Killing vector field

From Wikipedia, the free encyclopedia
  (Redirected from Killing vectors)
Jump to: navigation, search

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 generators of isometries; that is, flows 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 will not distort distances on the object.

Definition[edit]

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

In terms of the Levi-Civita connection, this is

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

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[edit]

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 General Relativity[edit]

A typical use of the Killing Field is to express a symmetry in General relativity (in which the geometry of spacetime as distorted by gravitational fields is viewed as a 4-dimensional Riemannian manifold). 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[edit]

If the metric coefficients in some coordinate basis are independent of one of the coordinatees , then is a Killing vector, where is the Kronecker delta.[1]

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 (when time passes), 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[edit]

A Killing field is determined uniquely by a vector at some point and its gradient (i.e. all covariant derivatives 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 the isometry group of the manifold if M is complete.

For compact manifolds

  • Negative Ricci curvature implies there are no nontrivial (nonzero) 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.

The divergence of every Killing vector field vanishes.

If is a Killing vector field and is a harmonic vector field, then is a harmonic function.

If is a Killing vector field and is a harmonic p-form, then

Geodesics[edit]

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.[2]

Generalizations[edit]

  • 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 tensor fields are symmetric tensor fields T such that the trace-free part of the symmetrization of vanishes. Examples of manifolds with Killing tensors include the rotating black hole and the FRW cosmology.[3]
  • Killing vector fields can also be defined on any (possibly nonmetric) manifold M if we take any Lie group G acting on it instead of the group of isometries.[4] 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.

See also[edit]

Notes[edit]

  1. ^ Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0. 
  2. ^ Carrol, Sean (2004). An Introduction to General Relativity Spacetime and Geometry. Addison Wesley. pp. 133–139. 
  3. ^ Carrol, Sean (2004). An Introduction to General Relativity Spacetime and Geometry. Addison Wesley. pp. 263, 344. 
  4. ^ Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0-7204-0494-4 

References[edit]

  • Jost, Jurgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. ISBN 3-540-42627-2. .
  • Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975). Introduction to General Relativity (Second ed.). New York: McGraw-Hill. ISBN 0-07-000423-4. . See chapters 3,9