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# 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.

 (1)

so let

 (2)
 (3)
 (4) (5) (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) (18)

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

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.

Killing Vectors

## Cite this as:

Weisstein, Eric W. "Killing Vectors." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KillingVectors.html