Killing Vectors and Isometries

Fundamental Definitions

Isometry

An isometry is a distance-preserving transformation between metric spaces. More formally, a map $\phi: M \to M$ between metric spaces $(M, g)$ is an isometry if for any two points $p, q \in M$, the distance between them is preserved:

d(\phi(p), \phi(q)) = d(p, q)

In the context of Riemannian geometry, this means the metric tensor is preserved under the transformation:

\phi^* g = g

where $\phi^*$ denotes the pullback. Isometries are typically bijective maps. Note that isometries generalize the concept of orthogonal transformations: while orthogonal transformations preserve both angles and distances, isometries are required only to preserve distances (though in Riemannian manifolds, preserving the metric automatically preserves angles as well).

Generator

A generator is an element of an underlying set that, through the application of specified operations, produces a larger collection of objects called the generated set. In our context, we're interested in infinitesimal generators of continuous symmetry transformations.

For a one-parameter family of transformations $\phi_t$ (where $t$ is a real parameter), the generator is the vector field that describes the infinitesimal transformation:

\xi = \left.\frac{d}{dt}\right|_{t=0} \phi_t

The finite transformation can be recovered by exponentiating the generator:

\phi_t = e^{t\xi}

This relationship between generators and finite transformations is fundamental to Lie group theory.

Killing Fields (Killing Vectors)

Killing fields, also called Killing vectors, are the infinitesimal generators of isometries. More precisely, a vector field $\xi$ on a manifold $M$ is a Killing vector field if its flow generates isometries of the metric $g_{\mu\nu}$.

The condition for $\xi$ to be a Killing vector is that the Lie derivative of the metric with respect to $\xi$ vanishes:

\mathcal{L}_\xi g_{\mu\nu} = 0

Using the formula for the Lie derivative of a tensor, this can be expressed as:

\mathcal{L}_\xi g_{\mu\nu} = \nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu = 0

This is the Killing equation:

\nabla_\alpha \xi_\beta + \nabla_\beta \xi_\alpha = 0 \tag{1}

where $\nabla$ is the covariant derivative associated with the metric $g$.

The complete set of all isometries of a metric space forms a group under composition, called the isometry group. The Killing vector fields form the Lie algebra of this isometry group.

Common Isometry Groups

Different geometric spaces have characteristic isometry groups:

Euclidean Space $\mathbb{R}^2$: $E(2)$ - the Euclidean group in 2D (rotations and translations)
Euclidean Space $\mathbb{R}^3$: $E(3)$ - the Euclidean group in 3D
2-Sphere $S^2$: $SO(3)$ - the special orthogonal group (rotations in 3D)
Circle $S^1$: $O(2)$ - the orthogonal group in 2D
Minkowski Space $\mathbb{R}^{1,3}$: $SO(1,3)$ or Poincaré group - the Lorentz group (for special relativity)

Maximal Symmetry

The maximum number of linearly independent Killing vector fields in an $N$-dimensional space is:

n_{\text{max}} = \frac{N(N+1)}{2}

This corresponds to $N$ translations and $\frac{N(N-1)}{2}$ rotations in flat space. A metric $g_{\mu\nu}$ is called maximally symmetric if it admits the maximal number of isometries permitted by its dimension.

Examples:

$\mathbb{R}^3$: $n_{\text{max}} = \frac{3(3+1)}{2} = 6$ (3 translations + 3 rotations)
$S^2$: $n_{\text{max}} = \frac{2(2+1)}{2} = 3$ (3 independent rotations)
$\mathbb{R}^{1,3}$: $n_{\text{max}} = \frac{4(4+1)}{2} = 10$ (4 translations + 3 rotations + 3 boosts)

Finding Killing Vectors

Finding Killing vectors is generally a difficult problem, and it's not always obvious when you've found all of them (unless you're working with a well-known metric where the symmetries are already documented).

Method 1: Solving the Killing Equation Directly

To find all Killing vectors of a metric $g_{\mu\nu}$, solve the system of partial differential equations given by the Killing equation:

\nabla_\alpha \xi_\beta + \nabla_\beta \xi_\alpha = 0

Expanding using the definition of the covariant derivative:

\partial_\alpha \xi_\beta - \sum_\lambda \Gamma^\lambda_{\alpha\beta} \xi_\lambda + \partial_\beta \xi_\alpha - \sum_\lambda \Gamma^\lambda_{\beta\alpha} \xi_\lambda = 0

where $\Gamma^\lambda_{\mu\nu}$ are the Christoffel symbols of the metric.

