Killing vector field

In mathematics and theoretical physics, a Killing vector field or Killing field (named after Wilhelm Killing) is a vector field on a Riemannian manifold or pseudo-Riemannian manifold that preserves the metric.

Flows generated by Killing vector fields are continuous isometries of the manifold. This means that the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.

Definitions

A vector field

X

{\displaystyle X}

on a Riemannian or pseudo-Riemannian manifold

(

M

,

g

)

{\displaystyle (M,g)}

is called a Killing vector if the Lie derivative with respect to

X

{\displaystyle X}

of the metric tensor

g

{\displaystyle g}

vanishes:

L

X

g

=

0.

{\displaystyle {\mathcal {L}}_{X}g=0.}

Equivalently, the flow of

X

{\displaystyle X}

consists of local isometries of

g

{\displaystyle g}

; for this reason, Killing vector fields are called by some authors infinitesimal isometries.

In terms of the Levi-Civita connection, the condition of being a Killing vector field is

g

(

∇

Y

X

,

Z

)

+

g

(

Y

,

∇

Z

X

)

=

0

{\displaystyle g\left(\nabla _{Y}X,Z\right)+g\left(Y,\nabla _{Z}X\right)=0}

for all vectors

Y

{\displaystyle Y}

and ⁠

Z

{\displaystyle Z}

⁠. In local coordinates, this amounts to the Killing equation

∇

μ

X

ν

+

∇

ν

X

μ

=

0

.

{\displaystyle \nabla _{\mu }X_{\nu }+\nabla _{\nu }X_{\mu }=0\,.}

This condition is expressed in covariant form. Therefore, it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.

Examples

Circle

The vector field on a circle that points counterclockwise and has the same magnitude at each point is a Killing vector field, since moving each point on the circle along this vector field simply rotates the circle.