18.1 Linear connections and geodesics

中文

There is no natural way of comparing tangent vectors $Y_p$ and $Y_q$ at $p$ and $q_{:}$ , for if they had identical components in one coordinate system this will not generally be true in a different coordinate chart covering the two points. In a slightly different light, consider the partial derivatives $Y_{,j}^i=\partial Y^i/\partial x^i$ of a vector field $Y$ in a coordinate chart $(U;x^i)$ . On performing a transformation to coordinates $(U^{\prime };x^{\prime }{}^{i^{\prime }})$ , we have from Eq. (15.13)

$$Y_{,j^{\prime }}^{\prime i^{\prime }}=\frac{\partial Y^{\prime i^{\prime }}}{\partial x^{\prime j^{\prime }}}=\frac{\partial x^j}{\partial x^{\prime j^{\prime }}}\frac{\partial }{\partial x^j}\left(Y^i\frac{\partial x^{\prime i^{\prime }}}{\partial x^i}\right),$$

whence

$$Y_{,j^{\prime }}^{\prime i^{\prime }}=\frac{\partial x^{\prime i^{\prime }}}{\partial x^i}\frac{\partial x^j}{\partial x^{\prime j^{\prime }}}{Y}_{,j}^i+Y^i\frac{\partial x^j}{\partial x^{\prime j^{\prime }}}\frac{{\partial }^2{x}^{\prime i^{\prime }}}{\partial x^i\partial x^j}.\text{\hspace{1em}(18.1)}$$

The first term on the right-hand side has the form of a tensor transformation term, as in $Eq.$ (15.15), but the second term is definitely not tensorial in character. Thus, if $Y^i$ has constant components in the chart $(U;x^i)$ (so that $Y_{,j}^i=0)$ this will not be true in the char $(U^{\prime };x^{\prime }{}^{i^{\prime }})$ ) unless the coordinate transformation functions are linear,

$$x^{\prime i^{\prime }}=A_j^{i^{\prime }}{x}^j⟺\frac{{\partial }^2{x}^{\prime i^{\prime }}}{\partial x^i\partial x^j}=0.\text{\hspace{1em}(18.2)}$$

Suppose we had a well-defined notion of ‘directional derivative’ of a vector field $Y$ with respect to a tangent vector $X_p$ at ${}^{p,}$ rather like the concept of directional derivative of a function $f$ with respect to $X_p$ . It would then be possible to define a ‘constant’ vector field $Y(t)$ along a parametrized curve $\gamma$ connecting $p$ and $q$ by requiring that the directional derivative with respect to the tangent vector $\dot{\gamma }$ to the curve be zero for all $t$ in the parameter range. The resulting tangent vector $Y_q$ at $q$ may, however, be dependent on the choice of connecting curve $\gamma .$ . If we write the action of a vector field $X$ on a scalar field $f$ as $D_{X}f\equiv Xf$ , then for any real-valued function $g$

$$D_{gX}f=gXf=g{D}_{X}f.\text{\hspace{1em}(18.3)}$$

This property is essential for $D_X$ to be a local action, as the action of $D_{gX}$ at $p$ only depends on the value $g(p)$ , not on the behaviour of the function $g$ in an entire neighbourhood of $p,$

$$\left(D_{gX}f\right)(p)=g(p)D_{X}f(p).$$

We may therefore write $D_{X_p}f=(D_{X}f)(p)$ without ambiguity, for if $X_p=Y_p$ then $(D_{X}f)(p)=(D_{Y}f)(p)$

Exercise

Show that the last assertion follows from Eq. (18.3)

Extending this idea to vector fields $Y$, we seek a derivative $D_{X}Y$ having the property

$$D_{gX}Y=g{D}_{X}Y$$

for any function $g$ : $M\to \mathbb{R}$ . We will show that the derivative of $Y$ relative to a tangent vector $X_p$ at $p$ can then be defined by setting $D_{X_p}Y=(D_{X}Y{)}_p-$ the result will be independent of the choice of vector field $X$ reducing to $X_p$ at $p$.

The Lie derivative ${\mathcal{L}}_{X}Y=[X,Y]$ (see Section 15.5) is not a derivative in the sense required here since, for a general function $g\in \mathcal{F}(M)$ ,

$${\mathcal{L}}_{gX}Y=[gX,Y]=g[X,Y]-Y(g)X\ne g{\mathcal{L}}_{X}Y.$$

To calculate the Lie derivative $[X,Y{]}_p$ at a point $p$ it is not sufficient to know the tangent vector $X_p$ at $p$ – we must know the behaviour of the vector field $X$ in an entire neighbourhood of a point $p$.

