7.8 Non-coordinate bases

7.8.1 Definitions

In the coordinate basis, $T_{p}M$ is spanned by $\{e_{\mu }\}=\{\partial /\partial x^{\mu }\}$ and $T_p^{\ast }M$ by $\{\mathrm{d}{x}^{\mu }\}$ . If, moreover, $M$ is endowed with a metric $g$ , there may be an alternative choice. Let us consider the linear combination,

$${\hat{e}}_{\alpha }=e_{\alpha }^{\mu }\frac{\partial }{\partial x^{\mu }}\{e_{\alpha }^{\mu }\}\in \mathrm{GL}(m,\mathbb{R})\text{\hspace{1em}(7.131)}$$

where $\det e_{\alpha }^{\mu }>0$ . In other words, $\{{\hat{e}}_{\alpha }\}$ is the frame of basis vectors which is obtained by a $\mathrm{GL}(m,\mathbb{R})$ -rotation of the basis $\{e_{\mu }\}$ preserving the orientation. We require that $\{{\hat{e}}_{\alpha }\}$ be orthonormal with respect to $g$ ,

$$g\left({\hat{e}}_{\alpha },{\hat{e}}_{\beta }\right)=e_{\alpha }^{\mu }{e}_{\beta }^v{g}_{\mu v}={\delta }_{\alpha \beta }.\text{\hspace{1em}(7.132a)}$$

If the manifold is Lorentzian, ${\delta }_{\alpha \beta }$ should be replaced by ${\eta }_{\alpha \beta }$ . We easily reverse (7.132a),

$$g_{\mu v}=e_{\mu }^{\alpha }{e}_v^{\beta }{\delta }_{\alpha \beta }\text{\hspace{1em}(7.132b)}$$

where $e^{\alpha }{}_{\mu }$ is the inverse of $e_{\alpha }^{\mu }$ ; $e^{\alpha }{}_{\mu }{e}_{\alpha }^{\nu }={\delta }_{\mu }^{\nu }$ , $e^{\alpha }{}_{\mu }{e}_{\beta }^{\mu }={\delta }^{\alpha }{}_{\beta }$ . [We have used the same symbols for a matrix and its inverse. So long as the indices are written explicitly it does not cause confusion.] Since a vector $V$ is independent of the basis chosen, we have $V=V^{\mu }{e}_{\mu }=V^{\alpha }{\hat{e}}_{\alpha }=V^{\alpha }{e}_{\alpha }^{\mu }{e}_{\mu }$ . It follows that

$$V^{\mu }=V^{\alpha }{e}_{\alpha }^{\mu }{V}^{\alpha }=e_{\mu }^{\alpha }{V}^{\mu }.\text{\hspace{1em}(7.133)}$$

Let us introduce the dual basis $\{{\hat{\theta }}^{\alpha }\}$ defined by $⟨{\hat{\theta }}^{\alpha },{\hat{e}}_{\beta }⟩={\delta }_{\beta }^{\alpha }$ . ${\hat{\theta }}^{\alpha }$ is given by

$${\hat{\theta }}^{\alpha }=e_{\mu }^{\alpha }\mathrm{d}{x}^{\mu }.\text{\hspace{1em}(7.134)}$$

In terms of $\{{\hat{\theta }}^{\alpha }\}$ , the metric is

$$g=g_{\mu \nu }\mathrm{d}{x}^{\mu }\otimes \mathrm{d}{x}^{\nu }={\delta }_{\alpha \beta }{\hat{\theta }}^{\alpha }\otimes {\hat{\theta }}^{\beta }.\text{\hspace{1em}(7.135)}$$

The bases $\{{\hat{e}}_{\alpha }\}$ and $\{{\hat{\theta }}^{\alpha }\}$ are called the non-coordinate bases. We use $\kappa ,\lambda ,\mu ,\nu ,\dots$ ( $\alpha ,\beta ,\gamma ,\delta ,\dots$ ) to denote the coordinate (non-coordinate) basis. The coefficients $e_{\alpha }^{\mu }$ are called the vierbeins if the space is four dimensional and vielbeins if it is many dimensional. The non-coordinate basis has a non-vanishing Lie bracket. If the $\{{\hat{e}}_{\alpha }\}$ are given by (7.131), they satisfy

$${\left[{\hat{e}}_{\alpha },{\hat{e}}_{\beta }\right]|}_p=c_{\alpha \beta }^{\gamma }(p){\hat{e}}_{\gamma }{|}_p\text{\hspace{1em}(7.136a)}$$

where

$$c_{\alpha \beta }^{\gamma }(p)=e_v^{\gamma }\left[e_{\alpha }^{\mu }{\partial }_{\mu }{e}_{\beta }^v-e_{\beta }^{\mu }{\partial }_{\mu }{e}_{\alpha }^v\right](p).\text{\hspace{1em}(7.136b)}$$

