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Killing Vectors

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

g_{ab} = {\partial {x'}^c\over\partial x^a} {\partial {x'}^d\over\partial x^b} g_{cd}(x'),
\end{displaymath} (1)

so let
{x'}^a=x^a+\epsilon x^a
\end{displaymath} (2)

{\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
\end{displaymath} (5)

g_{ab}g^{bc} = \delta_a^c,
\end{displaymath} (6)

which gives Killing's Equation
{\mathcal L}_X g_{ab} = X_{a;b}+X_{b;a} = 2X_{(a;b)}=0.
\end{displaymath} (7)

A Killing vector $X^b$ satisfies
\end{displaymath} (8)

X_{a;bc} = R_{abcd}X^d
\end{displaymath} (9)

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

\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 .''

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© 1996-9 Eric W. Weisstein