Killing field

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

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

L

X

# g
0

.

{\displaystyle \mathcal{L}\_{X} g = 0 \,.}

In terms of the Levi-Civita connection, this is

g
(

∇
Y

X
,
Z
)
g
(
Y
,

∇
Z

X
# )
0

{\displaystyle g(\nabla\_Y X, Z) + g(Y, \nabla\_Z X) = 0 \,}

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

∇
μ

X
ν

∇

ν

X
μ

=
0

.

{\displaystyle \nabla\_\mu X\_\nu + \nabla\_{\nu} 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

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

A toy example for a Killing vector field is on the upper-half plane

# M

R

y
>
0

2

{\displaystyle M=\mathbb{R}^2\_{y > 0}}

 equipped metric

# g

y

−
2

(
d

x
2

d

y
2

)

{\displaystyle g=y^{-2}(dx^2 + dy^2)}

. The pair

(
M
,
g
)

{\displaystyle (M,g)}

 is typically called the **hyperbolic plane** and has Killing vector field

∂
x

{\displaystyle \partial\_x}

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

∇

∂
x

g

{\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).

### Killing vector in general relativity

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.

### Derivation

If the metric coefficients

g

μ
ν

{\displaystyle g\_{\mu \nu}\,}

 in some coordinate basis

d

x

a

{\displaystyle dx^{a} \,}

 are independent of one of the coordinates

x

κ

{\displaystyle x^{\kappa} \,}

, then

K

μ

=

δ

κ

μ

{\displaystyle K^{\mu} = \delta^{\mu}\_{\kappa} \,}

 is a Killing vector, where

δ

κ

μ

{\displaystyle \delta^{\mu}\_{\kappa} \,}

 is the Kronecker delta.

To prove this, let us assume

g

μ
ν

,
0

=
0

{\displaystyle g\_{\mu \nu},\_0=0 \,}

. Then

K
μ

=

δ
0
μ

{\displaystyle K^\mu=\delta^\mu\_0 \,}

 and

K

μ

=

g

μ
ν

K
ν

=

g

μ
ν

δ
0
ν

=

g

μ
0

{\displaystyle K\_{\mu}=g\_{\mu \nu} K^\nu = g\_{\mu \nu} \delta^\nu\_0= g\_{\mu 0} \,}

 Now let us look at the Killing condition

K

μ
;
ν

K

ν
;
μ

=

K

μ
,
ν

K

ν
,
μ

−
2

Γ

μ
ν

ρ

K
ρ

=

g

μ
0
,
ν

g

ν
0
,
μ

−

g

ρ
σ

(

g

σ
μ
,
ν

g

σ
ν
,
μ

−

g

μ
ν
,
σ

)

g

ρ
0

{\displaystyle K\_{\mu;\nu}+K\_{\nu;\mu}=K\_{\mu,\nu}+K\_{\nu,\mu}-2\Gamma^\rho\_{\mu\nu}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

g

ρ
0

g

ρ
σ

=

δ
0
σ

{\displaystyle g\_{\rho 0}g^{\rho \sigma} = \delta\_0^\sigma \,}

. The Killing condition becomes

g

μ
0
,
ν

g

ν
0
,
μ

−
(

g

0
μ
,
ν

g

0
ν
,
μ

−

g

μ
ν
,
0

# )
0

{\displaystyle g\_{\mu 0,\nu}+g\_{\nu 0,\mu} - ( g\_{0\mu,\nu}+g\_{0\nu,\mu}-g\_{\mu\nu,0} ) = 0 \,}

that is

g

μ
ν
,
0

=
0

{\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.

## 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.

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

X

{\displaystyle X}

 is a Killing vector field and

Y

{\displaystyle Y}

 is a harmonic vector field, then

g
(
X
,
Y
)

{\displaystyle g(X,Y)}

 is a harmonic function.

If

X

{\displaystyle X}

 is a Killing vector field and

ω

{\displaystyle \omega}

 is a harmonic p-form, then

L

X

# ω
0

.

{\displaystyle \mathcal{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. That is, along a geodesic with some affine parameter

λ
,

{\displaystyle \lambda,
}

 the equation

d

d
λ

(

K
μ

d

x
μ

d
λ

)

=
0

{\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.

## Generalizations

* Killing vector fields can be generalized to conformal Killing vector fields defined by

L

X

# g
λ
g

{\displaystyle \mathcal{L}\_{X} g = \lambda g \,}

 for some scalar

λ
.

{\displaystyle \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

∇
T

{\displaystyle \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

g

{\displaystyle \mathfrak g}

 of *G*.