Example 7.13

The standard metric on $S^2$ is

$$g=\mathrm{d}\theta \otimes \mathrm{d}\theta +{\sin }^2\theta \mathrm{d}\varphi \otimes \mathrm{d}\varphi ={\hat{\theta }}^1\otimes {\hat{\theta }}^1+{\hat{\theta }}^2\otimes {\hat{\theta }}^2\text{\hspace{1em}(7.137)}$$

where ${\hat{\theta }}^1=\mathrm{d}\theta$ and ${\hat{\theta }}^2=\sin \theta \mathrm{d}\varphi$ . The ‘zweibeins’ are

$$\begin{array}{rl}& e_{\theta }^1=1e_{\varphi }^1=0\\ & e_{\theta }^2=0e_{\varphi }^2=\sin \theta .\end{array}\text{\hspace{1em}(7.138)}$$

The non-vanishing components of $c_{\alpha \beta }^{\gamma }$ are $c_{12}^2=-c_{21}^2=-\cot \theta$

Exercise 7.19

(a) Verify the identities,

$${\delta }^{\alpha \beta }=g^{\mu \nu }{e}_{\mu }^{\alpha }{e}_{\nu }^{\beta },g^{\mu \nu }={\delta }^{\alpha \beta }{e}_{\alpha }^{\mu }{e}_{\beta }^{\nu }.\text{\hspace{1em}(7.139)}$$

(b) Let ${\gamma }^{\alpha }$ be the Dirac matrices in Minkowski spacetime, which satisfy $\{{\gamma }^{\alpha },{\gamma }^{\beta }\}=2{\eta }^{\alpha \beta }$ . Define the curved spacetime counterparts of the Dirac matrices by ${\gamma }^{\mu }\equiv e_{\alpha }^{\mu }{\gamma }^{\alpha }$ . Show that

$$\left\{{\gamma }^{\mu },{\gamma }^v\right\}=2g^{\mu v}.\text{\hspace{1em}(7.140)}$$

7.8.2 Cartan’s structure equations

In section 7.3 the curvature tensor $R$ and the torsion tensor $T$ have been defined by

$$\begin{array}{rl}& R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z\\ & T(X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y].\end{array}$$

Let $\{{\hat{e}}_{\alpha }\}$ be the non-coordinate basis and $\{{\hat{\theta }}^{\alpha }\}$ the dual basis. The vector fields $\{{\hat{e}}_{\alpha }\}$ satisfy $[{\hat{e}}_{\alpha },{\hat{e}}_{\beta }]=c_{\alpha \beta }{}^{\gamma }{\hat{e}}_{\gamma }$ . Define the connection coefficients with respect to the basis $\{{\hat{e}}_{\alpha }\}$ by

$${\nabla }_{\alpha }{\hat{e}}_{\beta }\equiv {\nabla }_{{\hat{e}}_{\alpha }}{\hat{e}}_{\beta }={\Gamma }_{\alpha \beta }^{\gamma }{\hat{e}}_{\gamma }.\text{\hspace{1em}(7.141)}$$

Let ${\hat{e}}_{\alpha }=e_{\alpha }^{\mu }{e}_{\mu }$ . Then (7.141) becomes $e_{\alpha }^{\mu }({\partial }_{\mu }{e}_{\beta }{}^{\nu }+e_{\beta }^{\lambda }{\Gamma }_{\mu \lambda }^v)e_{\nu }={\Gamma }_{\alpha \beta }^{\gamma }{e}_{\gamma }{}^{\nu }{e}_{\nu }$ , from which we find that

$${\Gamma }_{\alpha \beta }^{\gamma }=e_v^{\gamma }{e}_{\alpha }^{\mu }\left({\partial }_{\mu }{e}_{\beta }^v+e_{\beta }^{\lambda }{\Gamma }_{\mu \lambda }^{\nu }\right)=e_v^{\gamma }{e}_{\alpha }^{\mu }{\nabla }_{\mu }{e}_{\beta }^v.\text{\hspace{1em}(7.142)}$$

