19.5 Groups of isometries

中文

Let $(M,g)$ be a pseudo-Riemannian manifold. An isometry of $M$ is a transformation $\phi :M\to M$ such that $\tilde{\phi }g=g$ , where $\tilde{\phi }$ is the map induced on tensor fields as defined in Section 15.5. This condition amounts to requiring

$$g_{\phi (p)}({\phi }_{\ast }{X}_p,{\phi }_{\ast }{Y}_p)=g_p(X_p,Y_p)$$

for all $X_p,Y_p\in T_p(\mathcal{M})$

Let $G$ be a Lie group of isometries of $(M,g)$ , and its Lie algebra of left-invariant vector fields. If $A\in \mathcal{G}$ is a left-invariant vector field then the induced vector field $X=\tilde{A}$ is called a Killing vector on $M$. If ${\sigma }_t$ is the one-parameter group of isometries generated by $A$ then, by Eq. (15.33), we have

$${\mathcal{L}}_{X}g=\underset{t\to 0}{\lim }\frac{1}{t}(g-{\tilde{\sigma }}_{t}g)=0.$$

In any coordinate chart $(U;x^i)$ let $X={\xi }^i{\partial }_{x^i}$ and, by Eq. (15.39), this equation becomes

$${\mathcal{L}}_X{g}_{ij}=g_{ij,k}{\xi }^k+{\xi }_{,i}^k{g}_{kj}+{\xi }_{,j}^k{g}_{ik}=0,\text{\hspace{1em}(19.17)}$$

known as Killing’s equations. In a local chart such that ${\xi }^i=(1,0,\dots ,0)$ (see Theorem 15.3), Eq. (19.17) reads

$$g_{ij,1}=\frac{\partial g_{ij}}{\partial x^1}=0$$

and the components of $g$ are independent of the coordinate $x^1,g_{ij}=g_{ij}(x^2,\dots ,x^n)$ ). By direct computation from the Christoffel symbols or by considering the equation in geodesic coordinates, ordinary derivatives may be replaced by covariant derivatives in Eq. (19.17)

$$g_{ij;k}{\xi }^k+{\xi }_{;i}^k{g}_{kj}+{\xi }_{;j}^k{g}_{ik}=0,$$

and since $g_{ij;k}=0$ Killing’s equations may be written in the covariantform:

$${\xi }_{i;j}+{\xi }_{j;i}=0.\text{\hspace{1em}(19.18)}$$

By Theorem 19.6, if $X=\tilde{A}$ and $Y=\tilde{B}$ then $[X,Y]=\widetilde{[A,B]}$ . We also conclude from Problem 15.18 that if $X$ and $Y$ satisfy Killing’s equations then so does $[X,Y]$ . In fact, there

can be at most a finite number of linearly independent Killing vectors. For, from Eq. (19.18) and the Ricci identities (18.29), ${\xi }_{k;ij}-{\xi }_{k;ji}={\xi }_a{R}^a{}_{kij}$ (no torsion), we have

$${\xi }_{k;ij}+{\xi }_{j;ki}={\xi }_a{R}_{kij}^a.$$

From the cyclic first Bianchi identity (18.26), ${R^i}_{jkl}+{R^i}_{klj}+{R^i}_{ljk}=0$ , we have ${\xi }_{i;jk}+{\xi }_{j;ki}+{\xi }_{k;ij}=0$ , whence

$${\xi }_{i;jk}=-{\xi }_{j;ki}-{\xi }_{k;ij}=-{\xi }_a{R}_{kij}^a={\xi }_a{R}_{kji}^a.\text{\hspace{1em}(19.19)}$$

