Killing vector fields

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 of an object the same distance in the direction of the Killing vector will not distort distances on the object.

Definition

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

:$\mathcal_ g = 0 \,.$

In terms of the Levi-Civita connection, this is

:$g\left(\nabla_Y X, Z\right) + g\left(Y, \nabla_Z X\right) = 0 \,$

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

:$\nabla_\mu X_\nu + \nabla_ X_\mu = 0 \,.$

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.

Examples

Killing field on the circle

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 fields in flat space

Here we derive the Killing fields for general flat space.
From Killing's equation and the Ricci identity for a covector $K_a$,
:$\nabla_a\nabla_b K_c - \nabla_b\nabla_a K_c = R^d_K_d$
(using abstract index notation) where $R^a_$ is the Riemann curvature tensor, the following identity may be proven for a Killing field $X^a$:
:$\nabla_a\nabla_b X_c = R^d_X_d.$
When the base manifold $M$ is flat space, that is, Euclidean space or possibly 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
:$\partial_\mu\partial_\nu X_\rho = 0.$
Integrating and imposing the Killing equation allows us to write the general solution to $X_\rho$ as
:$X^\rho = \omega^ x_\sigma + c^\rho$
where $\omega^ = -\omega^$ is antisymmetric. By taking appropriate values of $\omega^$ and $c^\rho$, we get a basis for the generalised Poincaré algebra of isometries of flat space:
:$M_ = x_\mu\partial_\nu - x_\nu\partial_\mu$
:$P_\rho = \partial_\rho.$
These generate pseudo-rotations (rotations and boosts) and translations respectively. Intuitively these preserve the (pseudo)-metric at each point.

For (Pseudo-)Euclidean space of total dimension, in total there are $n(n+1)/2$ 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 $n$, this group of symmetries has the expected dimension (as a Lie group).

Heuristically, we can derive the dimension of the Killing field algebra. Treating Killing's equation $\nabla_a X_b + \nabla_b X_a = 0$ together with the identity $\nabla_a\nabla_b X_c = R^c_X_c.$ as a system of second order differential equations for $X_a$, we can determine the value of $X_a$ at any point given initial data at a point $p$. The initial data specifies $X_a(p)$ and $\nabla_a X_b(p)$, but Killing's equation imposes that the covariant derivative is antisymmetric. In total this is $n - n(n-1)/2 = n(n+1)/2$ independent values of initial data.

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

Killing fields on the hyperbolic plane

A toy example for a Killing vector field is on the upper half-plane $M = \mathbb^2_$ equipped with the Poincaré metric $g = y^\left(dx^2 + dy^2\right)$. The pair $(M, g)$ is typically called the hyperbolic plane and has Killing vector field $\partial_x$ (using standard coordinates). This should be intuitively clear since the covariant derivative $\nabla_g$ transports the metric along an integral curve generated by the vector field (whose image is parallel to the x-axis).

Furthermore, the metric is independent of $x$ from which we can immediately conclude that $\partial_x$ is a Killing field using one of the results below in this article.

The isometry group of the upper half-plane model (or rather, the component connected to the identity) is $\text(2, \mathbb)$ (see Poincaré half-plane model), and the other two Killing fields may be derived from considering the action of the generators of $\text(2, \mathbb)$ on the upper half-plane. The other two generating Killing fields are dilatation $D = x\partial_x + y\partial_y$ and the special conformal transformation $K = (x^2 - y^2)\partial_x + 2xy \partial_y$.

Killing fields on a 2-sphere

The Killing fields of the two-sphere $S^2$, or more generally the $n$-sphere $S^n$ should be "obvious" from ordinary intuition: spheres, having rotational symmetry, should possess Killing fields which generate rotations about any axis. That is, we expect $S^2$ to have symmetry under the action of the 3D rotation group SO(3).

When explicitly expressed in terms of the standard coordinate chart for $S^n$, 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 chart for the 2-sphere embedded in $\mathbb^3$ in Cartesian coordinates $(x,y,z)$ is given by
:$x = \sin\theta\cos\phi,\qquad y = \sin\theta\sin\phi,\qquad z = \cos\theta$
so that $\theta$ parametrises the height, and $\phi$ parametrises rotation about the $z$-axis.

Pulling back the standard Cartesian metric $ds^2 = dx^2 + dy^2 + dz^2$ to these coordinates gives the standard metric on the sphere,
:$ds^2 = d\theta^2 + \sin^2\theta d\phi^2$.

