Killing vector field

A toy example for a Killing vector field is on the upper half-plane ${\displaystyle M=\mathbb {R} _{y>0}^{2}}$ equipped with the Poincaré metric ${\displaystyle g=y^{-2}(dx^{2}+dy^{2})}$. The pair ${\displaystyle (M,g)}$ is typically called the hyperbolic plane and has Killing vector field ${\displaystyle \partial _{x}}$ (using standard coordinates). This should be intuitively clear since the covariant derivative ${\displaystyle \nabla _{\partial _{x}}g}$ transports the metric along an integral curve generated by the vector field (whose image is parallel to the x-axis).

The Killing fields on the two-sphere ${\displaystyle S^{2}}$, or any sphere, should be, in a sense, "obvious" from ordinary intuition: spheres, being sphere-symmetric, should posses 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

${\displaystyle g(\Omega )=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}$.

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

${\displaystyle U=\partial _{\phi }}$

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

${\displaystyle V=\partial _{\theta }.}$

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 ${\displaystyle e_{1},\cdots ,e_{n}}$ of this algebra, what are the numbers ${\displaystyle {f_{ij}}^{k}}$ appearing in ${\displaystyle [e_{i},e_{j}]={f_{ij}}^{k}e_{k}?}$

Direct computation of ${\displaystyle W=[U,V]}$ leads to an unenlightening explosion of sines and cosines. This is perhaps not obvious; certainly, at the equator ${\displaystyle \theta =\pi /2}$, one has that ${\displaystyle [\partial _{\phi },\partial _{\theta }]=0.}$ However, moving off of the equator, the two vector fields ${\displaystyle \partial _{\phi }}$ and ${\displaystyle \partial _{\theta }}$ are no longer orthonormal, and so, in general one has ${\displaystyle [\partial _{\phi },\partial _{\theta }]\neq 0}$ for a point ${\displaystyle (\theta ,\phi )\in S^{2}}$ in general position. Worse, to obtain the dimension of the algebra, one must either determine that ${\displaystyle U,V,W}$ 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 ${\displaystyle [U,W]}$ and ${\displaystyle [U,[U,W]]}$ 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 ${\displaystyle x,y,z}$ where commutators are straight-forward. The conventional 3-space coordinate system is given by

${\displaystyle x=\sin \theta \cos \phi \qquad y=\sin \theta \sin \phi \qquad z=\cos \theta }$

The generator ${\displaystyle \partial _{\phi }}$ is recognized as a rotation about the ${\displaystyle z}$-axis

${\displaystyle R=x\partial _{y}-y\partial _{x}=\sin ^{2}\theta \,\partial _{\phi }}$

A second generator, rotations about the ${\displaystyle x}$-axis, is plainly

${\displaystyle S=z\partial _{y}-y\partial _{z}}$

Commuting these two, ${\displaystyle T=[R,S]}$ one promptly finds a third generator for rotations about the ${\displaystyle y}$-axis

${\displaystyle T=z\partial _{x}-x\partial _{z}}$

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

${\displaystyle [R,S]=T\quad [S,T]=R\quad [T,R]=S}$

One concludes that the Lie algebra for the Killing fields on the two-sphere is three-dimensional, and that the set ${\displaystyle R,S,T}$ provide a complete basis for the algebra. That these satisfy ${\displaystyle {\mathcal {L}}_{R}g={\mathcal {L}}_{S}g={\mathcal {L}}_{T}g=0}$ 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 ${\displaystyle x,y,z}$ coordinate system, one can apply the Hodge dual (the cross-product in three dimensions). The resulting vectors do not lie in the tangent space ${\displaystyle TS^{2}}$, and so are "outside of the manifold". They are everywhere normal to the sphere; the coordinates ${\displaystyle x,y,z}$ are extrinsic, as compared to the intrinsic coordinates ${\displaystyle \theta ,\phi }$. The utility of doing this is that now, in the embedding space ${\displaystyle \mathbb {R} ^{3}}$, the Hodge duals ${\displaystyle *R,*S,*T}$ are globally orthonormal (i.e. are orthonormal at every point on the sphere.)

Working in the intrinsic coordinate system ${\displaystyle \theta ,\phi }$, 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 ${\displaystyle z}$-axis. Normalizing this, and expressing these in spherical coordinates, one has

${\displaystyle R^{\prime }={\frac {R}{\sin ^{2}\theta }}=\partial _{\phi }}$

${\displaystyle S^{\prime }={\frac {S}{\sin ^{2}\theta }}=\sin \phi \,\partial _{\theta }+\cot \theta \,\cos \phi \,\partial _{\phi }}$

${\displaystyle T^{\prime }={\frac {T}{\sin ^{2}\theta }}=\cos \phi \,\partial _{\theta }-\cot \theta \,\sin \phi \,\partial _{\phi }}$

and one may readily verify that the commutators still hold:

${\displaystyle [R^{\prime },S^{\prime }]=T^{\prime }\quad [S^{\prime },T^{\prime }]=R^{\prime }\quad [T^{\prime },R^{\prime }]=S^{\prime }}$

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).