The components of $T$ and $R$ in this basis are given by

$$\begin{array}{rl}{T}_{\beta \gamma }^{\alpha }& =⟨{\hat{\theta }}^{\alpha },T({\hat{e}}_{\beta },{\hat{e}}_{\gamma })⟩=⟨{\hat{\theta }}^{\alpha },{\nabla }_{\beta }{\hat{e}}_{\gamma }-{\nabla }_{\gamma }{\hat{e}}_{\beta }-[{\hat{e}}_{\beta },{\hat{e}}_{\gamma }]⟩\\ & ={\Gamma }_{\beta \gamma }^{\alpha }-{\Gamma }_{\gamma \beta }^{\alpha }-c_{\beta \gamma }^{\alpha }.\end{array}\text{\hspace{1em}(7.143)}$$

$$\begin{array}{rl}{R}_{\beta \gamma \delta }^{\alpha }& =⟨{\hat{\theta }}^{\alpha },{\nabla }_{\gamma }{\nabla }_{\delta }{\hat{e}}_{\beta }-{\nabla }_{\delta }{\nabla }_{\gamma }{\hat{e}}_{\beta }-{\nabla }_{[{\hat{e}}_{\gamma },{\hat{e}}_{\delta }]}{\hat{e}}_{\beta }⟩\\ & =⟨{\hat{\theta }}^{\alpha },{\nabla }_{\gamma }({\Gamma }_{\delta \beta }^{\epsilon }{\hat{e}}_{\epsilon })-{\nabla }_{\delta }{\Gamma }_{\gamma \beta }^{\epsilon }{\hat{e}}_{\epsilon })-c_{\gamma \delta }^{\epsilon }{\nabla }_{\epsilon }{\hat{e}}_{\beta }⟩\\ & ={\hat{e}}_{\gamma }\left[{\Gamma }_{\delta \beta }^{\alpha }\right]-{\hat{e}}_{\delta }\left[{\Gamma }_{\gamma \beta }^{\alpha }\right]+{\Gamma }_{\delta \beta }^{\epsilon }{\Gamma }_{\gamma \epsilon }^{\alpha }-{\Gamma }_{\gamma \beta }^{\epsilon }{\Gamma }_{\delta \epsilon }^{\alpha }-c_{\gamma \delta }^{\epsilon }{\Gamma }_{\epsilon \beta }^{\alpha }.\end{array}\text{\hspace{1em}(7.144)}$$

We define a matrix-valued one-form $\{{\omega }_{\beta }^{\alpha }\}$ called the connection one-form by

$${\omega }_{\beta }^{\alpha }\equiv {\Gamma }_{\gamma \beta }^{\alpha }{\hat{\theta }}^{\gamma }.\text{\hspace{1em}(7.145)}$$

Theorem 7.3

The connection one-form ${\omega }_{\beta }^{\alpha }$ satisfies Cartan’s structure equations,

$$\mathrm{d}{\hat{\theta }}^{\alpha }+{\omega }_{\beta }^{\alpha }\wedge {\hat{\theta }}^{\beta }=T^{\alpha }\text{\hspace{1em}(7.146a)}$$

$$\mathrm{d}{\omega }_{\beta }^{\alpha }+{\omega }_{\gamma }^{\alpha }\wedge {\omega }_{\beta }^{\gamma }=R_{\beta }^{\alpha }\text{\hspace{1em}(7.146b)}$$

where we have introduced the torsion two-form $T^{\alpha }\equiv \frac{1}{2}{T}_{\beta \gamma }^{\alpha }{\hat{\theta }}^{\beta }\wedge {\hat{\theta }}^{\gamma }$ and the curvature two-form $R_{\beta }^{\alpha }\equiv \frac{1}{2}{R}_{\beta \gamma \delta }^{\alpha }{\hat{\theta }}^{\gamma }\wedge {\hat{\theta }}^{\delta }$ .