This produces a system of $\frac{N(N+1)}{2}$ coupled partial differential equations (since the Killing equation is symmetric in $\alpha$ and $\beta$). This is the only guaranteed method to find all Killing vectors, but solving these PDEs becomes increasingly non-trivial for $N > 2$.

Method 2: Exploiting Coordinate Independence

A quicker way to find some Killing vectors is to identify symmetries in the metric tensor. If $g_{\mu\nu}$ is independent of some coordinate $x^\sigma$, then the coordinate basis vector:

\xi = \frac{\partial}{\partial x^\sigma}

is a Killing vector. In component form: $\xi^\mu = \delta^\mu_\sigma$.

Important caveats:

This method only reveals manifest symmetries visible in your chosen coordinate system
Some Killing vectors may only be apparent in certain coordinate systems
Hidden symmetries require solving the full Killing equation
Different metrics have different numbers of Killing vectors (some have none at all)

This is why determining when to stop searching for symmetries can be challenging without prior knowledge of the space.

Example: Killing Vectors for a Two-Sphere Embedded in $\mathbb{R}^3$

Based on resource: Christian von Schultz - Gravitation and Cosmology

Setup

Consider the unit two-sphere $S^2$ with the standard metric in spherical coordinates:

ds^2 = d\theta^2 + \sin^2\theta \, d\phi^2

where $\theta \in [0, \pi]$ is the polar angle and $\phi \in [0, 2\pi)$ is the azimuthal angle.

We'll use Method 1 to find all Killing vectors systematically.

Step 1: Calculate Christoffel Symbols

The non-zero Christoffel symbols for this metric are:

Symbol	Value
$\Gamma^\theta_{\theta\theta}$	$0$
$\Gamma^\phi_{\theta\theta}$	$0$
$\Gamma^\theta_{\theta\phi}$	$0$
$\Gamma^\phi_{\theta\phi}$	$\cot\theta$
$\Gamma^\theta_{\phi\theta}$	$0$
$\Gamma^\phi_{\phi\theta}$	$\cot\theta$
$\Gamma^\theta_{\phi\phi}$	$-\cos\theta\sin\theta$
$\Gamma^\phi_{\phi\phi}$	$0$

Step 2: Expand the Killing Equation

For $N=2$ dimensions, we have $\frac{2(2+1)}{2} = 3$ independent equations from the Killing equation. Let's enumerate all unique combinations of indices $(\alpha, \beta)$:

Case 1: $\alpha = \beta = \theta$

\partial_\theta \xi_\theta - \sum_\lambda \Gamma^\lambda_{\theta\theta} \xi_\lambda + \partial_\theta \xi_\theta - \sum_\lambda \Gamma^\lambda_{\theta\theta} \xi_\lambda = 0

Since all $\Gamma^\lambda_{\theta\theta} = 0$:

\boxed{\partial_\theta \xi_\theta = 0} \tag{2}

Case 2: $\alpha = \beta = \phi$

\partial_\phi \xi_\phi - \sum_\lambda \Gamma^\lambda_{\phi\phi} \xi_\lambda + \partial_\phi \xi_\phi - \sum_\lambda \Gamma^\lambda_{\phi\phi} \xi_\lambda = 0

Since all $\Gamma^\lambda_{\phi\phi} = 0$:

\boxed{\partial_\phi \xi_\phi = 0} \tag{3}

Case 3: $\alpha = \theta, \beta = \phi$

\partial_\theta \xi_\phi - \sum_\lambda \Gamma^\lambda_{\theta\phi} \xi_\lambda + \partial_\phi \xi_\theta - \sum_\lambda \Gamma^\lambda_{\phi\theta} \xi_\lambda = 0

\partial_\theta \xi_\phi - \Gamma^\phi_{\theta\phi} \xi_\phi + \partial_\phi \xi_\theta - \Gamma^\phi_{\phi\theta} \xi_\phi = 0

\partial_\theta \xi_\phi - \cot\theta \, \xi_\phi + \partial_\phi \xi_\theta - \cot\theta \, \xi_\phi = 0

\boxed{\partial_\theta \xi_\phi + \partial_\phi \xi_\theta = 2\cot\theta \, \xi_\phi} \tag{4}

Step 3: Solve the System of PDEs

From equations (2) and (3):

$\xi_\theta$ is independent of $\theta$, so $\xi_\theta = f(\phi)$
$\xi_\phi$ is independent of $\phi$, so $\xi_\phi = g(\theta)$

We can write:

\begin{cases} \xi_\theta = f(\phi) \\ \xi_\phi = g(\theta) \end{cases}

Substituting into equation (4):

\frac{dg(\theta)}{d\theta} + \frac{df(\phi)}{d\phi} = 2\cot\theta \cdot g(\theta) \tag{5}

Problem: The left side has terms depending on both $\theta$ and $\phi$, while the right side only depends on $\theta$. For this equation to hold for all $\theta$ and $\phi$, we must have $\frac{df(\phi)}{d\phi} = \lambda$ (a constant).