Intuitively, a rotation about any axis should be an isometry. In this chart, we can immediately write down the vector field which generates rotations about the $z$-axis:
:$\frac.$
Note that in these coordinates, the metric components are all independent of $\phi$, which is another way for us to see that $\partial_\phi$ is a Killing vector (see below).

Naïvely we might hope to find another Killing vector
:$\frac.$
The metric is dependent on $\theta$, so we cannot immediately deduce that it is a Killing vector. In fact, explicitly evaluating Killing's equation reveals it is not a Killing field. Intuitively, the flow generated by $\partial_\theta$ moves points downwards. Near $z=1$, points move apart, thus distorting the metric, and we can see it is not an isometry, and therefore not a Killing field.

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'' form a Lie subalgebra of vector fields on ''M''. For example, we may wish to compute the dimension of this algebra and its structure constants, and possibly identify the Lie algebra.

Since $\$ are a coordinate basis, their commutator vanishes. Unfortunately, as earlier mentioned, $\partial_\theta$ is not a Killing field. In these coordinates, it is difficult to guess other Killing fields.

We can realize the algebra of Killing fields by considering $S^2$ as an embedded manifold of $\mathbb^3$, and work in orthonormal Cartesian coordinates $x, y, z$ where commutators will turn out to be straight-forward.

The generator $\partial_\phi$ is recognized as a rotation about the $z$-axis

:$Z = x\partial_y - y\partial_x = 2\partial_\phi$

A second generator, rotations about the $x$-axis, is

:$X = z\partial_y - y\partial_z$

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

:Y = ,Z= z\partial_x - x\partial_z

The algebra given by linear combinations of these three generators closes, and we have the relations
: ,Y= Z \quad ,Z= X \quad ,Y= X. We identify this as the Lie algebra $\mathfrak(3)$

Expressing $X$ and $Y$ in terms of spherical coordinates gives

:$X = \sin\phi\partial_\theta + \cot\theta\cos\phi\partial_\phi$
and
:$Y = \cos\phi\partial_\theta - \cot\theta\sin\phi\partial_\phi$

We'd like to say these three Killing fields are a complete set of generators for the algebra. One way to see this is that as a submanifold of $\mathbb^3$, its symmetries are a strict subset of those of $\mathbb^3$. The symmetries of $\mathbb^3$ are generated by rotations and translations. This can be shown using Killing's equation. Translations do not preserve the unit sphere, but rotations do. Hence this is the complete set of generators of the algebra of Killing fields, which we can identify with $\mathfrak(3)$ the algebra of rotations.

Killing fields in 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
*:$\partial_t ~, \qquad \partial_x ~, \qquad \partial_y ~, \qquad \partial_z ~;$
* Vector fields generating three rotations, often called the ''J'' generators,
*:$-y \partial_x + x \partial_y ~, \qquad -z \partial_y + y \partial_z ~, \qquad -x \partial_z + z \partial_x ~;$
* Vector fields generating three boosts, the ''K'' generators,
*:$x \partial_t + t \partial_x~, \qquad y \partial_t + t \partial_y ~, \qquad z \partial_t + t \partial_z.$

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

Killing fields in 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 is independent of $t$, hence $\partial_t$ 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 are maximally symmetric spaces, with the $n$-dimensional versions of each possessing $\frac$ Killing fields.

Killing field of a constant coordinate

If the metric coefficients $g_ \,$ in some coordinate basis $dx^ \,$ are independent of one of the coordinates $x^ \,$, then $K^ = \delta^_ \,$ is a Killing vector, where $\delta^_ \,$ is the Kronecker delta.

To prove this, let us assume $g_,_0 = 0 \,$. Then $K^\mu = \delta^\mu_0 \,$ and $K_ = g_ K^\nu = g_ \delta^\nu_0 = g_ \,$

Now let us look at the Killing condition
:$K_ + K_ = K_ + K_ - 2\Gamma^\rho_K_\rho = g_ + g_ - g^(g_ + g_ - g_)g_ \,$

and from $g_g^ = \delta_0^\sigma \,$. The Killing condition becomes
:$g_ + g_ - (g_ + g_ - g_) = 0 \,$

that is $g_ = 0$, 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.

