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.
 | (1) |
so let
 | (2) |
 | (3) |
 |  |  | (4) |
 |  | ![(delta_a^c+epsilonX^c_(,a))(delta_b^d+epsilonX^d_(,b))[g_(cd)(x)+epsilonX^eg_(cd)(x)_(,e)+...]](https://mathworld.wolfram.com/images/equations/KillingVectors/Inline7.svg) | (5) |
 |  | ![g_(ab)(x)+epsilon[g_(ad)X^d_(,b)+g_(bd)X^d_(,a)+X^eg_(ab,e)]+O(epsilon^2)](https://mathworld.wolfram.com/images/equations/KillingVectors/Inline10.svg) | (6) |
 |  |  | (7) |
 |  |  | (8) |
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
 | (9) |
 | (10) |
which gives Killing's equation
 | (11) |
where 
denotes the symmetric tensor part and 
is a covariant derivative.
A Killing vector 
satisfies
 | (12) |
 | (13) |
 | (14) |
where 
is the Ricci curvature tensor and 
is the Riemann tensor.
In Minkowski space, there are 10 Killing vectors
 |  |  | (15) |
 |  |  | (16) |
 |  |  | (17) |
 |  | ![delta_mu^([0_xk]) for k=1,2,3.](https://mathworld.wolfram.com/images/equations/KillingVectors/Inline34.svg) | (18) |
The first group is translation, the second rotation, and the final corresponds to a "boost."
