Killing vector field

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In mathematics, a Killing vector field (often called a 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:[1]

In terms of the Levi-Civita connection, this is

for all vectors Y and Z. In local coordinates, this amounts to the Killing equation[2]

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 hyperbolic plane[edit]

A toy example for a Killing vector field is on the upper half-plane equipped with the Poincaré metric . The pair is typically called the hyperbolic plane and has Killing vector field (using standard coordinates). This should be intuitively clear since the covariant derivative transports the metric along an integral curve generated by the vector field (whose image is parallel to the x-axis).

Killing fields on a 2-sphere[edit]

The Killing fields on the two-sphere , or any sphere, should be, in a sense, "obvious" from ordinary intuition: spheres, being sphere-symmetric, should possess Killing fields that are generated by infinitessimal rotations about any axis. This is even straight-forward at an appropriate level of abstraction. However, when explicitly expressed in terms of the coordinate charts, the Killing fields have a non-obvious structure that obscures their nature. This is articulated below. This "non-obvious" structure is generic to manifolds that are not spheres, and thus the 2-sphere provides a good toy model on which to explore the intuitive interpretation of Killing fields.

The conventional metric on the sphere is

.

and obviously, a rotation about the pole should be an isometry. The infinitesimal generator of a rotation can then be identified as a generator of the Killing field. This can be immediately written down: it is

Note that it is normalized to unit length. The surface of the sphere is two-dimensional, and so obviously, there is another generator of isometries; it can be taken as

Killing fields have the property that the Lie bracket of two Killing fields is still a Killing field. Thus, the Killing fields on a manifold M thus form a Lie subalgebra of vector fields on M. Of some interest is the dimension of this algebra (how many generators does it have?) and its structure constants — given an orthonormal basis of this algebra, what are the numbers appearing in

Direct computation of leads to an unenlightening explosion of sines and cosines. This is perhaps not obvious; certainly, at the equator , one has that However, moving off of the equator, the two vector fields and are no longer orthonormal, and so, in general one has for a point in general position. Worse, to obtain the dimension of the algebra, one must either determine that form a complete, linearly independent basis for the algebra (making the algebra three-dimensional), or that possibly there is a fourth, fifth, ... (linearly-independent) vector field obtained by computing and and so-forth. There is no particular a priori reason to believe that the algebra is two dimensional or three dimensional; this must somehow be proven. The sphere coordinate system is not amenable to such calculations.

The simplest solution is to embed the sphere in 3D Euclidean space, and then work in orthonormal Cartesian coordinates where commutators are straight-forward. The conventional 3-space coordinate system is given by

The generator is recognized as a rotation about the -axis

A second generator, rotations about the -axis, is plainly

Commuting these two, one promptly finds a third generator for rotations about the -axis

That this forms a complete basis is readily verified by noting that

One concludes that the Lie algebra for the Killing fields on the two-sphere is three-dimensional, and that the set provide a complete basis for the algebra. That these satisfy should be either readily apparent from the construction, or can be directly validated post factum. As vector fields, they are not globally orthonormal; they are neither orthogonal, nor of unit length for points in general position. They cannot be globally normalized by the "hairy ball theorem", in that one "cannot comb the hair on a sphere without leaving a tuft or a bald spot".

Attempts to further orthogonalize or normalize these vector fields are not fruitful, and there are no particular further simplifications possible, other than to work in a vielbein coordinate system. In this particular case, working in the coordinate system, one can apply the Hodge dual (the cross-product in three dimensions). The resulting vectors do not lie in the tangent space , and so are "outside of the manifold". They are everywhere normal to the sphere; the coordinates are extrinsic, as compared to the intrinsic coordinates . The utility of doing this is that now, in the embedding space , the Hodge duals are globally orthonormal (i.e. are orthonormal at every point on the sphere.)

Working in the intrinsic coordinate system , it is easy enough to make one of the vector fields be of unit length. By common convention in general relativity, e.g. in Schwarzschild coordinates, it is the generator of rotations about the -axis. Normalizing this, and expressing these in spherical coordinates, one has

and one may readily verify that the commutators still hold:

These are three generators of the algebra. Of course, any other (non-degenerate) linear combination of these will also generate the algebra. Note the somewhat unintuitive counting: although the surface of the sphere is two-dimensional, and so one expects two distinct isometries, one has, in fact, more. This somewhat surprising result is a generic property of symmetric spaces. This is described further, below, as the Cartan decomposition: at each point on the manifold, the algebra of the Killing fields splits naturally into two parts, one of which is tangent to the manifold, and one of which is vanishing (at the chosen point).

