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}](k_524.gif) |
(1) |
so let
![\begin{displaymath}
{x'}^a=x^a+\epsilon x^a
\end{displaymath}](k_525.gif) |
(2) |
![\begin{displaymath}
{\partial {x'}^a\over \partial x^b} = \delta^a_b +\epsilon x^{a}{}_{,b}
\end{displaymath}](k_526.gif) |
(3) |
where
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}](k_531.gif) |
(5) |
![\begin{displaymath}
g_{ab}g^{bc} = \delta_a^c,
\end{displaymath}](k_532.gif) |
(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}](k_533.gif) |
(7) |
A Killing vector
satisfies
![\begin{displaymath}
g^{bc}X_{c;ab}-R_{ab}X^b=0
\end{displaymath}](k_535.gif) |
(8) |
![\begin{displaymath}
X_{a;bc} = R_{abcd}X^d
\end{displaymath}](k_536.gif) |
(9) |
![\begin{displaymath}
X^{a;b}{}_{;b} +R_c^a X^c = 0,
\end{displaymath}](k_537.gif) |
(10) |
where
is the Ricci Tensor and
is the Riemann Tensor.
A 2-sphere with Metric
![\begin{displaymath}
ds^2=d\theta^2+\sin^2\theta\,d\phi^2
\end{displaymath}](k_540.gif) |
(11) |
has three Killing vectors, given by the angular momentum operators
The Killing vectors in Euclidean 3-space are
![$\displaystyle x^1$](k_547.gif) |
![$\textstyle =$](k_36.gif) |
![$\displaystyle {\partial\over\partial x}$](k_548.gif) |
(15) |
![$\displaystyle x^2$](k_549.gif) |
![$\textstyle =$](k_36.gif) |
![$\displaystyle {\partial\over\partial y}$](k_550.gif) |
(16) |
![$\displaystyle x^3$](k_551.gif) |
![$\textstyle =$](k_36.gif) |
![$\displaystyle {\partial\over\partial z}$](k_552.gif) |
(17) |
![$\displaystyle x^4$](k_553.gif) |
![$\textstyle =$](k_36.gif) |
![$\displaystyle y{\partial\over\partial z}-z{\partial\over\partial y}$](k_554.gif) |
(18) |
![$\displaystyle x^5$](k_555.gif) |
![$\textstyle =$](k_36.gif) |
![$\displaystyle z{\partial\over\partial x}-x{\partial\over\partial z}$](k_556.gif) |
(19) |
![$\displaystyle x^6$](k_557.gif) |
![$\textstyle =$](k_36.gif) |
![$\displaystyle x{\partial\over\partial y}-y{\partial\over\partial x}.$](k_558.gif) |
(20) |
In Minkowski Space, there are 10 Killing vectors
The first group is Translation, the second Rotation, and the final corresponds to a
``boost
.''
© 1996-9 Eric W. Weisstein
1999-05-26