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Killing Vectors


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

 g_(ab)=(partialx^'^c)/(partialx^a)(partialx^'^d)/(partialx^b)g_(cd)(x^'),
(1)

so let

 x^'^a=x^a+epsilonX^a
(2)
 (partialx^'^a)/(partialx^b)=delta_b^a+epsilonX^a_(,b)
(3)
g_(ab)(x)=(delta_a^c+epsilonX^c_(,a))(delta_b^d+epsilonX^d_(,b))g_(cd)(x^e+epsilonX^e)
(4)
=(delta_a^c+epsilonX^c_(,a))(delta_b^d+epsilonX^d_(,b))[g_(cd)(x)+epsilonX^eg_(cd)(x)_(,e)+...]
(5)
=g_(ab)(x)+epsilon[g_(ad)X^d_(,b)+g_(bd)X^d_(,a)+X^eg_(ab,e)]+O(epsilon^2)
(6)
=g_(ab)+L_Xg_(ab)
(7)
=g_(ab)^',
(8)

where 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

 g_(ab;c)=0
(9)
 g_(ab)g^(bc)=delta_a^c,
(10)

which gives Killing's equation

 L_Xg_(ab)=X_(a;b)+X_(b;a)=2X_((a;b))=0,
(11)

where X_((a;b)) denotes the symmetric tensor part and X_(a;b) is a covariant derivative.

A Killing vector X^b satisfies

 g^(bc)X_(c;ab)-R_(ab)X^b=0
(12)
 X_(a;bc)=R_(abcd)X^d
(13)
 X^(a;b)_(;b)+R_c^aX^c=0,
(14)

where R_(ab) is the Ricci curvature tensor and R_(abcd) is the Riemann tensor.

In Minkowski space, there are 10 Killing vectors

X_i^mu=a_i^mu   for i=1,2,3,4
(15)
X_k^0=0
(16)
X_k^l=epsilon^(lkm)x_m  for k=1,2,3
(17)
X_mu^k=delta_mu^([0_xk])  for k=1,2,3.
(18)

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.

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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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