# Killing vector 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:

${\mathcal {L}}_{X}g=0\,.$ In terms of the Levi-Civita connection, this is

$g(\nabla _{Y}X,Z)+g(Y,\nabla _{Z}X)=0\,$ for all vectors Y and Z. In local coordinates, this amounts to the Killing equation

$\nabla _{\mu }X_{\nu }+\nabla _{\nu }X_{\mu }=0\,.$ 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 $M=\mathbb {R} _{y>0}^{2}$ equipped metric $g=y^{-2}(dx^{2}+dy^{2})$ . 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 $\nabla _{\partial _{x}}g$ 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 the 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 $g_{\mu \nu }\,$ in some coordinate basis $dx^{a}\,$ are independent of one of the coordinates $x^{\kappa }\,$ , then $K^{\mu }=\delta _{\kappa }^{\mu }\,$ is a Killing vector, where $\delta _{\kappa }^{\mu }\,$ is the Kronecker delta.

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

$K_{\mu ;\nu }+K_{\nu ;\mu }=K_{\mu ,\nu }+K_{\nu ,\mu }-2\Gamma _{\mu \nu }^{\rho }K_{\rho }=g_{\mu 0,\nu }+g_{\nu 0,\mu }-g^{\rho \sigma }(g_{\sigma \mu ,\nu }+g_{\sigma \nu ,\mu }-g_{\mu \nu ,\sigma })g_{\rho 0}\,$ and from $g_{\rho 0}g^{\rho \sigma }=\delta _{0}^{\sigma }\,$ . The Killing condition becomes

$g_{\mu 0,\nu }+g_{\nu 0,\mu }-(g_{0\mu ,\nu }+g_{0\nu ,\mu }-g_{\mu \nu ,0})=0\,$ that is $g_{\mu \nu ,0}=0$ , 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 $X$ 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 ${\mathcal {L}}_{X}\omega =0\,.$ ### 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 $\lambda ,$ the equation

${\frac {d}{d\lambda }}\left(K_{\mu }{\frac {dx^{\mu }}{d\lambda }}\right)=0$ 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 ${\mathcal {L}}_{X}g=\lambda g\,$ 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 ${\mathfrak {g}}$ of G.