A connection, also called a linear or affine connection, on a differentiable manifold $M$ is a map $D:\mathcal{T}(M)\times \mathcal{T}(M)\mathcal{T}(M)$ , where ($M$) is the module of differentiable vector fields on $M$, such that the map $D_X:T(M)\to T(M)$ defined by $D_{X}Y=D(X,Y)$ satisfies the following conditions for arbitrary vector fields $X,Y,Z$ and scalar fields $f,g:$

(Con1) $D_{X+Y}Z=D_{X}Z+D_{Y}Z,$

(Con2) $D_{gX}Y=g{D}_{X}Y,$

(Con3) $D_X(Y+Z)=D_{X}Y+D_{X}Z,$

$$(\text{Con4})D_X(fY)=(Xf)Y+f{D}_{X}Y=(D_{X}f)Y+f{D}_{X}Y.$$

A linear connection is not inherent in the original manifold structure – it must be imposed as an extra structure on the manifold. Given a linear connection $D$ on a manifold $M$, for every vector field $Y$ there exists a tensor field $DY$ of type (1, 1) defined by

$$DY(\omega ,X)=⟨\omega ,D_{X}Y⟩\text{\hspace{1em}(18.4)}$$

for every 1-form $\omega$ and vector field $X$. The tensor nature of $DY$ follows from linearity in both arguments. Linearity in $\omega$ is trivial, while linearity in $X$ follows immediately from (Con1) and (Con2). The tensor field $DY$ is called the covariant derivative of the vector field $Y$. The theory of connections as described here is called a Koszul connection [1–6], while the ‘old-fashioned’ coordinate version that will be deduced below appears in texts such as [7, 8].

A connection can be restricted to any open submanifold $U\subset M$ in a natural way. For example, if $(U;x^i)$ is a coordinate chart on $M$ and ${\partial }_{x^i}$ the associated local basis of vector fields, we may set $D_k=D_{{\partial }_{\mathrm{r}}k}$ . Expanding the vector fields $D_k{\partial }_{x}j$ in terms of the local basis,

$$D_k{\partial }_{x^j}={\Gamma }_{jk}^i{\partial }_{x^i}\text{\hspace{1em}(18.5)}$$

where ${\Gamma }_{jk}^i$ are real-valued functions on $U_{:}$ , known as the components of the connection $D$ with respect to the coordinates $\{x^i\}$ . Using (Con3) and (Con4) we can compute the covariant derivative of any vector field $Y=Y^i{\partial }_{x^i}$ on $U!$ :

$$\begin{array}{c}{D}_{k}Y=D_k\left(Y^j{\partial }_{x^j}\right)=\left({\partial }_{x^k}{Y}^j\right){\partial }_{x^j}+Y^j{\Gamma }_{jk}^i{\partial }_{x^i}\\ =Y_{;k}^i{\partial }_{x^i}\end{array}$$

where

$$Y_{;k}^i=Y_{,k}^i+{\Gamma }_{jk}^i{Y}^j.\text{\hspace{1em}(18.6)}$$

The coefficients ${Y^i}_{;k}$ are the components of the covariant derivative with respect to these coordinates since, by Eq. (18.4),

$$\begin{array}{rl}(DY{)}_k^i=DY(\mathrm{d}{x}^i,{\partial }_{x^k})& =⟨\mathrm{d}{x}^i,D_{k}Y⟩\\ & =⟨\mathrm{d}{x}^i,Y_{;k}^j⟩{\partial }_{x^j}\\ & =Y_{;k}^j{\delta }_j^i=Y_{;k}^i.\end{array}$$

Thus

$$DY=Y_{;k}^i{\partial }_{x^i}\otimes \mathrm{d}{x}^k$$

and the components of $D_{X}Y$ with respect to the coordinates $x^i$ are

$${\left(D_{X}Y\right)}^i=DY(\mathrm{d}{x}^i,X)=Y_{;k}^i{X}^k.\text{\hspace{1em}(18.7)}$$

