Killing Vectors, Spacetime Symmetries, and Noether's Theorem

1. Defining Spacetime Symmetry: Isometries and the Lie Derivative

A symmetry of a spacetime is a transformation that preserves its geometry. Since the geometry is encoded in the metric tensor $g_{\mu \nu }$, a symmetry is a continuous transformation of the spacetime manifold onto itself that leaves the metric unchanged. Such a transformation is called an isometry.

How can we describe these continuous transformations? We use the concept of a flow generated by a vector field, $\xi$. Imagine the vector field $\xi$ as defining a velocity field on the manifold. The integral curves of this field are the paths that points follow under the flow. An isometry, then, corresponds to a flow along a special vector field $\xi$ that does not alter the metric.

The mathematical tool that precisely captures the change of a tensor field along the flow of a vector field is the Lie derivative, ${\mathcal{L}}_{\xi }$. You may recall this from our first module on mathematical foundations. The Lie derivative ${\mathcal{L}}_{\xi }T$ of a tensor $T$ essentially compares the value of $T$ at a point to the value of $T$ at a nearby point, after "dragging" it back along the flow of $\xi$.

A spacetime symmetry is generated by a vector field $\xi$ if the metric tensor is "Lie transported" along $\xi$, meaning its Lie derivative with respect to $\xi$ vanishes:
${\mathcal{L}}_{\xi }{g}_{\mu \nu }=0$
Any vector field $\xi$ that satisfies this condition is called a Killing vector field.

To better understand the Lie derivative and its role in defining symmetries, the following video provides a rigorous and visual derivation.

2. Killing's Equation

As derived in the video, the condition ${\mathcal{L}}_{\xi }{g}_{\mu \nu }=0$ can be expanded using the general formula for the Lie derivative of a (0,2) tensor:

This equation, involving partial derivatives, is a system of linear partial differential equations for the components of the vector field ${\xi }^{\mu }$.

While correct, this form is not manifestly covariant. A more elegant and useful form of Killing's equation uses the covariant derivative. By making use of the metric compatibility condition (${\nabla }_{\alpha }{g}_{\mu \nu }=0$), one can show that the above equation is exactly equivalent to:

This is Killing's equation. It states that the symmetric part of the covariant derivative of the covector ${\xi }_{\nu }$ must be zero. The detailed proof of this equivalence is found in Appendix A of the Chapter 9 Symmetries resource, which we will use shortly.

A very practical consequence arises from the partial derivative form:
If the components of the metric tensor $g_{\mu \nu }$ are independent of a particular coordinate $x^k$ in some coordinate system, then the basis vector field $\xi =\frac{\partial }{\partial x^k}$ is a Killing vector field.

For example, if a metric is time-independent (${\partial }_t{g}_{\mu \nu }=0$), then ${\xi }^{\mu }=(1,0,0,0)$ (in coordinates $(t,x,y,z)$) is a timelike Killing vector. If a metric is axially symmetric (${\partial }_{\varphi }{g}_{\mu \nu }=0$), then $\xi =\frac{\partial }{\partial \varphi }$ is a spacelike Killing vector. This gives us a powerful tool to immediately identify symmetries when we are given a metric in a convenient coordinate system.

3. Killing Vectors and Conserved Quantities

The profound physical importance of Killing vectors is that they directly correspond to conserved quantities. This is the GR equivalent of Noether's theorem.

To see how this works, we will use the following resource which provides clear, self-contained derivations.

Let's summarize the key results from these sections.

3.1 Conserved Quantities for Freely Falling Particles

Consider a test particle of mass $m$ moving along a geodesic. Its path is described by the geodesic equation, and its 4-velocity is $U^{\mu }=d{x}^{\mu }/d\tau$. If the spacetime possesses a Killing vector field ${\xi }^{\mu }$, then the quantity
$C={\xi }_{\mu }{U}^{\mu }=g_{\mu \nu }{\xi }^{\mu }{U}^{\nu }$
is conserved along the geodesic. That is, $\frac{dC}{d\tau }=0$.