Thus if we know the components ${\xi }_i$ and ${\xi }_{i;j}$ in a given pseudo-Riemannian space, al covariant derivatives of second order of ${\xi }_i$ may be calculated from Eq. (19.19). All highe orders may then be found by successively forming higher order covariant derivatives of this equation. Assuming that ${\xi }_i$ can be expanded in a power series in a neighbourhood of any point of $M$ (this is not actually an additional assumption as it turns out), we only need to know ${\xi }_i$ and ${\xi }_{i;j}=-{\xi }_{j;i}$ at a specified point $p$ to define the entire Killing vector field in a neighbourhood of $p$ . As there are $n+\left(\genfrac{}{}{0ex}{}{n}{2}\right)=n(n+1)/2$ linearly independent initial values at $p,$ , the maximum number of linearly independent Killing vectors in any neighbourhood of $M$ is $n(n+1)/2$ . In general of course there are fewer than these, say $r_{;}$ , and the genera Killing vector is expressible as a linear combination of $r$ Killing vectors $X_1,\dots ,X_r$

$$X=\sum _{i=1}^r{a}^i{X}_i,(a^i=\text{ const. })$$

generating a Lie algebra of dimension $r$ with structure constants $C_{ij}^k=-C_{ji}^k$

$$[X_i,X_j]=C_{ij}^k{X}_k.$$

Maximal symmetries and cosmology

A pseudo-Riemannian space is said to have maximal symmetry if it has the maximum number $n(n+1)/2$ of Killing vectors. Taking a covariant derivative of Eq. (19.19),

$${\xi }_{i;jkl}={\xi }_{a;l}{R}_{kji}^a+{\xi }_a{R}_{kji;l}^a,$$

and using the generalized Ricci identities given in Problem 18.9,

$$\begin{array}{r}{\xi }_{i;jkl}-{\xi }_{i:jlk}={\xi }_{a;j}{R}_{ikl}^a+{\xi }_{i;a}{R}_{jkl}^a\\ ={\xi }_{a;j}{R}_{ikl}^a-{\xi }_{a;i}{R}_{jkl}^a,\end{array}$$

we have

$${\xi }_a(R_{kji;l}^a-R_{lji;k}^a)={\xi }_{a;b}(R_{ikl}^a{\delta }_j^b-R_{jkl}^a{\delta }_i^b-R_{kji}^a{\delta }_l^b+R_{lji}^a{\delta }_k^b).$$

Since for maximal symmetry ${\xi }_a$ and ${\xi }_{a;b}=-{\xi }_{b;a}$ are arbitrary at any point, the antisymmet ric part with respect to $a$ and $b$ of the term in parentheses on the right-hand side vanishes,

$$R_{ikl}^a{\delta }_j^b-R_{jkl}^a{\delta }_i^b-R_{kji}^a{\delta }_l^b+R_{lji}^a{\delta }_k^b=R_{ikl}^b{\delta }_j^a-R_{jkl}^b{\delta }_i^a-R_{kji}^b{\delta }_l^a+R_{lji}^b{\delta }_k^a.$$

Contracting this equation with respect to indices $b$ and $l$, we find on using the cyclic symmetry (18.26),

$$(n-1)R_{kji}^a=R_{ik}{\delta }_j^a-R_{jk}{\delta }_i^a.$$

Another contraction with respect to $k$ and $i$ gives $n$ $R_j^a=R{\delta }_j^a$ and substituting back in the expression for $R^a{}_{kji}$ , we find on lowering the index and making a simple permutation of index symbols

$$R_{ijkl}=\frac{R}{n(n-1)}(g_{ik}{g}_{jl}-g_{il}{g}_{jk}).\text{\hspace{1em}(19.20)}$$

The contracted Bianchi identity (18.60) implies that the Ricci scalar is constant for $n>2$ since

$$R_{j;a}^a=\frac{1}{2}{R}_{,j}⟹n{R}_{,j}=\frac{1}{2}{R}_{,j}⟹R_{,j}=0.$$

Spaces whose Riemann tensor has this form are known as spaces of constant curvature. Example 18.4 provides another motivation for this nomenclature and shows that the 3-sphere of radius $a$ is a space of constant curvature, with $R=6/a^2$ . The converse is in fact true – every space of constant curvature has maximal symmetry. We give a few instances of this statement in the following examples.

