{\beta }\right]$ is the spinor representation of the generators of the Lorentz transformation. ${\Sigma }_{\alpha \beta }$ satisfies the $\mathfrak{o}(1,3)$ Lie algebra

$$\mathrm{i}\left[{\Sigma }_{\alpha \beta },{\Sigma }_{\gamma \delta }\right]={\eta }_{\gamma \beta }{\Sigma }_{\alpha \delta }-{\eta }_{\gamma \alpha }{\Sigma }_{\beta \delta }+{\eta }_{\delta \beta }{\Sigma }_{\gamma \alpha }-{\eta }_{\delta \alpha }{\Sigma }_{\gamma \beta }.\text{\hspace{1em}(7.224)}$$

Under the same Lorentz transformation, ${\Omega }_{\mu }$ transforms as

$$\begin{array}{rl}& {\Omega }_{\mu }\to (1+\frac{1}{2}\mathrm{i}{\epsilon }^{\alpha \beta }{\Sigma }_{\alpha \beta }){\Omega }_{\mu }(1-\frac{1}{2}\mathrm{i}{\epsilon }^{\gamma \delta }{\Sigma }_{\gamma \delta })-\frac{1}{2}\mathrm{i}{\partial }_{\mu }{\epsilon }^{\alpha \beta }{\Sigma }_{\alpha \beta }(1-\frac{1}{2}\mathrm{i}{\epsilon }^{\gamma \delta }{\Sigma }_{\gamma \delta })\\ & ={\Omega }_{\mu }+\frac{1}{2}\mathrm{i}{\epsilon }^{\alpha \beta }\left[{\Sigma }_{\alpha \beta },{\Omega }_{\mu }\right]-\frac{1}{2}\mathrm{i}{\partial }_{\mu }{\epsilon }^{\alpha \beta }{\Sigma }_{\alpha \beta }.\end{array}\text{\hspace{1em}(7.225)}$$

We recall that the connection one-form ${\omega }_{\beta }^{\alpha }$ transforms under an infinitesimal Lorentz transformation as (see (7.152))

$${\omega }_{\beta }^{\alpha }\to {\omega }_{\beta }^{\alpha }+{\epsilon }_{\gamma }^{\alpha }{\omega }_{\beta }^{\gamma }-{\omega }_{\gamma }^{\alpha }{\epsilon }_{\beta }^{\gamma }-\mathrm{d}{\epsilon }_{\beta }^{\alpha }\text{\hspace{1em}(7.226a)}$$

or in components,

$${\Gamma }_{\mu \beta }^{\alpha }\to {\Gamma }_{\mu \beta }^{\alpha }+{\epsilon }_{\gamma }^{\alpha }{\Gamma }_{\mu \beta }^{\gamma }-{\Gamma }_{\mu \gamma }^{\alpha }{\epsilon }_{\beta }^{\gamma }-{\partial }_{\mu }{\epsilon }_{\beta }^{\alpha }.\text{\hspace{1em}(7.226b)}$$

From (7.224), (7.225) and (7.226b), we find that the combination

$${\Omega }_{\mu }\equiv \frac{1}{2}\mathrm{i}{\Gamma }^{\alpha }{}_{\mu }{}^{\beta }{\Sigma }_{\alpha \beta }=\frac{1}{2}\mathrm{i}{e}_{\nu }^{\alpha }{\nabla }_{\mu }{e}^{\beta \nu }{\Sigma }_{\alpha \beta }\text{\hspace{1em}(7.227)}$$