As anticipated above, it is possible to define the covariant derivative of a vector field $Y$ with respect to a tangent vector $X_p$ at a point $p$ as $D_{X_p}Y={\left(D_{X}Y\right)}_p\in T_p(M)$ , where $X$ is any vector field that ‘reduces’ to $X_p$ at $p.$ . For this definition to make sense, we must show that it is independent of the choice of vector field $X$. Suppose $X^{\prime }$ is a second vector field such that $X_p^{\prime }=X_p$ . The vector field $Z=X-X^{\prime }$ vanishes at $p,$ , and we have

$$\begin{array}{rl}(D_{X}Y{)}_p-(D_{X^{\prime }}Y{)}_p& =(D_{X-X^{\prime }}Y{)}_p\\ & =(D_{Z}Y{)}_p\\ & =Y_{;k}^i(p)(Z_p{)}^k{\partial }_{x^i}=0\text{ by Eq. (18.7). }\end{array}$$

The covariant derivative of a vector field $Y$ along a curve $\gamma :\mathbb{R}M$ is defined to be

$$\frac{DY}{\mathrm{d}t}=D_{\dot{\gamma }}Y$$

where $X=\dot{\gamma }$ is the tangent vector to the curve. By Eq. (18.7), the components are

$$\begin{array}{c}\frac{D{Y}^i}{\mathrm{d}t}\equiv {\left(\frac{DY}{\mathrm{d}t}\right)}^i=\frac{\mathrm{d}{x}^k}{\mathrm{d}t}(\frac{\partial Y^i}{\partial x^k}+{\Gamma }_{jk}^i{Y}^j)\\ =\frac{\mathrm{d}{Y}^i(t)}{\mathrm{d}t}+{\Gamma }_{jk}^i{Y}^j\frac{\mathrm{d}{x}^k}{\mathrm{d}t}.\end{array}\text{\hspace{1em}(18.8)}$$

We will say the vector field $Y$ is parallel along the curve $\gamma$ if $DY(t)/\mathrm{d}t=0$ for all $t$ in the curve’s parameter range. A curve will be called a geodesic if its tangent vector is everywhere parallel along the curve, $D_{\dot{\gamma }}\dot{\gamma }=0-$ note that the expression on the right-hand side of

Eq. (18.8) depends only on the values of the components ${\mathcal{Y}}^i(t)\equiv {\mathcal{Y}}^i(\gamma (t))$ along the curve. By Eq. (18.8) a geodesic can be written locally as a set of differential equations

$$\frac{D}{\mathrm{d}t}\frac{\mathrm{d}{x}^i}{\mathrm{d}t}=\frac{{\mathrm{d}}^2{x}^i}{\mathrm{d}{t}^2}+{\Gamma }_{jk}^i\frac{\mathrm{d}{x}^j}{\mathrm{d}t}\frac{\mathrm{d}{x}^k}{\mathrm{d}t}=0.\text{\hspace{1em}(18.9)}$$

The above discussion can also be reversed. Let $p$ be any point of $M$ and $\gamma :[a,b]\to M$ a curve such that $p=\gamma (a)$ . In local coordinates $(U;x^i)$ the equations for a vector field to be parallel along the curve are a linear set of differential equations

$$\frac{\mathrm{d}{Y}^i(t)}{\mathrm{d}t}+{\Gamma }_{jk}^i{Y}^j(t)\frac{\mathrm{d}{x}^k}{\mathrm{d}t}=0.\text{\hspace{1em}(18.10)}$$

By the existence and uniqueness theorem of differential equations, for any tangent vector $Y_p$ at $p$ there exists a unique vector field $Y(t)$ parallel along $\gamma \cap U$ such that $Y(a)=Y_p$ The curve segment is a compact set and can be covered by a finite family of charts, so that existence and uniqueness extends over the entire curve $a\le t\le b$ . Furthermore, as the differential equations are linear the map $P_t:T_p(M)\to T_{\gamma (t)}(M)$ such that $P_t(Y_p)=Y(t)$ is a linear map, called parallel transport along $\gamma$ from $p=\gamma (a)$ to $\gamma (t)$ . Since the parallel transport map can be reversed by changing the parameter to $t^{\prime }=-t$ , the map $P_t$ is one-to one and must be a linear isomorphism.