Example 19.13

Euclidean 3-space $\mathrm{d}{s}^2={\delta }_{ij}\mathrm{d}{x}^i$ $d{x}^j$ of constant curvature zero. To find its Killing vectors, we must find all solutions of Killing’s equations Eq. (19.18),

$${\xi }_{i,j}+{\xi }_{j,i}=0.$$

Since this implies ${\xi }_{1,1}={\xi }_{2,2}={\xi }_{3,3}=0$ we have

$${\xi }_{i,11}=-{\xi }_{1,1i}=0,{\xi }_{i,22}={\xi }_{i,33}=0,$$

whence there exist constants $a_{ij}=-a_{ji}$ and $b_i$ such that

$${\xi }_i=a_{ij}{x}^j+b_i.$$

Setting $a_{ij}=-{ϵ}_{ijk}{a}^k$ and $b^k=b_k$ we can express the general Killing vector in the form

$$X=a^1{X}_1+a^2{X}_2+a^3{X}_3+b^1{Y}_1+b^2{Y}_2+b^3{Y}_3$$

where $X_1=x^2{\partial }_3-x^2{\partial }_3,X_2=x^3{\partial }_1-x^1{\partial }_3,X_3=x^1{\partial }_2-x^2{\partial }_1$ and $Y_1={\partial }_1,Y_2={\partial }_2{Y}_3={\partial }_3$ . As these are six independent Killing vectors, the space has maximal symmetry. Their Lie algebra commutators are

$$\begin{array}{c}[X_1,X_2]=-X_3,[X_2,X_3]=-X_1,[X_3,X_1]=-X_2,\\ [Y_1,Y_2]=[Y_1,Y_3]=[Y_2,Y_3]=0,\end{array}$$ $$[X_1,Y_1]=0,[X_2,Y_1]=Y_3,[X_3,Y_1]=-Y_2,$$ $$[X_1,Y_2]=-Y_3,[X_2,Y_2]=0,[X_3,Y_2]=Y_1,$$ $$[X_1,Y_3]=Y_2,[X_2,Y_3]=-Y_1,[X_3,Y_3]=0.$$

This is known as the Lie algebra of the Euclidean group.

Example 19.14

The 3-sphere of Example 18.4,

$$\mathrm{d}{s}^2=a^2(\mathrm{d}{\chi }^2+{\sin }^2\chi (\mathrm{d}{\theta }^2+{\sin }^2\theta \mathrm{d}{\varphi }^2)),$$

has Killing’s equations

$${\xi }_{1,1}=0,\text{\hspace{1em}(19.21)}$$

$${\xi }_{1,2}+{\xi }_{2,1}-2\cot \chi {\xi }_2=0,\text{\hspace{1em}(19.22)}$$

$${\xi }_{1,3}+{\xi }_{3,1}-2\cot \chi {\xi }_3=0,\text{\hspace{1em}(19.23)}$$

$${\xi }_{2,2}+\sin \chi \cos \chi {\xi }_1=0,\text{\hspace{1em}(19.24)}$$

$${\xi }_{2,3}+{\xi }_{3,2}-2\cot \chi {\xi }_3=0,\text{\hspace{1em}(19.25)}$$

$${\xi }_{3,3}+\sin \chi \cos \chi {\sin }^2\theta {\xi }_1+\sin \theta \cos \theta {\xi }_2=0.\text{\hspace{1em}(19.26)}$$

From Eq. (19.21) we have ${\xi }_1=F(\theta ,\varphi )$ and differentiating Eq. (19.22) with respect to $x^1=\chi$ we have a differential equation for ${\xi }_2$ ,

$${\xi }_{2,11}-2\cot \chi {\xi }_{2,1}+2{\sec }^2\chi {\xi }_2=0.$$

The general solution of this linear differential equation is not hard to find:

$${\xi }_2=-\sin \chi \cos \chi f(\theta ,\varphi )+{\sin }^2\chi G(\theta ,\varphi ).$$