satisfies the transformation property (7.222). In fact,

$$\begin{array}{rl}& \frac{1}{2}\mathrm{i}{\Gamma }_{\mu }^{\alpha \beta }{\Sigma }_{\alpha \beta }\to \frac{1}{2}\mathrm{i}({\Gamma }_{\mu }^{\alpha \beta }+{\epsilon }_{\gamma }^{\alpha }{\Gamma }_{\mu }^{\gamma \beta }-{\Gamma }_{\mu \gamma }^{\alpha }{\epsilon }^{\gamma \beta }-{\partial }_{\mu }{\epsilon }^{\alpha \beta }){\Sigma }_{\alpha \beta }\\ & =\frac{1}{2}\mathrm{i}{\Gamma }_{\mu }^{\alpha \beta }{\Sigma }_{\alpha \beta }+\frac{1}{2}\mathrm{i}({\epsilon }_{\gamma }^{\alpha }{\Gamma }_{\mu }^{\gamma \beta }{\Sigma }_{\alpha \beta }-{\Gamma }_{\mu \gamma }^{\alpha }{\epsilon }^{\gamma \beta }{\Sigma }_{\alpha \beta })\\ & -\frac{1}{2}\mathrm{i}{\partial }_{\mu }{\epsilon }^{\alpha \beta }{\Sigma }_{\alpha \beta }\\ & =\frac{1}{2}\mathrm{i}{\Gamma }_{\mu }^{\alpha \beta }{\Sigma }_{\alpha \beta }+\frac{1}{2}\mathrm{i}{\epsilon }^{\alpha \beta }[{\Sigma }_{\alpha \beta },\frac{1}{2}\mathrm{i}{\Gamma }_{\mu }^{\gamma \delta }{\Sigma }_{\gamma \delta }]-\frac{1}{2}\mathrm{i}{\partial }_{\mu }{\epsilon }^{\alpha \beta }{\Sigma }_{\alpha \beta }.\end{array}$$

We finally obtain the Lagrangian which is a scalar both under coordinate changes and local Lorentz rotations,

$$\mathcal{L}\equiv \bar{\psi }\left[\mathrm{i}{\gamma }^{\alpha }{e}_{\alpha }^{\mu }\left({\partial }_{\mu }+\frac{1}{2}\mathrm{i}{\Gamma }_{\mu }{}^{\beta }{}^{\gamma }{\Sigma }_{\beta \gamma }\right)+m\right]\psi \text{\hspace{1em}(7.228)}$$

and the scalar action

$$S_{\psi }\equiv {\int }_M{d}^{4}x\sqrt{-g}\bar{\psi }\left[i{\gamma }^{\alpha }{e}_{\alpha }^{\mu }\left({\partial }_{\mu }+\frac{1}{2}i{\Gamma }_{\mu }{}^{\beta }{}^{\gamma }{\Sigma }_{\beta \gamma }\right)+m\right]\psi .\text{\hspace{1em}(7.229a)}$$

If $\psi$ is coupled to the gauge field $\mathcal{A}$ , the action is given by

$$S_{\psi }={\int }_M{d}^{4}x\sqrt{-g}\bar{\psi }[i{\gamma }^{\alpha }{e}_{\alpha }^{\mu }\left({\partial }_{\mu }+{\mathcal{A}}_{\mu }+\frac{1}{2}i{\Gamma }_{\mu }{}^{\beta }{}^{\gamma }{\Sigma }_{\beta \gamma }\right)+m]\psi .\text{\hspace{1em}(7.229b)}$$

It is interesting to note that the spin connection term vanishes if $\dim M=2$ . To see this, we rewrite (7.229a) as

$$S_{\psi }=\frac{1}{2}{\int }_M{\mathrm{d}}^{2}x\sqrt{-g}\bar{\psi }[\mathrm{i}{\gamma }^{\mu }\overleftrightarrow{{\partial }_{\mu }}+\frac{1}{2}\mathrm{i}{\Gamma }^{\beta }{}_{\mu }{}^{\gamma }\{\mathrm{i}{\gamma }^{\mu },{\Sigma }_{\beta \gamma }\}+m]\psi \text{\hspace{1em}(7.229a’)}$$

where ${\gamma }^{\mu }={\gamma }^{\alpha }{e}_{\alpha }^{\mu }$ and we have added total derivatives to the Lagrangian to make it Hermitian. The non-vanishing components of $\Sigma$ are ${\Sigma }_{01}\propto [{\gamma }_0,{\gamma }_1]\propto {\gamma }_3$ where ${\gamma }_3$ is the two-dimensional analogue of ${\gamma }_5$ . Since $\{{\gamma }^{\mu },{\gamma }_3\}=0$ , the spin connection term drops out from $S_{\psi }$ .

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