Proof

Let the LHS of (7.146a) act on the basis vectors ${\hat{e}}_{\gamma }$ and ${\hat{e}}_{\delta }$ ,

$$\begin{array}{rl}& \mathrm{d}{\hat{\theta }}^{\alpha }({\hat{e}}_{\gamma },{\hat{e}}_{\delta })+[⟨{\omega }_{\beta }^{\alpha },{\hat{e}}_{\gamma }⟩⟨{\hat{\theta }}^{\beta },{\hat{e}}_{\delta }⟩-⟨{\hat{\theta }}^{\beta },{\hat{e}}_{\gamma }⟩⟨{\omega }_{\beta }^{\alpha },{\hat{e}}_{\delta }⟩]\\ & =\left\{{\hat{e}}_{\gamma }\left[⟨{\hat{\theta }}^{\alpha },{\hat{e}}_{\delta }⟩\right]-{\hat{e}}_{\delta }\left[⟨{\hat{\theta }}^{\alpha },{\hat{e}}_{\gamma }⟩\right]-⟨{\hat{\theta }}^{\alpha },\left[{\hat{e}}_{\gamma },{\hat{e}}_{\delta }\right]⟩\right\}+\left\{⟨{\omega }_{\delta }^{\alpha },{\hat{e}}_{\gamma }⟩-⟨{\omega }_{\gamma }^{\alpha },{\hat{e}}_{\delta }⟩\right\}\\ & =-c_{\gamma \delta }^{\alpha }+{\Gamma }_{\gamma \delta }^{\alpha }-{\Gamma }_{\delta \gamma }^{\alpha }=T_{\gamma \delta }^{\alpha }\end{array}$$

where use has been made of (5.70). The RHS acting on ${\hat{e}}_{\gamma }$ and ${\hat{e}}_{\delta }$ yields

$$\frac{1}{2}{T}_{\beta \epsilon }^{\alpha }[⟨{\hat{\theta }}^{\beta },{\hat{e}}_{\gamma }⟩⟨{\hat{\theta }}^{\epsilon },{\hat{e}}_{\delta }⟩-⟨{\hat{\theta }}^{\epsilon },{\hat{e}}_{\gamma }⟩⟨{\hat{\theta }}^{\beta },{\hat{e}}_{\delta }⟩]=T_{\gamma \delta }^{\alpha }$$

which verifies (7.146a).

Equation (7.146b) may be proved similarly (exercise).

Taking the exterior derivative of (7.146a) and (7.146b), we have the Bianchi identities

$$\mathrm{d}{T}^{\alpha }+{\omega }_{\beta }^{\alpha }\wedge T^{\beta }=R_{\beta }^{\alpha }\wedge {\hat{\theta }}^{\beta }\text{\hspace{1em}(7.147a)}$$

$$\mathrm{d}{R}_{\beta }^{\alpha }+{\omega }_{\gamma }^{\alpha }\wedge R_{\beta }^{\gamma }-R_{\gamma }^{\alpha }\wedge {\omega }_{\beta }^{\gamma }=0.\text{\hspace{1em}(7.147b)}$$

These are the non-coordinate basis versions of (7.81a) and (7.81b).

7.8.3 The local frame

In an $m$ -dimensional Riemannian manifold, the metric tensor $g_{\mu \nu }$ has $m(m+1)/2$ degrees of freedom while the vielbein $e_{\alpha }^{\mu }$ has $m^2$ degrees of freedom. There are many non-coordinate bases which yield the same metric, $g$ , each of which is related to the other by the local orthogonal rotation,

$${\hat{\theta }}^{\alpha }⟶{\hat{\theta }}^{\prime \alpha }(p)={\Lambda }_{\beta }^{\alpha }(p){\hat{\theta }}^{\beta }(p)\text{\hspace{1em}(7.148)}$$

at each point $p$ . The vielbein transforms as

$$e_{\mu }^{\alpha }(p)⟶e_{\mu }^{\prime \alpha }(p)={\Lambda }_{\beta }^{\alpha }(p)e_{\mu }^{\beta }(p).\text{\hspace{1em}(7.149)}$$