Substituting back into Eq. (19.22) we find $f=-F_{,2}$ where $x^2=\theta$ . Similarly,

$${\xi }_3=F_{,3}\sin \chi \cos \chi +H(\theta ,\varphi ){\sin }^2\chi .$$

Substituting these expressions in the remaining equations results in the following gen eral solution of Killing’s equations dependent on six arbitrary constants $a^1,a^2,a^3,b^1{b}^2,b^3$

$$X={\xi }^i{\partial }_i=a^i{X}_i+b^j{Y}_j$$

where ${\xi }^1={\xi }_1,{\xi }^2={\xi }_2/{\sin }^2\chi ,{\xi }^3={\xi }_3/{\sin }^2\chi {\sin }^2\theta$ , and

$$X_1=\cos \varphi {\partial }_{\theta }-\cot \theta \sin \varphi {\partial }_{\varphi },$$ $$X_2=\sin \varphi {\partial }_{\theta }+\cot \theta \cos \varphi {\partial }_{\varphi },$$ $$X_3={\partial }_{\varphi },$$ $$Y_1=\sin \theta \sin \varphi {\partial }_{\chi }+\cot \chi \cos \theta \sin \varphi {\partial }_{\theta }+\frac{\cot \chi }{\sin \theta }\cos \varphi {\partial }_{\varphi },$$ $$Y_2=-\sin \theta \cos \varphi {\partial }_{\chi }-\cot \chi \cos \theta \cos \varphi {\partial }_{\theta }+\frac{\cot \chi }{\sin \theta }\sin \varphi {\partial }_{\varphi },$$ $$Y_3=\cos \theta {\partial }_{\chi }-\sin \theta \cot \chi {\partial }_{\theta }.$$

The Lie algebra brackets are tedious to calculate compared with those in the previous

example, but the results have similarities to those of the Euclidean group:

$$\begin{array}{lll}[X_1,X_2]=-X_3,& [X_2,X_3]=-X_1,& [X_3,X_1]=-X_2,\\ [Y_1,Y_2]=-X_3,& [Y_2,Y_3]=-X_1,& [Y_3,Y_1]=-X_2,\\ [X_1,Y_1]=0,& [X_1,Y_2]=-Y_3,& [X_1,Y_3]=Y_2,\\ [X_2,Y_1]=Y_3,& [X_2,Y_2]=0,& [X_2,Y_3]=-Y_1,\\ [X_3,Y_1]=-Y_2,& [X_3,Y_2]=Y_1,& [X_3,Y_3]=0.\end{array}$$

Not surprisingly this Lie algebra is isomorphic to the Lie algebra of the four-dimensiona rotation group, $SO(4)$.

The Robertson–Walker cosmologies of Section 18.9 all have maximally symmetric spatial sections. The sections $t$ const. of the open model (18.105) are 3-spaces of constant negative curvature, calledpseudo-spheres. These models are called homogeneous and isotropic. It is not hard to see that these space-times have the same number of independen Killing vectors as their spatial sections. In general they have six Killing vectors, but some special cases may have more. Of particular interest is the de Sitter universe, which is a maximally symmetric space-time, having 10 independent Killing vectors:

$$\mathrm{d}{s}^2=-\mathrm{d}{t}^2+a^2{\cosh }^2(t/a)\left[\mathrm{d}{\chi }^2+{\sin }^2\chi (\mathrm{d}{\theta }^2+{\sin }^2\theta \mathrm{d}{\varphi }^2)\right].$$

