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)

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

The first group is translation , the second rotation ,
and the final corresponds to a "boost."

See also Killing's Equation ,

Killing
Form ,

Lie Derivative
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References Weinberg, S. "Killing Vectors." §13.1 in Gravitation
and Cosmology: Principles and Applications of the General Theory of Relativity.
New York: Wiley, pp. 375-381, 1972. Referenced on Wolfram|Alpha Killing Vectors
Cite this as:
Weisstein, Eric W. "Killing Vectors."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/KillingVectors.html

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