Killing vector field explained

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:[1]

l{L}Xg=0.

In terms of the Levi-Civita connection, this is

g\left(\nablaYX,Z\right)+g\left(Y,\nablaZX\right)=0

for all vectors

Y

and

Z

. In local coordinates, this amounts to the Killing equation[2]

\nabla\muX\nu+\nabla\nuX\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 counterclockwise 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 on the hyperbolic plane

M=

2
R
y>0
equipped with the Poincaré metric

g=y-2\left(dx2+dy2\right)

. The pair

(M,g)

is typically called the hyperbolic plane and has Killing vector field

\partialx

(using standard coordinates). This should be intuitively clear since the covariant derivative
\nabla
\partialx

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

\partialx

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

SL(2,R)

(see Poincaré half-plane model), and the other two Killing fields may be derived from considering the action of the generators of

SL(2,R)

on the upper half-plane. The other two generating Killing fields are dilatation

D=x\partialx+y\partialy

and the special conformal transformation

K=(x2-

2)\partial
y
x

+2xy\partialy

.

Killing fields on a 2-sphere

The Killing fields of the two-sphere

S2

, or more generally the

n

-sphere

Sn

should be obvious from ordinary intuition: spheres, having rotational symmetry, should possess Killing fields which generate rotations about any axis. That is, we expect

S2

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. This is not possible in general, and so this example is of very limited educational value.

The conventional chart for the 2-sphere embedded in

R3

in Cartesian coordinates

(x,y,z)

is given by

x=\sin\theta\cos\phi,    y=\sin\theta\sin\phi,    z=\cos\theta

so that

\theta

parametrises the height, and

\phi

parametrises rotation about the

z

-axis.

The pullback of the standard Cartesian metric

ds2=dx2+dy2+dz2

gives the standard metric on the sphere,

ds2=d\theta2+\sin2\thetad\phi2

.

Intuitively, a rotation about any axis should be an isometry. In this chart, the vector field which generates rotations about the

z

-axis:
\partial
\partial\phi

.

In these coordinates, the metric components are all independent of

\phi

, which shows that

\partial\phi

is a Killing field.

The vector field

\partial
\partial\theta
is not a Killing field; the coordinate

\theta

explicitly appears in the metric. The flow generated by

\partial\theta

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.

The generator

\partial\phi

is recognized as a rotation about the

z

-axis

Z=x\partialy-y\partialx=\sin2\theta\partial\phi

A second generator, for rotations about the

x

-axis, is

X=z\partialy-y\partialz

The third generator, for rotations about the

y

-axis, is

Y=z\partialx-x\partialz

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

[X,Y]=Z[Y,Z]=X[Z,X]=Y.

This is the Lie algebra

ak{so}(3)

.

Expressing

X

and

Y

in terms of spherical coordinates gives

X=\sin2\theta(\sin\phi\partial\theta+\cot\theta\cos\phi\partial\phi)

and

Y=\sin2\theta(\cos\phi\partial\theta-\cot\theta\sin\phi\partial\phi)

That these three vector fields are actually Killing fields can be determined in two different ways. One is by explicit computation: just plug in explicit expressions for

l{L}Xg

and chug to show that

l{L}Xg=l{L}Yg=l{L}Zg=0.

This is a worth-while exercise. Alternately, one can recognize

X,Y

and

Z

are the generators of isometries in Euclidean space, and since the metric on the sphere is inherited from metric in Eucliden 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.

Z

vanishes at the north and south poles; likewise,

X

and

Y

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).

X,Y

and

Z

are not of unit length. One can normalize by dividing by the common factor of

\sin2\theta

appearing in all three expressions. However, in that case, the fields are no longer smooth: for example,

\partial\phi=X/\sin2\theta

is singular (non-differentiable) at the north and south poles.

X,Y

and

Z

that vanishes: these three vectors are an over-complete basis for the two-dimensional tangent plane at that point.

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

\partialt~,    \partialx~,    \partialy~,    \partialz~;

-y\partialx+x\partialy~,    -z\partialy+y\partialz~,    -x\partialz+z\partialx~;

x\partialt+t\partialx~,    y\partialt+t\partialy~,    z\partialt+t\partialz.

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 flat space

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

Ka

,

\nablaa\nablabKc-\nablab\nablaaKc=

d{}
R
cab

Kd

(using abstract index notation) where
a{}
R
bcd
is the Riemann curvature tensor, the following identity may be proven for a Killing field

Xa

:

\nablaa\nablabXc=

d{}
R
acb

Xd.

When the base manifold

M

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

\partial\mu\partial\nuX\rho=0.

Integrating and imposing the Killing equation allows us to write the general solution to

X\rho

as

X\rho=\omega\rho\sigmax\sigma+c\rho

where

\omega\mu\nu=-\omega\nu\mu

is antisymmetric. By taking appropriate values of

\omega\mu\nu

and

c\rho

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

M\mu\nu=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

\{x\inRp,q:η(x,x)=\pm

1
\kappa2

\}

which have O(pq) 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

\nablaaXb+\nablabXa=0

together with the identity

\nablaa\nablabXc=

c{}
R
bad

Xc.

as a system of second order differential equations for

Xa

, we can determine the value of

Xa

at any point given initial data at a point

p

. The initial data specifies

Xa(p)

and

\nablaaXb(p)

, but Killing's equation imposes that the covariant derivative is antisymmetric. In total this is

n2-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 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

\partialt

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
n(n+1)
2
Killing fields.