Unlike $\kappa ,\lambda ,\mu ,\nu ,\dots$ which transform under coordinate changes, the indices $\alpha ,\beta ,\gamma ,\dots$ transform under the local orthogonal rotation and are inert under coordinate changes. Since the metric tensor is invariant under the rotation, ${\Lambda }^{\alpha }{}_{\beta }$ satisfies

$${\Lambda }_{\beta }^{\alpha }{\delta }_{\alpha \delta }{\Lambda }_{\gamma }^{\delta }={\delta }_{\beta \gamma }\text{i f}M\text{i s R i e m a n n i a n}\text{\hspace{1em}(7.150a)}$$

$${\Lambda }_{\beta }^{\alpha }{\eta }_{\alpha \delta }{\Lambda }_{\gamma }^{\delta }={\eta }_{\beta \gamma }\text{i f}M\text{i s L o r e n t z i a n .}\text{\hspace{1em}(7.150b)}$$

This implies that $\{{\Lambda }_{\beta }^{\alpha }(p)\}\in \mathrm{SO}(m)$ if $M$ is Riemannian with $\dim M=m$ and $\{{\Lambda }_{\beta }^{\alpha }(p)\}\in \mathrm{SO}(m-1,1)$ if $M$ is Lorentzian. The dimension of these Lie groups is $m(m-1)/2=m^2-m(m+1)/2$ , that is the difference between the degrees of freedom of $e_{\alpha }^{\mu }$ and $g_{\mu \nu }$ . Under the local frame rotation ${\Lambda }_{\beta }^{\alpha }(p)$ , the indices $\alpha ,\beta ,\gamma ,\delta ,\dots$ are rotated while $\kappa ,\lambda ,\mu ,\nu ,\dots$ (world indices) are not affected. Under the rotation (7.148), the basis vector transforms as

$${\hat{e}}_{\alpha }⟶{\hat{e}}_{\alpha }^{\prime }={\hat{e}}_{\beta }({\Lambda }^{-1}{)}_{\alpha }^{\beta }.\text{\hspace{1em}(7.151)}$$

Let $t=t_{\nu }^{\mu }{e}_{\mu }\otimes \mathrm{d}{x}^{\nu }$ be a tensor field of type (1,1). In the bases $\{{\hat{e}}_{\alpha }\}$ and $\{{\hat{\theta }}^{\alpha }\}$ , we have $t=t_{\beta }^{\alpha }{\hat{e}}_{\alpha }\otimes {\hat{\theta }}^{\beta }$ , where $t_{\beta }^{\alpha }=e_{\mu }^{\alpha }{e}_{\beta }{}^{\nu }{t}_{\nu }^{\mu }$ . If the new frames $\{{\hat{e}}_{\alpha }^{\prime }\}=\{{\hat{e}}_{\beta }({\Lambda }^{-1}{)}_{\alpha }^{\beta }\}$ and $\{{\hat{\theta }}^{\prime \alpha }\}=\{{\Lambda }_{\beta }^{\alpha }{\hat{\theta }}^{\beta }\}$ are employed, the tensor $t$ is expressed as

$$t=t_{\beta }^{\prime \alpha }{\hat{e}}_{\alpha }^{\prime }\otimes {\hat{\theta }}^{\prime \beta }=t_{\beta }^{\prime \alpha }{\hat{e}}_{\gamma }({\Lambda }^{-1}{)}_{\alpha }^{\gamma }\otimes {\Lambda }_{\delta }^{\beta }{\hat{\theta }}^{\delta }$$

from which we find the transformation rule,

$$t_{\beta }^{\alpha }⟶t_{\beta }^{\prime \alpha }={\Lambda }_{\gamma }^{\alpha }{t}_{\delta }^{\gamma }({\Lambda }^{-1}{)}_{\beta }^{\delta }.$$

To summarize, the upper (lower) non-coordinate indices are rotated by $\Lambda ({\Lambda }^{-1})$ . The change from the coordinate basis to the non-coordinate basis is carried out by multiplications of vielbeins.