This is a space-time of constant curvature, which may be thought of as a hyperboloid embedded in five-dimensional space,

$$x^2+y^2+z^2+w^2-v^2=a^2.$$

Since it is a space of constant curvature, $\begin{array}{r}{R}_{\mu \nu }=\frac{1}{4}R{g}_{\mu \nu }\end{array}$ , the Einstein tensor is

$$G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=-\frac{1}{4}R{g}_{\mu \nu }=\frac{3}{a^2}{g}_{\mu \nu }.$$

This can be thought of in two ways. It can be interpreted as a solution of Einstein’s field equations $G_{\mu \nu }=\kappa T_{\mu \nu }$ with a perfect fluid $T_{\mu \nu }\propto g_{\mu \nu }$ having negative pressure $P=-\rho c^2$ However it is more common to interpret it as a vacuum solution $T_{\mu \nu }=0$ of the modified Einstein field equations with cosmological constant $\Lambda$,

$${\mathbb{G}}_{\mu \nu }=\kappa T_{\mu \nu }-\Lambda g_{\mu \nu }(\Lambda =3a^{-2}).$$

This model is currently popular with advocates of the inflationary cosmology. Interesting aspects of its geometry are described in [8, 9].

Sometimes cosmologists focus on cosmologies having fewer symmetries. A common technique is to look for homogeneous models that are not necessarily isotropic, equivalen to relaxing the Lie algebra of Killing vectors from six to three, and assuming the orbits are three-dimensional subspaces of space-time. All three-dimensional Lie algebras may be categorized into one of nine Bianchi types, usually labelled by Roman numerals. A detailed discussion may be found in [10]. The Robertson–Walker models all fall into this classification, the flat model being of Bianchi type I, the closed model of type IX, and the open model of type V. To see how such a relaxation of symmetry gives rise to more genera models, consider type I, which is the commutative Lie algebra

$$[X_1,X_2]=[X_2,X_3]=[X_1,X_3]=0.$$

It is not hard to show locally that a metric having these symmetries must have the form

$$\mathrm{d}{s}^2={\mathrm{e}}^{2{\alpha }_1(t)}(\mathrm{d}{x}^1{)}^2+{\mathrm{e}}^{2{\alpha }_2(t)}(\mathrm{d}{x}^2{)}^2+{\mathrm{e}}^{2{\alpha }_3(t)}(\mathrm{d}{x}^3{)}^2-c^2\mathrm{d}{t}^2.$$

The vacuum solutions of this metric are (see [11])

$${\alpha }_i(t)=a_i\ln t(a_1+a_2+a_3=a_1^2+a_2^2+a_3^2=1),$$

called Kasner solutions. The pressure-free dust cosmologies of this type are called Heckmann–Schücking solutions (see the article by Ernst Heckmann and Otto Schücking in [12]) and have the form

$${\alpha }_i=a_i\ln (t-t_1)+b_i\ln (t-t_2)\left(b_i=\frac{2}{3}-a_i,\sum _{i=1}^3{a}_i=\sum _{i=1}^3(a_i{)}^2=1\right).$$

It is not hard to show that $\begin{array}{r}{\sum }_{i=1}^3{b}_i={\sum }_{i=1}^3(b_i{)}^2=1\end{array}$ . The density in these solutions evolves as

$$\rho =\frac{1}{6\pi G(t-t_1)(t-t_2)}.$$

The flat Friedmann model arises as the limit $t_1=t_2$ of this model.

Spherical symmetry

A space-time is said to be spherically symmetric if it has three spacelike Killing vectors $X_1,X_2,X_3$ such that they span a Lie algebra isomorphic with (3),

$$[X_i,X_j]=-{ϵ}_{ijk}{X}_k(i,j,k\in \{1,2,3\})$$

and such that the orbits of all points are two-dimensional surfaces, or possibly isolated points. The idea is that the orbits generated by the group of transformations are in general 2-spheres that could be represented as $r$ const. in appropriate coordinates. There should therefore be coordinates $x=r,x^2=\theta ,x^3=\varphi$ such that the $X_i$ are spanned by ${\partial }_{\theta }$ and ${\partial }_{\varphi }$ and using Theorem 15.3 it should be locally possible to choose these coordinates such that

$$X_3={\partial }_{\varphi },X_1={\xi }^1{\partial }_{\theta }+{\xi }^2{\partial }_{\varphi },X_2={\eta }^1{\partial }_{\theta }+{\eta }^2{\partial }_{\varphi }.$$