The uniqueness of a maximal solution to a set of differential equations also shows that if $p$ is any point of $M$ there exists a unique maximal geodesic $\sigma :[0,a)$ where $a\le$ starting with any specified tangent vector $\dot{\sigma }(0)=X_p$ at $p=\sigma (0)$ ). The parameter $t$ such that a geodesic satisfies Eq. (18.9) is called an affine parameter. Under a parameter transformation $t^{\prime }=f(t)$ ) the tangent vector becomes

$${\dot{\sigma }}^{\prime }=\frac{\mathrm{d}{x}^i}{\mathrm{d}{t}^{\prime }}{\partial }_{x^i}=\frac{1}{f^{\prime }(t)}\frac{\mathrm{d}{x}^i}{\mathrm{d}t}{\partial }_{x^i}$$

where $f^{\prime }(t)=\mathrm{d}f/\mathrm{d}t$ and, using Eq. (18.9), we have

$$\begin{array}{c}\frac{{\mathrm{d}}^2{x}^i}{\mathrm{d}{t}^{\prime 2}}+{\Gamma }_{jk}^i\frac{\mathrm{d}{x}^j}{\mathrm{d}{t}^{\prime }}\frac{\mathrm{d}{x}^k}{\mathrm{d}{t}^{\prime }}=\frac{1}{f^{\prime }(t)}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{f^{\prime }(t)}\right)\frac{\mathrm{d}{x}^i}{\mathrm{d}t}\\ =-\frac{f^{\prime \prime }(t)}{(f^{\prime }(t){)}^2}\frac{\mathrm{d}{x}^i}{\mathrm{d}{t}^{\prime }}.\end{array}$$

The new parameter $t^{\prime }$ is an affine parameter if and only if $f^{\prime \prime }(t)=0-$ that is, an affine transformation, $t^{\prime }=at+b$ . Herein lies the reason behind the term affine parameter.

Coordinate transformations

Consider a coordinate transformation from a chart $(U;x^i)$ to a chart $(U^{\prime };x^{\prime }{}^{i^{\prime }})$ . In the overlap $U\cap U^{\prime }$ we have, using the transformations between coordinate bases given by

Eq. (15.12),

$$\begin{array}{l}{D}_{{\partial }_{x^{\prime }{k}^{\prime }}}{\partial }_{x^{\prime }{j}^{\prime }}=\frac{\partial x^k}{\partial x^{\prime k^{\prime }}}{D}_k\left(\frac{\partial x^j}{\partial x^{\prime j^{\prime }}}{\partial }_{x^j}\right)\\ =\frac{\partial x^k}{\partial x^{\prime k^{\prime }}}\frac{\partial x^j}{\partial x^{\prime j^{\prime }}}{\Gamma }_{jk}^i\frac{\partial x^{\prime i^{\prime }}}{\partial x^i}{\partial }_{x^{\prime i^{\prime }}}+\frac{{\partial }^2{x}^j}{\partial x^{\prime k^{\prime }}\partial x^{\prime j^{\prime }}}{\partial }_{x^j}\\ ={\Gamma }_{j^{\prime }{k}^{\prime }}^{\prime i^{\prime }}{\partial }_{x^{\prime i^{\prime }}}\end{array}$$

where

$${\Gamma }_{j^{\prime }{k}^{\prime }}^{\prime i^{\prime }}=\frac{\partial x^k}{\partial x^{\prime k^{\prime }}}\frac{\partial x^j}{\partial x^{\prime j^{\prime }}}\frac{\partial x^{\prime i^{\prime }}}{\partial x^i}{\Gamma }_{jk}^i+\frac{{\partial }^2{x}^i}{\partial x^{\prime k^{\prime }}\partial x^{\prime j^{\prime }}}\frac{\partial x^{\prime i^{\prime }}}{\partial x^i}.\text{\hspace{1em}(18.11)}$$

This is the law of transformation of components of a connection.

The first term on the right-hand side of Eq. (18.11) is tensorial in nature, but the second term adds a complication that only vanishes for linear transformations. It is precisely the expression needed to counteract the non-tensorial part of the transformation of the derivative of a vector field given in Eq. (18.1) – see Problem 18.1.

Problems

Problem 18.1

Show directly from the transformation laws Eq. (18.1) and Eq. (18.11) that the components of the covariant derivative Eq. (18.6) of a vector field transform as a tensor of type (1, 1).

Problem 18.2

Show that the transformation law Eq. (18.11) can be written in the form

$${\Gamma }_{j^{\prime }{k}^{\prime }}^{\prime i^{\prime }}=\frac{\partial x^k}{\partial x^{\prime k^{\prime }}}\frac{\partial x^j}{\partial x^{\prime j^{\prime }}}\frac{\partial x^{\prime i^{\prime }}}{\partial x^i}{\Gamma }_{jk}^i-\frac{{\partial }^2{x}^{\prime k^{\prime }}}{\partial x^i\partial x^j}\frac{\partial x^i}{\partial x^{\prime i^{\prime }}}\frac{\partial x^j}{\partial x^{\prime j^{\prime }}}.$$