Killing fields in Minkowski space[edit]

The Killing fields of Minkowski space are the three generators of rotations (the little group) and the three generators of boosts. These are

  • Vector fields generating three rotations, often called the J generators,
  • Vector fields generating three boosts, the K generators,

Together, they generate the Lorentz group. See that article for an extensive discussion.

Killing fields in general relativity[edit]

A typical use of a 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 pseudo-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. For example, the Schwarzschild metric has four Killing fields: one time-like, and two isometries originating from its spherical symmetry; these split into the three shown for the sphere coordinate system above. The Kerr metric has only two Killing fields: the timelike field, and an axis-symmetric field (Kerr solutions correspond to rotating black holes, and are not spherically symmetric; they are only axially symmetric, about the rotation axis.) See Schwarzschild coordinates#Killing vector fields for an example.

Killing field of a constant coordinate[edit]

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.[3]

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. A Riemannian manifold with a transitive group of isometries is a homogenous space.

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.[4]

Cartan decomposition[edit]

As noted above, the Lie bracket of two Killing fields is still a Killing field. The Killing fields on a manifold thus form a Lie subalgebra of all vector fields on Selecting a point the algebra can be decomposed into two parts:

and

where is the covariant derivative. These two parts are orthogonal, and split the algebra, in that and

Intuitively, the isometries of locally define a submanifold of the total space, and the Killing fields show how to "slide along" that submanifold. They span the tangent space of that submanifold. The tangent space should have the same dimension as the isometries acting effectively at that point. That is, one expects Yet, in general, the number of Killing fields is larger than the dimension of that tangent space. How can this be? The answer is that the "extra" Killing fields are redundant. Taken all together, the fields provide an over-complete basis for the tangent space at any particular selected point; linear combinations can be made to vanish at that particular point. This was seen in the example of the Killing fields on a 2-sphere: there are 3 Killing fields; at any given point, two span the tangent space at that point, and the third one is a linear combination of the other two. Picking any two defines the remaining degenerate linear combinations define an orthogonal space

Cartan involution[edit]

The Cartan involution is defined as the mirroring or reversal of the direction of a geodesic. Its differential flips the direction of the tangents to a geodesic. It is a linear operator of norm one; it has two invariant subspaces, of eigenvalue +1 and −1. These two subspaces correspond to and respectively.

This can be made more precise. Fixing a point consider a geodesic passing through , with The involution is defined as

This map is an involution, in that When restricted to geodesics along the Killing fields, it is also clearly an isometry. It is uniquely defined.

Let be the group of isometries generated by the Killing fields. The function defined by

is a homomorphism of . Its infinitesimal is

The Cartan involution is a Lie algebra homomorphism, in that

for all The subspace has odd parity under the Cartan involution, while has even parity. That is, denoting the Cartan involution at point as one has

and

where is the identity map. From this, it follows that the subspace is a Lie subalgebra of , in that As these are even and odd parity subspaces, the Lie brackets split, so that and

The above decomposition holds at all points for a symmetric space ; proofs can be found in Jost.[5] They also hold in more general settings, but not necessarily at all points of the manifold.[citation needed]

For the special case of a symmetric space, one explicitly has that that is, the Killing fields span the entire tangent space of a symmetric space. Equivalently, the curvature tensor is covariantly constant on locally symmetric spaces, and so these are locally parallelizable; this is the Cartan–Ambrose–Hicks theorem.

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.[6]
  • 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.[7] 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]

References[edit]

  1. ^ Jost, Jurgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. ISBN 3-540-42627-2.
  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.
  3. ^ Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.CS1 maint: multiple names: authors list (link)
  4. ^ Carroll, Sean (2004). Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley. pp. 133–139.
  5. ^ Jurgen Jost, (2002) "Riemmanian Geometry and Geometric Analysis" (Third edition) Springer. (See section 5.2 pages 241-251.}
  6. ^ Carroll, Sean (2004). Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley. pp. 263, 344.
  7. ^ Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0-7204-0494-4