We then have

$$[X_3,X_1]=-X_2⟹{\eta }^1=-{\xi }_{,\varphi }^1,{\eta }^2=-{\xi }_{,\varphi }^2$$ $$[X_3,X_2]=X_1⟹{\xi }^1={\eta }_{,\varphi }^1,{\xi }^2={\eta }_{,\varphi }^2$$

whence ${\xi }_{,\varphi \varphi }^i=-{\xi }^i\left(i=1,2\right)$ , so that

$${\xi }^1=f\sin \varphi +g\cos \varphi ,{\xi }^2=h\sin \varphi +k\cos \varphi$$ $${\eta }^1=-f\cos \varphi +g\sin \varphi ,{\eta }^2=-h\cos \varphi +k\sin \varphi$$

where the functions $f,g,h,k$ are arbitrary functions of $\theta ,r$ and $t$. The remaining commutation relation $[X_1,X_2]=-X_3$ implies, after some simplification,

$$f{g}_{\theta }-g{f}_{\theta }+gk+fh=0\text{\hspace{1em}(19.27)}$$

$$f{k}_{\theta }-g{h}_{\theta }+h^2+k^2=-1\text{\hspace{1em}(19.28)}$$

where $g_{\theta }\equiv \partial g/\partial \theta$ , etc. A coordinate transformation ${\varphi }^{\prime }=\varphi +F(\theta ,r,t),\theta =G(\theta ,r,t)$ has the effect

$${\partial }_{\varphi }={\partial }_{{\varphi }^{\prime }},{\partial }_{\theta }=F_{\theta }{\partial }_{{\varphi }^{\prime }}+G_{\theta }{\partial }_{{\theta }^{\prime }}$$

and therefore

$$X_1={\xi }^1{G}_{\theta }{\partial }_{{\theta }^{\prime }}+\left({\xi }^1{F}_{\theta }+{\xi }^2\right){\partial }_{{\varphi }^{\prime }}.$$

Hence, using addition of angle identities for the functions $\sin$ and $\cos$,

$$\begin{array}{c}({\xi }^1{)}^{\prime }={\xi }^1{G}_{\theta }=(f\sin ({\varphi }^{\prime }-F)+g\cos ({\varphi }^{\prime }-F))G_{\theta }\\ =(f\cos F+g\sin F)G_{\theta }\sin {\varphi }^{\prime }+(-f\sin F+g\cos F)G_{\theta }\cos {\varphi }^{\prime }.\end{array}$$

Choosing

$$\tan F=-\frac{f}{g}\text{ and }{G}_{\theta }=\frac{1}{g\cos F-f\sin F}$$

we have $({\xi }^1{)}^{\prime }=\cos {\varphi }^{\prime }$ . We have thus arrived at the possibility of selecting coordinates $\theta$ and $\varphi$ such that $f=0,g=1$ . Substituting in Eqs. Eq. (19.27) and Eq. (19.28) gives $k=0$ and $h=-\cot (\theta -{\theta }_0(r,t))$ . Making a final coordinate transformation $\theta \theta -{\theta }_0(r,t)$ , which has no effect on ${\xi }^1$ , we have

$$X_1=\cos \varphi {\partial }_{\theta }-\cot \theta \sin \varphi {\partial }_{\varphi },X_2=\sin \varphi {\partial }_{\theta }+\cot \theta \cos \varphi {\partial }_{\varphi },X_3={\partial }_{\varphi }.$$