From these facts we find the transformation rule of the connection one-form ${\omega }_{\beta }^{\alpha }$ . The torsion two-form transforms as

$$T^{\alpha }⟶T^{\prime \alpha }=\mathrm{d}{\hat{\theta }}^{\prime \alpha }+{\omega }_{\beta }^{\prime \alpha }\wedge {\hat{\theta }}^{\prime \beta }={\Lambda }_{\beta }^{\alpha }[\mathrm{d}{\hat{\theta }}^{\beta }+{\omega }_{\gamma }^{\beta }\wedge {\hat{\theta }}^{\gamma }].$$

Substituting ${\hat{\theta }}^{\prime \alpha }={\Lambda }_{\beta }^{\alpha }{\hat{\theta }}^{\beta }$ into this equation, we find that

$${\omega }_{\beta }^{\prime \alpha }{\Lambda }_{\gamma }^{\beta }={\Lambda }_{\delta }^{\alpha }{\omega }_{\gamma }^{\delta }-\mathrm{d}{\Lambda }_{\gamma }^{\alpha }.$$

Multiplying both sides by ${\Lambda }^{-1}$ from the right, we have

$${\omega }_{\beta }^{\prime \alpha }={\Lambda }_{\gamma }^{\alpha }{\omega }_{\delta }^{\gamma }({\Lambda }^{-1}{)}_{\beta }^{\delta }+{\Lambda }_{\gamma }^{\alpha }(\mathrm{d}{\Lambda }^{-1}{)}_{\beta }^{\gamma }\text{\hspace{1em}(7.152)}$$

where use has been made of the identity $\mathrm{d}\Lambda {\Lambda }^{-1}+\Lambda \mathrm{d}{\Lambda }^{-1}=0$ , which is derived from $\Lambda {\Lambda }^{-1}=I_m$ .

The curvature two-form transforms homogeneously as

$$R_{\beta }^{\alpha }⟶R_{\beta }^{\prime \alpha }={\Lambda }_{\gamma }^{\alpha }{R}_{\delta }^{\gamma }({\Lambda }^{-1}{)}_{\beta }^{\delta }\text{\hspace{1em}(7.153)}$$

under a local frame rotation $\Lambda$ .

7.8.4 The Levi-Civita connection in a non-coordinate basis

Let $\nabla$ be a Levi-Civita connection on $(M,g)$ , which is characterized by the metric compatibility ${\nabla }_{X}g=0$ , and the vanishing torsion ${\Gamma }_{\mu \nu }^{\lambda }-{\Gamma }_{\nu \mu }^{\lambda }=0$ . It is interesting to see how these conditions are expressed in the present approach. The components ${\Gamma }_{\mu \nu }^{\lambda }$ and ${\Gamma }_{\beta \gamma }^{\alpha }$ are related to each other by (7.142). Let $(M,g)$ be a Riemannian manifold (if $(M,g)$ is Lorentzian, we simply replace ${\delta }_{\alpha \beta }$ all below by ${\eta }_{\alpha \beta }$ ). If we define the Ricci rotation coefficient ${\Gamma }_{\alpha \beta \gamma }$ by ${\delta }_{\alpha \delta }{\Gamma }_{\beta \gamma }^{\delta }$ the metric compatibility is expressed as

$$\begin{array}{rl}{\Gamma }_{\alpha \beta \gamma }& ={\delta }_{\alpha \delta }{e}_{\lambda }^{\delta }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\gamma }^{\lambda }=-{\delta }_{\alpha \delta }{e}_{\gamma }^{\lambda }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\lambda }^{\delta }\\ & =-{\delta }_{\gamma \delta }{e}_{\lambda }^{\delta }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\alpha }^{\lambda }=-{\Gamma }_{\gamma \beta \alpha }\end{array}\text{\hspace{1em}(7.154)}$$

where ${\nabla }_{\mu }g=0$ has been used. In terms of the connection one-form ${\omega }_{\alpha \beta }\equiv$ ${\delta }_{\alpha \gamma }{\omega }_{\beta }^{\gamma }$ , this becomes

$${\omega }_{\alpha \beta }=-{\omega }_{\beta \alpha }.\text{\hspace{1em}(7.155)}$$

The torsion-free condition is

$$\mathrm{d}{\hat{\theta }}^{\alpha }+{\omega }_{\beta }^{\alpha }\wedge {\hat{\theta }}^{\beta }=0.\text{\hspace{1em}(7.156)}$$

