- Killing vector field
In
mathematics , a Killing vector field, named afterWilhelm Killing , is avector field on aRiemannian manifold (orpseudo-Riemannian manifold ) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of themanifold . More simply, the flow generates asymmetry , in the sense that moving each point on an object in the direction of the Killing vector field will not distort distances on the object. For example, 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 g_{mu u} , in some coordinate basis dx^{a} , are independent of x^{K} ,, then x^{mu} = delta^{mu}_{K} , is automatically a Killing vector, where delta^{mu}_{K} , is the
Kronecker delta . (Misner, et al, 1973). For example if none of the metric coefficients are functions of time, the manifold must automatically have a time-like Killing vector.Explanation
Specifically, a vector field "X" is a Killing field if the
Lie derivative with respect to "X" of the metric "g" vanishes::mathcal{L}_{X} g = 0 ,.
In terms of the
Levi-Civita connection , this is:g( abla_{Y} X, Z) + g(Y, abla_{Z} X) = 0 ,
for all vectors "Y" and "Z". In
local coordinates , this amounts to the Killing equation:abla_{mu} X_{ u} + abla_{ u} 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.
A Killing field is determined uniquely by a vector at some point and its gradient (i.e. all
covariant derivative s of the field at the point).The
Lie bracket of two Killing fields is still a Killing field. The Killing fields on a manifold "M" thus form a Lie subalgebra of vector fields on "M". This is the Lie algebra of theisometry group of the manifold.For compact manifolds
* NegativeRicci curvature implies there are no nontrivial (nonzero) Killing fields.
* NonpositiveRicci curvature implies that any Killing field is parallel. i.e. covariant derivative along any vector field is identically zero.
* If thesectional curvature is positive and the dimension of "M" is even, a Killing field must have a zero.Killing vector fields can be generalized to conformal Killing vector fields defined by:mathcal{L}_{X} g = lambda g ,for some scalar lambda ,. The derivatives of one parameter families of
conformal map s are conformal Killing fields. Another generalization is to conformal Killing tensor fields. These are symmetrictensor fields "T" such that the trace-free part of the symmetrization of abla T , vanishes.References
*.
*. "See chapters 3,9"
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