Killing Vector Field

# Killing vector field

to get instant updates about 'Killing Vector Field' on your MyPage. Meet other similar minded people. Its Free!

X

Description:
In mathematics, a Killing vector field (often just Killing field) , named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generator of isometries; that is, flow generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object.

## Explanation

Specifically, a vector field X is a Killing field if the Lie derivative with respect to X of the metric g vanishes:

[itex]mathcal_ g = 0 ,.[/itex]

In terms of the Levi-Civita connection, this is

[itex]g(nabla_ X, Z) + g(Y, nabla_ X) = 0 ,[/itex]

for all vectors Y and Z. In local coordinates, this amounts to the Killing equation

[itex]nabla_ X_ + nabla_ X_ = 0 ,.[/itex]

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

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

• If the metric coefficients [itex]g_ ,[/itex] in some coordinate basis [itex]dx^ ,[/itex] are......
• ...

No feeds found

All