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,

  ${\displaystyle -y\partial _{x}+x\partial _{y}~,\qquad -z\partial _{y}+y\partial _{z}~,\qquad -x\partial _{z}+z\partial _{x}~;}$

• Vector fields generating three boosts, the K generators,

  ${\displaystyle x\partial _{t}+t\partial _{x}~,\qquad y\partial _{t}+t\partial _{y}~,\qquad z\partial _{t}+t\partial _{z}.}$

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

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.

If the metric coefficients ${\displaystyle g_{\mu \nu }\,}$ in some coordinate basis ${\displaystyle dx^{a}\,}$ are independent of one of the coordinates ${\displaystyle x^{\kappa }\,}$, then ${\displaystyle K^{\mu }=\delta _{\kappa }^{\mu }\,}$ is a Killing vector, where ${\displaystyle \delta _{\kappa }^{\mu }\,}$ is the Kronecker delta.[3]

To prove this, let us assume ${\displaystyle g_{\mu \nu },_{0}=0\,}$. Then ${\displaystyle K^{\mu }=\delta _{0}^{\mu }\,}$ and ${\displaystyle K_{\mu }=g_{\mu \nu }K^{\nu }=g_{\mu \nu }\delta _{0}^{\nu }=g_{\mu 0}\,}$
Now let us look at the Killing condition

${\displaystyle K_{\mu$ ;\nu }+K_{\nu ;\mu }=K_{\mu ,\nu }+K_{\nu ,\mu }-2\Gamma _{\mu \nu }^{\rho }K_{\rho }=g_{\mu 0,\nu }+g_{\nu 0,\mu }-g^{\rho \sigma }(g_{\sigma \mu ,\nu }+g_{\sigma \nu ,\mu }-g_{\mu \nu ,\sigma })g_{\rho 0}\,}

and from ${\displaystyle g_{\rho 0}g^{\rho \sigma }=\delta _{0}^{\sigma }\,}$. The Killing condition becomes

${\displaystyle g_{\mu 0,\nu }+g_{\nu 0,\mu }-(g_{0\mu ,\nu }+g_{0\nu ,\mu }-g_{\mu \nu ,0})=0\,}$

that is ${\displaystyle g_{\mu \nu ,0}=0}$, 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.

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 ${\displaystyle \lambda ,}$ the equation

${\displaystyle {\frac {d}{d\lambda }}\left(K_{\mu }{\frac {dx^{\mu }}{d\lambda }}\right)=0}$

is satisfied. This aids in analytically studying motions in a spacetime with symmetries.[4]

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

${\displaystyle {\mathfrak {h}}=\{X\in {\mathfrak {g}}:X(p)=0\}}$

and

${\displaystyle {\mathfrak {m}}=\{X\in {\mathfrak {g}}:\nabla X(p)=0\}}$

where ${\displaystyle \nabla }$ is the covariant derivative. These two parts are orthogonal, and split the algebra, in that ${\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}$ and ${\displaystyle {\mathfrak {m}}\cap {\mathfrak {h}}=\varnothing ~.}$

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

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 ${\displaystyle {\mathfrak {p}}}$ and ${\displaystyle {\mathfrak {m}},}$ respectively.

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

${\displaystyle \sigma _{p}(\gamma (\lambda ))=\gamma (-\lambda )}$

This map is an involution, in that ${\displaystyle \sigma _{p}^{2}=1~.}$ When restricted to geodesics along the Killing fields, it is also clearly an isometry. It is uniquely defined. Let ${\displaystyle G}$ be the group of isometries generated by the Killing fields. The function ${\displaystyle s_{p}:G\to G}$ defined by

${\displaystyle s_{p}(g)=\sigma _{p}\circ g\circ \sigma _{p}=\sigma _{p}\circ g\circ \sigma _{p}^{-1}}$

is a homomorphism of ${\displaystyle G}$. Its infinitesimal ${\displaystyle \theta _{p}:{\mathfrak {g}}\to {\mathfrak {g}}}$ is

${\displaystyle \theta _{p}(X)=\left.{\frac {d}{d\lambda }}s_{p}(e^{\lambda X})\right|_{\lambda =0}}$

The Cartan involution is a Lie algebra homomorphism, in that

${\displaystyle \theta _{p}[X,Y]=[\theta _{p}X,\theta _{p}Y]}$

for all ${\displaystyle X,Y\in {\mathfrak {g}}~.}$ The subspace ${\displaystyle {\mathfrak {m}}}$ has odd parity under the Cartan involution, while ${\displaystyle {\mathfrak {h}}}$ has even parity. That is, denoting the Cartan involution at point ${\displaystyle p\in M}$ as ${\displaystyle \theta _{p}}$ one has

${\displaystyle \left.\theta _{p}\right|_{\mathfrak {m}}=-Id}$

and

${\displaystyle \left.\theta _{p}\right|_{\mathfrak {h}}=+Id}$

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

The above decomposition holds at all points ${\displaystyle p\in M}$ for a symmetric space ${\displaystyle M}$; proofs can be found in Jost.[5] 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 ${\displaystyle T_{p}M\cong {\mathfrak {m}};}$ 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.