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Wikipedia / Killing vector field

In mathematics and theoretical physics, a Killing vector field or Killing field (named after Wilhelm Killing) is a vector field on a Riemannian manifold or pseudo-Riemannian manifold that preserves the metric.

Flowss generated by Killing vector fields are continuous isometries of the manifold. This means that the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.

Definitions

is called a Killing vector if the Lie derivative of the metric tensor vanishes:Equivalently, the flow consists of local isometries; for this reason, Killing vector fields are called by some authors infinitesimal isometries.

In terms of the Levi-Civita connection, the condition of being a Killing vector field is

and . 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

Circle

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

Hyperbolic plane

A toy example for a Killing vector field is on the upper half-plane equipped with the Poincaré metric is typically called the hyperbolic plane transports the metric tensor along an integral curve generated by the vector field (whose image is parallel to the x-axis).

is a Killing field using one of the results below in this article.

The isometry group (see Poincaré half-plane model and the special conformal transformation .

2-sphere

The Killing fields of the two-sphere should be obvious from ordinary intuition: spheres, having rotational symmetry to have symmetry under the action of the 3D rotation group SO(3). That is, by using the a priori knowledge that spheres can be embedded in Euclidean space, it is immediately possible to guess the form of the Killing fields.

is given by

-axis.

The pullback gives the standard metric on the sphere,

-axis:

In these coordinates, the metric components are all independent of is a Killing field.

The vector field

goes from north to south; points at the north pole spread apart, those at the south come together. Any transformation that moves points closer or farther apart cannot be an isometry; therefore, the generator of such motion cannot be a Killing field.

-axis

-axis, is

The algebra given by linear combinations of these three generators closes, and obeys the relations

This is the Lie algebra .

in terms of spherical coordinates gives

and

and chug to show that are the generators of isometries in Euclidean space, and since the metric on the sphere is inherited from metric in Euclidean space, the isometries are inherited as well.

These three Killing fields form a complete set of generators for the algebra. They are not unique: any linear combination of these three fields is still a Killing field.

There are several subtle points to note about this example.

vanish at antipodes on the equator. One way to understand this is as a consequence of the "hairy ball theorem". This property, of bald spots, is a general property of symmetric spaces in the Cartan decomposition. At each point on the manifold, the algebra of the Killing fields splits naturally into two parts, one part which is tangent to the manifold, and another part which is vanishing (at the point where the decomposition is being made).
is singular (non-differentiable) at the north and south poles.
that vanishes: these three vectors are an over-complete basis for the two-dimensional tangent plane at that point.
The a priori knowledge that spheres can be embedded into Euclidean space, and thus inherit a metric tensor from this embedding, leads to a confusing intuition about the correct number of Killing fields that one might expect. Without such an embedding, intuition might suggest that the number of linearly independent generators would be no greater than the dimension of the tangent bundle. After all, fixing any point on a manifold, one can only move in those directions that are tangent. The dimension of the tangent bundle for the 2-sphere is two, and yet three Killing fields are found. Again, this "surprise" is a generic property of symmetric spaces.

Minkowski space

The Killing fields of Minkowski space are the 3 space translations, time translation, three generators of rotations (the little group) and the three generators of boosts. These are

Time and space translations
Vector fields generating three rotations, often called the J generators,
Vector fields generating three boosts, the K generators,

The boosts and rotations generate the Lorentz group. Together with space-time translations, this forms the Lie algebra for the Poincaré group.

Flat space

Here we derive the Killing fields for general flat space. From Killing's equation and the Ricci identity for a covector ,

(using abstract index notation is the Riemann curvature tensor:

is flat space, that is, Euclidean space or pseudo-Euclidean space (as for Minkowski space), we can choose global flat coordinates such that in these coordinates, the Levi-Civita connection and hence Riemann curvature vanishes everywhere, giving

and , we get a basis for the generalised Poincaré algebra of isometries of flat space:

These generate pseudo-rotations (rotations and boosts) and translations respectively. Intuitively these preserve the metric tensor at each point.

generators, making flat space maximally symmetric. This number is generic for maximally symmetric spaces. Maximally symmetric spaces can be considered as sub-manifolds of flat space, arising as surfaces of constant proper distance

which have O(p, q) symmetry. If the submanifold has dimension , this group of symmetries has the expected dimension (as a Lie group).

together with the identity . as a system of second order differential equations for at any point given initial data at a point and independent values of initial data.

For concrete examples, see below for examples of flat space (Minkowski space) and maximally symmetric spaces (sphere, hyperbolic space).

General relativity

Killing fields are used to discuss isometries 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: the metric tensor is independent of is a time-like Killing field. The other three are the three generators of rotations discussed above. The Kerr metric for a rotating black hole has only two Killing fields: the time-like field, and a field generating rotations about the axis of rotation of the black hole.

De Sitter space and anti-de Sitter space Killing fields.

Of a constant coordinate

are independent of one of the coordinates is the Kronecker delta.

To prove this, let us assume 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 tensor 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.

admits a Killing field , then one can construct coordinates for which is nowhere tangent to on denotes the parameter along the integral curve on . In these coordinates, the Lie derivative reduces to the coordinate derivative, that is,

and by the definition of the Killing field the left-hand side vanishes.

Properties

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 thus form a Lie subalgebra of vector fields on . This is the Lie algebra of the isometry group of the manifold if is complete. A Riemannian manifold with a transitive group of isometries is a homogeneous 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 field is identically zero;
if the sectional curvature is positive and the dimension of is even, a Killing field must have a zero.

The covariant divergence of every Killing vector field vanishes.

is a harmonic vector field is a harmonic function.

is a harmonic p-form, then .

Geodesics

Each Killing vector corresponds to a quantity which is conserved along geodesics then given the Killing vector is conserved:

This aids in analytically studying motions in a spacetime with symmetries.

Stress-energy tensor

Given a conserved, symmetric tensor and , which are properties typical of a stress-energy tensor, and a Killing vector satisfying

Cartan decomposition

As noted above, the Lie bracket thus form a Lie subalgebra of all vector fields on . Selecting a point can be decomposed into two parts:

and

is the covariant derivative. These two parts intersect trivially but do not in general split if and only if is a Riemannian symmetric space.

of the total space, and the Killing fields show how to "slide along" that submanifold. They span 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 three Killing vector 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

The Cartan involution and , respectively.

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.

defined by

is a homomorphism of is

The Cartan involution is a Lie algebra homomorphism, in that

for all has odd parity under the Cartan involution one has

and

is a Lie subalgebra of , in that and .

for a symmetric space ; proofs can be found in Jost. They also hold in more general settings, but not necessarily at all points of the manifold.

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

Killing vector fields can be generalized to conformal Killing vector fields for some scalar . The derivatives of one parameter families of conformal maps are conformal Killing fields.
Killing tensor fields are symmetric tensor fields vanishes. Examples of manifolds with Killing tensors include the rotating black hole and the FRW cosmology.
Killing vector fields can also be defined on any manifold M (possibly without a metric tensor) if we take any Lie group 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 of .