Killing Vectors

If any set of points is displaced by where all distance relationships are unchanged (i.e., there is an Isometry), then the Vector field is called a Killing vector.

$\begin{displaymath} g_{ab} = {\partial {x'}^c\over\partial x^a} {\partial {x'}^d\over\partial x^b} g_{cd}(x'), \end{displaymath}$

(1)

so let

$\begin{displaymath} {x'}^a=x^a+\epsilon x^a \end{displaymath}$

(2)

$\begin{displaymath} {\partial {x'}^a\over \partial x^b} = \delta^a_b +\epsilon x^{a}{}_{,b} \end{displaymath}$

(3)

$\displaystyle g_{ab}(x)$	$\textstyle =$	$\displaystyle \left({\delta_a^c+\epsilon x^c{}_{,a}}\right)\left({\delta_b^d+\epsilon x^d{}_{,b}}\right)g_{cd} (x^e +\epsilon X^e)$
	$\textstyle =$	$\displaystyle \left({\delta_a^c +\epsilon x^c{}_{,a}}\right)\left({\delta_b^d +\epsilon x^d{}_{,b}}\right)[g_{cd}(x)+\epsilon X^e g_{cd}(x){}_{,e}+\ldots]$
	$\textstyle =$	$\displaystyle g_{ab}(x)+\epsilon [g_{ad}X^d{}_{,b}+g_{bd}X^d{}_{,a}+X^eg_{ab,e}]+{\mathcal O}(\epsilon^2) = {\mathcal L}_X g_{ab},$	(4)

where ${\mathcal L}$ is the Lie Derivative. An ordinary derivative can be replaced with a covariant derivative in a Lie Derivative, so we can take as the definition

$\begin{displaymath} g_{ab;c}=0 \end{displaymath}$

(5)

$\begin{displaymath} g_{ab}g^{bc} = \delta_a^c, \end{displaymath}$

(6)

which gives Killing's Equation

$\begin{displaymath} {\mathcal L}_X g_{ab} = X_{a;b}+X_{b;a} = 2X_{(a;b)}=0. \end{displaymath}$

(7)

A Killing vector

satisfies

$\begin{displaymath} g^{bc}X_{c;ab}-R_{ab}X^b=0 \end{displaymath}$

(8)

$\begin{displaymath} X_{a;bc} = R_{abcd}X^d \end{displaymath}$

(9)

$\begin{displaymath} X^{a;b}{}_{;b} +R_c^a X^c = 0, \end{displaymath}$

(10)

where $R_{ab}$ is the Ricci Tensor and $R_{abcd}$ is the Riemann Tensor.

A 2-sphere with Metric

$\begin{displaymath} ds^2=d\theta^2+\sin^2\theta\,d\phi^2 \end{displaymath}$

(11)

has three Killing vectors, given by the angular momentum operators

$\displaystyle \tilde L_x$	$\textstyle =$	$\displaystyle -\cos\phi{\partial\over\partial\theta}+\cot\theta\sin\phi{\partial\over\partial\phi}$	(12)
$\displaystyle \tilde L_y$	$\textstyle =$	$\displaystyle \sin\phi{\partial\over \partial\theta}+\cot\theta\cos\phi{\partial\over\partial\phi}$	(13)
$\displaystyle \tilde L_z$	$\textstyle =$	$\displaystyle {\partial\over\partial \phi}.$	(14)

The Killing vectors in Euclidean 3-space are

$\displaystyle x^1$	$\textstyle =$	$\displaystyle {\partial\over\partial x}$	(15)
$\displaystyle x^2$	$\textstyle =$	$\displaystyle {\partial\over\partial y}$	(16)
$\displaystyle x^3$	$\textstyle =$	$\displaystyle {\partial\over\partial z}$	(17)
$\displaystyle x^4$	$\textstyle =$	$\displaystyle y{\partial\over\partial z}-z{\partial\over\partial y}$	(18)
$\displaystyle x^5$	$\textstyle =$	$\displaystyle z{\partial\over\partial x}-x{\partial\over\partial z}$	(19)
$\displaystyle x^6$	$\textstyle =$	$\displaystyle x{\partial\over\partial y}-y{\partial\over\partial x}.$	(20)

In Minkowski Space, there are 10 Killing vectors

$\displaystyle X^\mu_i$	$\textstyle =$	$\displaystyle a_i{}^\mu \quad {\rm for\ } i=1, 2, 3, 4$	(21)
$\displaystyle X^0_k$	$\textstyle =$	$\displaystyle 0$	(22)
$\displaystyle X^l_k$	$\textstyle =$	$\displaystyle \epsilon^{lkm}x_m \quad {\rm for\ } k=1, 2, 3$	(23)
$\displaystyle X^k_\mu$	$\textstyle =$	$\displaystyle \delta_\mu{}^{[0_xk]} \quad {\rm for\ } k=1, 2, 3.$	(24)

The first group is Translation, the second Rotation, and the final corresponds to a ``boost

.''