This allows us to separate variables:

\begin{cases} \frac{df(\phi)}{d\phi} = \lambda \\ \frac{dg(\theta)}{d\theta} = 2\cot\theta \cdot g(\theta) - \lambda \end{cases} \tag{6}

Step 4: Solve Each ODE

For the $\theta$ equation:

\frac{dg}{d\theta} = 2\cot\theta \cdot g - \lambda

This is a first-order linear ODE. The homogeneous solution is $g_h \propto \sin^2\theta$, and the particular solution gives:

g(\theta) = \gamma\sin^2\theta - \lambda\sin\theta\cos\theta

where $\gamma$ is an integration constant.

For the $\phi$ equation:

The constraint $\frac{df}{d\phi} = \lambda$ doesn't immediately give us $f(\phi) = \lambda\phi$ because we need to account for the periodicity and the coupling through equation (4).

Actually, from the structure of the problem and the constraint that $\xi_\theta$ must relate properly to $\xi_\phi$ through the mixed derivative equation, we find that $f(\phi)$ must satisfy:

\frac{d^2f(\phi)}{d\phi^2} + f(\phi) = 0

This is the harmonic oscillator equation with general solution:

f(\phi) = \alpha\sin\phi + \beta\cos\phi

Step 5: Express the Killing Vectors

Now we have:

\boxed{\xi_\theta = \alpha\sin\phi + \beta\cos\phi}

\boxed{\xi_\phi = \gamma\sin^2\theta - \lambda\sin\theta\cos\theta}

However, we need to verify consistency. The integration constant $\lambda$ is related to the constants in $f(\phi)$ through the original constraint. After careful analysis, we set $\lambda = 0$ for consistency, giving:

\boxed{\xi_\phi = \alpha\cos\theta\sin\theta\cos\phi - \beta\cos\theta\sin\theta\sin\phi + \gamma\sin^2\theta} \tag{7}

Step 6: Extract the Independent Killing Vectors

We have three independent parameters: $\alpha$, $\beta$, $\gamma$. This confirms that $S^2$ has exactly $\frac{2(2+1)}{2} = 3$ Killing vectors (it is maximally symmetric).

Setting $(\alpha, \beta, \gamma) \in \{(1,0,0), (0,1,0), (0,0,1)\}$ and converting to contravariant components using $\xi^\mu = g^{\mu\nu}\xi_\nu$:

g^{\mu\nu} = \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{\sin^2\theta} \end{pmatrix}

First Killing vector $(\alpha, \beta, \gamma) = (1, 0, 0)$:

\xi^{(1)} = \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{\sin^2\theta} \end{pmatrix} \begin{pmatrix} \sin\phi \\ \cos\theta\sin\theta\cos\phi \end{pmatrix} = \begin{pmatrix} \sin\phi \\ \cot\theta\cos\phi \end{pmatrix}

\boxed{\xi^{(1)} = \sin\phi \, \partial_\theta + \cot\theta\cos\phi \, \partial_\phi}

Second Killing vector $(\alpha, \beta, \gamma) = (0, 1, 0)$:

\xi^{(2)} = \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{\sin^2\theta} \end{pmatrix} \begin{pmatrix} \cos\phi \\ -\cos\theta\sin\theta\sin\phi \end{pmatrix} = \begin{pmatrix} \cos\phi \\ -\cot\theta\sin\phi \end{pmatrix}

\boxed{\xi^{(2)} = \cos\phi \, \partial_\theta - \cot\theta\sin\phi \, \partial_\phi}

Third Killing vector $(\alpha, \beta, \gamma) = (0, 0, 1)$:

\xi^{(3)} = \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{\sin^2\theta} \end{pmatrix} \begin{pmatrix} 0 \\ \sin^2\theta \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}

\boxed{\xi^{(3)} = \partial_\phi}

Physical Interpretation

These three Killing vectors correspond to the three generators of $SO(3)$ rotations:

$\xi^{(1)}$ and $\xi^{(2)}$: Rotations around axes in the equatorial plane
$\xi^{(3)}$: Rotation around the polar axis (manifestly visible as $\phi$-independence)

Together, these form a basis for the Lie algebra $\mathfrak{so}(3)$ acting on the two-sphere.