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.
Killing vector in hyperbolic plane
A toy example for a Killing vector field is on the upper-half plane equipped metric. The pair is typically called the hyperbolic plane and has Killing vector field . This should be intuitively clear since the covariant derivative transports the metric along an integral curve generated by the vector field.
A typical use of a Killing Field is to express a symmetry in general relativity. In a static configuration, in which nothing changes with time, the time vector will be a Killing vector, and thus the Killing field will point in the direction of forward motion in time.
Derivation
If the metric coefficients in some coordinate basis are independent of one of the coordinates, then is a Killing vector, where is the Kronecker delta. To prove this, let us assume. Then and Now let us look at the Killing condition and from. The Killing condition becomes that is, which is true.
The physical meaning is, for example, that, if none of the metric coefficients is a function of time, the manifold must automatically have a time-like Killing vector.
In layman's terms, if an object doesn't transform or "evolve" in time, time passing won't change the measures of the object. Formulated like this, the result sounds like a tautology, but one has to understand that the example is very much contrived: Killing fields apply also to much more complex and interesting cases.
Properties
A Killing field is determined uniquely by a vector at some point and its gradient. 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 the isometry group of the manifold if M is complete. For compact manifolds
Negative Ricci curvature implies there are no nontrivial Killing fields.
Nonpositive Ricci curvature implies that any Killing field is parallel. i.e. covariant derivative along any vector j field is identically zero.
If the sectional curvature is positive and the dimension of M is even, a Killing field must have a zero.
Each Killing vector corresponds to a quantity which is conserved along geodesics. This conserved quantity is the metric product between the Killing vector and the geodesic tangent vector. That is, along a geodesic with some affine parameter the equation is satisfied. This aids in analytically studying motions in a spacetime with symmetries.
Generalizations
Killing vector fields can be generalized to conformal Killing vector fields defined by for some scalar The derivatives of one parameter families of conformal maps are conformal Killing fields.
Killing vector fields can also be defined on any manifold M if we take any Lie group G acting on it instead of the group of isometries. In this broader sense, a Killing vector field is the pushforward of a right invariant vector field on G by the group action. If the group action is effective, then the space of the Killing vector fields is isomorphic to the Lie algebra of G.