The proof, as shown in Section 9.3 of the reading, is elegant. We calculate the derivative of $C$ along the particle's worldline:

The second term vanishes by the geodesic equation ($U^{\nu }{\nabla }_{\nu }{U}^{\mu }=0$). The first term can be rewritten using the symmetry of $U^{\nu }{U}^{\mu }$:

The term in the parentheses is precisely the left-hand side of Killing's equation, which is zero. Therefore, $\frac{dC}{d\tau }=0$.

The physical interpretation depends on the nature of the Killing vector:

• If ${\xi }^{\mu }$ is a timelike Killing vector (associated with time-translation symmetry), the conserved quantity $C$ corresponds to the conserved energy of the particle.
• If ${\xi }^{\mu }$ is a spacelike Killing vector (associated with spatial translation or rotation), $C$ corresponds to a component of the particle's conserved linear or angular momentum.

3.2 Conserved Quantities for Fields

This principle extends beyond single particles to continuous matter and fields described by a stress-energy tensor $T^{\mu \nu }$. Recall the covariant conservation law ${\nabla }_{\nu }{T}^{\mu \nu }=0$. As we've discussed, this isn't a true integral conservation law on its own.

However, if a Killing vector ${\xi }_{\mu }$ exists, we can construct a vector field $J^{\nu }={\xi }_{\mu }{T}^{\mu \nu }$. Let's calculate its ordinary divergence. It can be shown (as in Section 9.4 of the reading) that the covariant divergence of this current is zero:

The second term is zero by the conservation of $T^{\mu \nu }$. The first term vanishes because ${\nabla }_{\nu }{\xi }_{\mu }$ is antisymmetric in its indices (due to Killing's equation), while $T^{\mu \nu }$ is symmetric. The contraction of an antisymmetric tensor with a symmetric one is always zero.

Since ${\nabla }_{\nu }{J}^{\nu }=0$, this implies the ordinary divergence of a related vector density is zero:

This is a genuine conservation law. It means we can define a conserved charge $Q={\int }_{\Sigma }\sqrt{-g}{J}^0{d}^{3}x$ over a spacelike slice $\Sigma$, which will be constant in time. This conserved charge represents the total energy, momentum, or angular momentum of the matter fields, depending on the nature of the Killing vector $\xi$.

Conclusion

Today, we have established a fundamental bridge between the geometry of spacetime and the laws of physics that unfold within it. Symmetries of the metric, known as isometries, are generated by Killing vector fields.

Key Takeaways:

• A spacetime symmetry (isometry) is a continuous transformation that leaves the metric tensor $g_{\mu \nu }$ unchanged.
• These symmetries are generated by Killing vector fields, $\xi$, which are defined by the condition that the Lie derivative of the metric along them vanishes: ${\mathcal{L}}_{\xi }{g}_{\mu \nu }=0$.
• This condition is equivalent to the covariant Killing's equation: ${\nabla }_{\mu }{\xi }_{\nu }+{\nabla }_{\nu }{\xi }_{\mu }=0$.
• For every Killing vector, there is an associated conserved quantity.
  • For a particle on a geodesic with 4-velocity $U^{\mu }$, the scalar $C={\xi }_{\mu }{U}^{\mu }$ is constant.
  • For fields with stress-energy tensor $T^{\mu \nu }$, the current $J^{\nu }={\xi }_{\mu }{T}^{\mu \nu }$ is conserved (${\nabla }_{\nu }{J}^{\nu }=0$), leading to an integrated conserved charge.
• Timelike Killing vectors correspond to energy conservation, while spacelike Killing vectors correspond to momentum or angular momentum conservation.

Preview of Next Module:

We have now completed our survey of the fundamental principles of General Relativity. We have the field equations, we know how to set them up as a dynamical system, and we have a powerful tool—symmetries—to help us find and interpret solutions.

In the next module, we will put all this machinery to work. Our first major goal will be to derive the most famous exact solution of the Einstein Field Equations: the Schwarzschild metric. This describes the spacetime outside a static, spherically symmetric, uncharged body, like a star or a non-rotating black hole. We will use the power of Killing vectors to make this derivation tractable, and then spend several lessons analyzing the fascinating and bizarre properties of this solution.