The reader should verify that (7.156) implies the symmetry of the connection coefficient ${\Gamma }_{\mu \nu }^{\lambda }={\Gamma }_{\nu \mu }^{\lambda }$ in the coordinate basis. The condition (7.156) enables us to compute the $c_{\alpha \beta }^{\gamma }$ of the basis $\{{\hat{e}}_{\alpha }\}$ . Let us look at the commutation relation

$$c_{\alpha \beta }^{\gamma }{\hat{e}}_{\gamma }=[{\hat{e}}_{\alpha },{\hat{e}}_{\beta }]={\nabla }_{\alpha }{\hat{e}}_{\beta }-{\nabla }_{\beta }{\hat{e}}_{\alpha }\text{\hspace{1em}(7.157)}$$

where the final equality follows from the torsion-free condition. From (7.141), we find that

$$c_{\alpha \beta }^{\gamma }={\Gamma }_{\alpha \beta }^{\gamma }-{\Gamma }_{\beta \alpha }^{\gamma }.\text{\hspace{1em}(7.158)}$$

Substituting (7.158) into (7.144) we may express the Riemann curvature tensor in terms of $\Gamma$ only,

$$\begin{array}{rl}& R_{\beta \gamma \delta }^{\alpha }={\hat{e}}_{\gamma }[{\Gamma }_{\delta \beta }^{\alpha }]-{\hat{e}}_{\delta }[{\Gamma }_{\gamma \beta }^{\alpha }]+{\Gamma }_{\delta \beta }^{\epsilon }{\Gamma }_{\gamma \epsilon }^{\alpha }-{\Gamma }_{\gamma \beta }^{\epsilon }{\Gamma }_{\delta \epsilon }^{\alpha }\\ & -\left({\Gamma }_{\gamma \delta }^{\epsilon }-{\Gamma }_{\delta \gamma }^{\epsilon }\right){\Gamma }_{\epsilon \beta }^{\alpha }.\end{array}\text{\hspace{1em}(7.159)}$$

Example 7.14

Let us take the sphere $S^2$ of example 7.13. The components of $e_{\mu }^{\alpha }$ are

$$e_{\theta }^1=1e_{\varphi }^1=0e_{\theta }^2=0e_{\varphi }^2=\sin \theta .\text{\hspace{1em}(7.160)}$$

We first note that the metric condition implies ${\omega }_{11}={\omega }_{22}=0$ , hence ${\omega }_1^1={\omega }_2^2=0$ . Other connection one-forms are obtained from the torsion-free conditions,

$$\begin{array}{rl}& \mathrm{d}(\mathrm{d}\theta )+{\omega }_2^1\wedge (\sin \theta \mathrm{d}\varphi )=0(7.161a)\\ & \mathrm{d}(\sin \theta \mathrm{d}\varphi )+{\omega }_1^2\wedge \mathrm{d}\theta =0.(7.161b)\end{array}$$

From the second equation of (7.161), we easily see that ${\omega }_1^2=\cos \theta \mathrm{d}\varphi$ and the metric condition ${\omega }_{12}=-{\omega }_{21}$ implies ${\omega }_2^1=-\cos \theta \mathrm{d}\varphi$ . The Riemann tensor is also found from Cartan’s structure equation,

$${\omega }_2^1\wedge {\omega }_1^2=\frac{1}{2}{R}_{1\alpha \beta }^1{\hat{\theta }}^{\alpha }\wedge {\hat{\theta }}^{\beta }\text{\hspace{1em}(7.162a)}$$

$$\mathrm{d}{\omega }_2^1=\frac{1}{2}{R}_{2\alpha \beta }^1{\hat{\theta }}^{\alpha }\wedge {\hat{\theta }}^{\beta }\text{\hspace{1em}(7.162b)}$$

$$\mathrm{d}{\omega }_1^2=\frac{1}{2}{R}_{1\alpha \beta }^2{\hat{\theta }}^{\alpha }\wedge {\hat{\theta }}^{\beta }\text{\hspace{1em}(7.162c)}$$

$${\omega }_1^2\wedge {\omega }_2^1=\frac{1}{2}{R}_{2\alpha \beta }^2{\hat{\theta }}^{\alpha }\wedge {\hat{\theta }}^{\beta }.\text{\hspace{1em}(7.162d)}$$

The non-vanishing components are $R_{212}^1=-R_{221}^1=\sin \theta$ , $R_{112}^2=-R_{121}^2=-\sin \theta$ . The transition to the coordinate basis expression is carried out with the help of $e_{\alpha }^{\mu }$ and $e_{\mu }^{\alpha }$ . For example,

$$R_{\varphi \theta \varphi }^{\theta }=e_{\alpha }^{\theta }{e}_{\varphi }^{\beta }{e}_{\gamma \theta }^{\gamma }{e}_{\varphi }^{\delta }{R}_{\beta \gamma \delta }^{\alpha }=\frac{1}{{\sin }^2\theta }{R}_{212}^1=\frac{1}{\sin \theta }.$$