From Killing’s equations Eq. (19.17) with $X=X_3$ we have $g_{\mu \nu }=g_{\mu \nu }(r,\theta ,t)$ and for $X=X_2{X}_3$ , we find that these equations have the form

$${\xi }^2{\partial }_{\theta }{g}_{\mu \nu }+{\xi }_{,\mu }^2{g}_{2\nu }+{\xi }_{,\mu }^3{g}_{3\nu }+{\xi }_{,\nu }^2{g}_{2\mu }+{\xi }_{,\nu }^3{g}_{3\mu }=0$$

and successively setting $\mu \nu =11,12,...$ . we obtain

$$\begin{array}{c}{g}_{11}=g_{11}(r,t),g_{14}=g_{14}(r,t),g_{44}=g_{44}(r,t),\\ g_{12}=g_{13}=g_{42}=g_{43}=g_{23}=0,\\ g_{22}=f(r,t),g_{33}=f(r,t){\sin }^2\theta .\end{array}$$

As there is still an arbitrary coordinate freedom in the radial and time coordinate,

$$r^{\prime }=F(r,t),t^{\prime }=G(r,t)$$

it is possible to choose the new radial coordinate to be such that $f=r^{\prime 2}$ and the time coordinate may then be found so that $g_{14}^{\prime }=0$ . The resulting form of the metric is tha

postulated in Eq. (18.90),

$$\mathrm{d}{s}^2=g_{11}(r,t)\mathrm{d}{r}^2+r^2(\mathrm{d}{\theta }^2+{\sin }^2\theta \mathrm{d}{\varphi }^2)-|g_{44}(r,t)|c^2\mathrm{d}{t}^2.$$

If $g_{11}$ and $g_{44}$ are independent of the time coordinate then the vector $X={\partial }_t$ is a Killing vector. Any space-time having a timelike Killing vector is called stationary. For the case considered here the Killing vector has the special property that it is orthogonal to the 3- surfaces $t=const.$ ., and is called a static space-time. The condition for a space-time to be static is that the covariant version of the Killing vector be proportional to a gradient ${\xi }_{\mu }=g_{\mu \nu }{\xi }^{\nu }=\lambda f_{,\mu }$ for some functions $\lambda$ and $f.$ . Equivalently, if $\xi$ is the 1-form $\xi ={\xi }_{\mu }\mathrm{d}{x}^{\mu }$ , then $\xi =\lambda \mathrm{d}f$ which, by the Frobenius theorem 16.4, can hold if and only if $\mathrm{d}\xi \wedge \xi =0$ . For the spherically symmetric metric above, $\xi =g_{44}c$ $\mathrm{d}t$ and $\mathrm{d}\xi \wedge \xi =\mathrm{d}{g}_{44}\wedge c\mathrm{d}t\wedge g_{44}c\mathrm{d}t=0:$ as required. An important example of a metric that is stationary but not static is the Ker solution, representing a rotating body in general relativity. More details can be found in [8, 13].

Problem

Problem 19.19

Show that the non-translational Killing vectors of pseudo-Euclidean space with metric tensor $g_{ij}={\eta }_{ij}$ are of the form

$$X=A_j^k{x}^j{\partial }_{x^k}\text{ where }{A}_{kl}=A_k^j{\eta }_{jl}=-A_{lk}.$$

Hence, with reference to Example 19.3, show that the Lie algebra of $SO(p,q)$ ) is generated by matrice $I_{ij}$ with $i<j$ , having matrix elements $(I_{ij}{)}_a^b={\eta }_{ia}{\delta }_j^b-{\delta }_i^b{\eta }_{ja}$ . Show that the commutators of these generators can be written (setting $I_{ij}=-I_{ji}\mathrm{if}i>j)$

$$[I_{ij},I_{kl}]=I_{il}{\eta }_{jk}+I_{jk}{\eta }_{il}-I_{ik}{\eta }_{jl}-I_{jl}{\eta }_{ik}.$$