Killing field of a constant coordinate

If the metric coefficients

g\mu

in some coordinate basis

dxa

are independent of one of the coordinates

x\kappa

, then

K\mu=

\mu
\delta
\kappa

is a Killing vector, where
\mu
\delta
\kappa

is the Kronecker delta.[3]

To prove this, let us assume

g\mu,0=0

. Then

K\mu=

\mu
\delta
0

and

K\mu=g\muK\nu=g\mu

\nu
\delta
0

=g\mu

Now let us look at the Killing condition

K\mu;\nu+K\nu;\mu=K\mu,\nu+K\nu,\mu-

\rho
2\Gamma
\mu\nu

K\rho=g\mu+g\nu-g\rho\sigma(g\sigma\mu,\nu+g\sigma\nu,\mu-g\mu\nu,\sigma)g\rho

and from

g\rhog\rho=

\sigma
\delta
0

. The Killing condition becomes

g\mu+g\nu-(g0\mu,\nu+g0\nu,\mu-g\mu\nu,0)=0

that is

g\mu\nu,0=0

, which is true.

Conversely, if the metric

g

admits a Killing field

Xa

, then one can construct coordinates for which

\partial0g\mu\nu=0

. These coordinates are constructed by taking a hypersurface

\Sigma

such that

Xa

is nowhere tangent to

\Sigma

. Take coordinates

xi

on

\Sigma

, then define local coordinates

(t,xi)

where

t

denotes the parameter along the integral curve of

Xa

based at

(xi)

on

\Sigma

. In these coordinates, the Lie derivative reduces to the coordinate derivative, that is,

l{L}Xg\mu\nu=\partial0g\mu\nu

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

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

l{L}X\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

Ua

then given the Killing vector

Xb

, the quantity
bX
U
b
is conserved:
bX
U
b)=0

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

Stress-energy tensor

Given a conserved, symmetric tensor

Tab

, that is, one satisfying

Tab=Tba

and

\nablaaTab=0

, which are properties typical of a stress-energy tensor, and a Killing vector

Xb

, we can construct the conserved quantity

Ja:=TabXb

satisfying

\nablaaJa=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

ak{g}

of all vector fields on

M.

Selecting a point

p\inM~,

the algebra

ak{g}

can be decomposed into two parts:

ak{h}=\{X\inak{g}:X(p)=0\}

and

ak{m}=\{X\inak{g}:\nablaX(p)=0\}

where

\nabla

is the covariant derivative. These two parts intersect trivially but do not in general split

ak{g}

. For instance, if

M

is a Riemannian homogeneous space, we have

ak{g}=ak{h}ak{m}

if and only if

M

is a Riemannian symmetric space.[5]

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

TpN

should have the same dimension as the isometries acting effectively at that point. That is, one expects

TpN\congak{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

ak{m};

the remaining degenerate linear combinations define an orthogonal space

ak{h}.

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

ak{p}

and

ak{m},

respectively.

This can be made more precise. Fixing a point

p\inM

consider a geodesic

\gamma:R\toM

passing through

p

, with

\gamma(0)=p~.

The involution

\sigmap

is defined as

\sigmap(\gamma(λ))=\gamma()

This map is an involution, in that

2
\sigma
p

=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

sp:G\toG

defined by

sp(g)=\sigmap\circg\circ\sigmap=\sigmap\circg\circ

-1
\sigma
p

is a homomorphism of

G

. Its infinitesimal

\thetap:ak{g}\toak{g}

is

\thetap(X)=\left.

d
dλ
λX
s
p\left(e

\right)\right|λ=0

The Cartan involution is a Lie algebra homomorphism, in that

\thetap[X,Y]=\left[\thetapX,\thetapY\right]

for all

X,Y\inak{g}~.

The subspace

ak{m}

has odd parity under the Cartan involution, while

ak{h}

has even parity. That is, denoting the Cartan involution at point

p\inM

as

\thetap

one has

\left.\thetap\right|ak{m

} = -Id

and

\left.\thetap\right|ak{h

} = +Id

where

Id

is the identity map. From this, it follows that the subspace

ak{h}

is a Lie subalgebra of

ak{g}

, in that

[ak{h},ak{h}]\subsetak{h}~.

As these are even and odd parity subspaces, the Lie brackets split, so that

[ak{h},ak{m}]\subsetak{m}

and

[ak{m},ak{m}]\subsetak{h}~.

The above decomposition holds at all points

p\inM

for a symmetric space

M

; proofs can be found in Jost.[6] 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

TpM\congak{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.

Generalizations

l{L}Xg=λg

for some scalar

λ.

The derivatives of one parameter families of conformal maps are conformal Killing fields.

\nablaT

vanishes. Examples of manifolds with Killing tensors include the rotating black hole and the FRW cosmology.[7]

ak{g}

of G.

See also

Notes and References

  1. Book: Jost, Jurgen. Riemannian Geometry and Geometric Analysis. Berlin . Springer-Verlag . 2002 . 3-540-42627-2.
  2. Book: Adler, Ronald . Bazin, Maurice . Schiffer, Menahem . Introduction to General Relativity . registration . Second . New York . McGraw-Hill . 1975 . 0-07-000423-4. . See chapters 3, 9.
  3. Book: Misner, Thorne, Wheeler . Gravitation . 1973 . W H Freeman and Company. 0-7167-0344-0.
  4. Book: Carroll, Sean. Spacetime and Geometry: An Introduction to General Relativity . limited . Addison Wesley. 2004. 133–139. 9780805387322 .
  5. Olmos, Carlos; Reggiani, Silvio; Tamaru, Hiroshi (2014). The index of symmetry of compact naturally reductive spaces. Math. Z. 277, 611–628. DOI 10.1007/s00209-013-1268-0
  6. Jurgen Jost, (2002) "Riemmanian Geometry and Geometric Analysis" (Third edition) Springer. (See section 5.2 pages 241-251.)
  7. Book: Carroll, Sean . Spacetime and Geometry: An Introduction to General Relativity . limited . Addison Wesley. 2004. 263, 344. 9780805387322 .