Example 7.15

The Schwarzschild metric is given by

$$\begin{array}{rl}\mathrm{d}{s}^2& =-\left(1-\frac{2M}{r}\right)\mathrm{d}{t}^2+\frac{1}{1-\frac{2M}{r}}\mathrm{d}{r}^2+r^2(\mathrm{d}{\theta }^2+{\sin }^2\theta \mathrm{d}{\varphi }^2)\\ & =-{\hat{\theta }}^0\otimes {\hat{\theta }}^0+{\hat{\theta }}^1\otimes {\hat{\theta }}^1+{\hat{\theta }}^2\otimes {\hat{\theta }}^2+{\hat{\theta }}^3\otimes {\hat{\theta }}^3\end{array}\text{\hspace{1em}(7.163)}$$

where

$$\begin{array}{rl}& {\hat{\theta }}^0={\left(1-\frac{2M}{r}\right)}^{1/2}\mathrm{d}t{\hat{\theta }}^1={\left(1-\frac{2M}{r}\right)}^{-1/2}\mathrm{d}r\\ & {\hat{\theta }}^2=r\mathrm{d}\theta {\hat{\theta }}^3=r\sin \theta \mathrm{d}\varphi .\end{array}\text{\hspace{1em}(7.164)}$$

The parameters run over the range $0<2M<r$ , $0\le \theta \le \pi$ and $0\le \varphi <2\pi$ . The metric condition yields ${\omega }_0^0={\omega }_1^1={\omega }_2^2={\omega }_3^3=0$ and the torsion-free conditions are:

$$\mathrm{d}\left[(1-2M/r{)}^{1/2}\mathrm{d}t\right]+{\omega }_{\beta }^0\wedge {\hat{\theta }}^{\beta }=0\text{\hspace{1em}(7.165a)}$$

$$\mathrm{d}\left[{\left(1-2M/r\right)}^{-1/2}\mathrm{d}r\right]+{\omega }_{\beta }^1\wedge {\hat{\theta }}^{\beta }=0\text{\hspace{1em}(7.165b)}$$

$$\mathrm{d}(r\mathrm{d}\theta )+{\omega }_{\beta }^2\wedge {\hat{\theta }}^{\beta }=0\text{\hspace{1em}(7.165c)}$$

$$\mathrm{d}(r\sin \theta \mathrm{d}\varphi )+{\omega }_{\beta }^3\wedge {\hat{\theta }}^{\beta }=0.\text{\hspace{1em}(7.165d)}$$

The non-vanishing components of the connection one-forms are

$$\begin{array}{rl}& {\omega }_1^0={\omega }_0^1=\frac{M}{r^2}dt{\omega }_1^2=-{\omega }_2^1={\left(1-\frac{2M}{r}\right)}^{1/2}d\theta \\ & {\omega }_1^3=-{\omega }_3^1={\left(1-\frac{2M}{r}\right)}^{1/2}\sin \theta d\varphi {\omega }_2^3=-{\omega }_3^2=\cos \theta d\varphi .\end{array}\text{\hspace{1em}(7.166)}$$

The curvature two-forms are found from the structure equations to be

$$\begin{array}{rl}& R_1^0=R_0^1=\frac{2M}{r^3}{\hat{\theta }}^0\wedge {\hat{\theta }}^1{R}_2^0=R_0^2=-\frac{2M}{r^3}{\hat{\theta }}^0\wedge {\hat{\theta }}^2\\ & R_3^0=R_0^3=-\frac{M}{r^3}{\hat{\theta }}^0\wedge {\hat{\theta }}^3{R}_2^1=-R_1^2=-\frac{M}{r^3}{\hat{\theta }}^1\wedge {\hat{\theta }}^2\\ & R_3^1=-R_1^3=-\frac{M}{r^3}{\hat{\theta }}^1\wedge {\hat{\theta }}^3{R}_3^2=-R_2^3=\frac{2M}{r^3}{\hat{\theta }}^2\wedge {\hat{\theta }}^3.\end{array}\text{\hspace{1em}(7.167)}$$

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