Conversely, if the metric $\mathbf$ admits a Killing field $X^a$, then one can construct coordinates for which $\partial_0 g_ = 0$. These coordinates are constructed by taking a hypersurface $\Sigma$ such that $X^a$ is nowhere tangent to $\Sigma$. Take coordinates $x^i$ on $\Sigma$, then define local coordinates $(t,x^i)$ where $t$ denotes the parameter along the integral curve of $X^a$ based at $(x^i)$ on $\Sigma$. In these coordinates, the Lie derivative reduces to the coordinate derivative, that is,
:$\mathcal_Xg_ = \partial_0 g_$
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 ''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 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 ''M'' is even, a Killing field must have a zero.

The covariant divergence of every Killing vector field vanishes.

If $X$ is a Killing vector field and $Y$ is a harmonic vector field, then $g(X, Y)$ is a harmonic function.

If $X$ is a Killing vector field and $\omega$ is a harmonic p-form, then $\mathcal_ \omega = 0 \,.$

Geodesics

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. Along an affinely parametrized geodesic with tangent vector $U^a$ then given the Killing vector $X_b$, the quantity $U^bX_b$ is conserved:
:$U^a\nabla_a(U^bX_b)=0$

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

Stress-energy tensor

Given a conserved, symmetric tensor $T^$, that is, one satisfying $T^ = T^$ and $\nabla_a T^=0$, which are properties typical of a stress-energy tensor, and a Killing vector $X_b$, we can construct the conserved quantity $J^a := T^X_b$ satisfying
:$\nabla_a J^a = 0.$

Cartan decomposition

As noted above, the Lie bracket of two Killing fields is still a Killing field. The Killing fields on a manifold $M$ thus form a Lie subalgebra $\mathfrak$ of all vector fields on $M.$ Selecting a point $p \in M~,$ the algebra $\mathfrak$ can be decomposed into two parts:
:$\mathfrak = \$
and
:$\mathfrak = \$

where $\nabla$ is the covariant derivative. These two parts intersect trivially but do not in general split $\mathfrak$. For instance, if $M$ is a Riemannian homogeneous space, we have $\mathfrak = \mathfrak \oplus \mathfrak$ if and only if $M$ is a Riemannian symmetric space.

Intuitively, the isometries of $M$ locally define a submanifold $N$ 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 $T_pN$ should have the same dimension as the isometries acting effectively at that point. That is, one expects $T_pN \cong \mathfrak~.$ 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 $\mathfrak;$ the remaining degenerate linear combinations define an orthogonal space $\mathfrak.$

Cartan involution

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 $\mathfrak$ and $\mathfrak,$ respectively.

This can be made more precise. Fixing a point $p \in M$ consider a geodesic $\gamma: \mathbb \to M$ passing through $p$, with $\gamma(0) = p~.$ The involution $\sigma_p$ is defined as

:$\sigma_p(\gamma(\lambda)) = \gamma(-\lambda)$

This map is an involution, in that $\sigma_p^2 = 1~.$ When restricted to geodesics along the Killing fields, it is also clearly an isometry. It is uniquely defined.

Let $G$ be the group of isometries generated by the Killing fields. The function $s_p: G \to G$ defined by
:$s_p(g) = \sigma_p \circ g \circ \sigma_p = \sigma_p \circ g \circ \sigma_p^$

is a homomorphism of $G$. Its infinitesimal $\theta_p: \mathfrak \to \mathfrak$ is
:$\theta_p(X) = \left. \frac s_p\left(e^\right) \_$

The Cartan involution is a Lie algebra homomorphism, in that
:$\theta_p$, Y= \left theta_p X, \theta_p Y\right/math>

for all $X, Y \in \mathfrak~.$ The subspace $\mathfrak$ has odd parity under the Cartan involution, while $\mathfrak$ has even parity. That is, denoting the Cartan involution at point $p \in M$ as $\theta_p$ one has
:$\left.\theta_p\_ = -Id$

and
:$\left.\theta_p\_ = +Id$

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

The above decomposition holds at all points $p \in M$ for a symmetric space $M$; 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 $T_pM \cong \mathfrak;$ 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 defined by $\mathcal_ g = \lambda g\,$ for some scalar $\lambda.$ 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 $\nabla T \,$ 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 (possibly nonmetric) 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 $\mathfrak$ of ''G''.

See also

* Affine vector field
* Curvature collineation
* Homothetic vector field
* Killing form
* Killing horizon
* Killing spinor
* Matter collineation
* Spacetime symmetries

References

{{DEFAULTSORT:Killing Vector Field
Riemannian geometry