4.3 Symmetries

We introduced the three spacetimes – Minkowski, de Sitter and anti-de Sitter – as simple examples of solutions to the Einstein equations. In fact, what makes them special are their symmetries.

The symmetries of Minkowski space are very familiar: they consist of translations and rotations in space and time, the latter splitting into genuine rotations and Lorentz boosts. It’s hard to overstate the importance of these symmetries: on a fixed Minkowski background they are responsible for the existence of energy, momentum and angular momentum.

The purpose of this section is to find a way to characterise the symmetries of a general metric.

4.3.1 Isometries

Intuitively, the notion of symmetry is clear. If you hold up a round sphere, it looks the same no matter what way you rotate it. In contrast, if the sphere has dimples and bumps, then the rotational symmetry is broken. The distinction between these two should be captured by the metric. Roughly speaking, the metric on a round sphere looks the same at all points, while the metric on a dimpled sphere will depend on where you sit. We want a way to state this mathematically.

To do this, we need the concept of a flow that we introduced in Section 2.2.3. Recall that a flow on a manifold M is a one-parameter family of diffeomorphisms ${\sigma }_t:M\to M$ A flow can be identified with a vector field $K\in \mathfrak{X}(M)$ which, at each point in M, points along tangent vectors to the flow

$$K^{\mu }=\frac{d{x}^{\mu }}{dt}$$

This flow is said to be an isometry, if the metric looks the same at each point along a given flow line. Mathematically, this means that an isometry satisfies

$${\mathcal{L}}_{K}g=0\iff {\nabla }_{\mu }{K}_{\nu }+{\nabla }_{\nu }{K}_{\mu }=0\text{\hspace{1em}(4.29)}$$

where the equivalence of the two expressions was shown in Section 4.1.3. This is the Killing equation and any K satisfying this equation is known as a Killing vector field.

Sometimes it is possible to stare at a metric and immediately write down a Killing vector. Suppose that the metric components $g_{\mu \nu }(x)$ do not depend on one particular coordinate, say $y\equiv x^1$ . Then the vector field $X=\partial /\partial y$ is a Killing vector, since

$$({\mathcal{L}}_{{\partial }_y}g{)}_{\mu \nu }=\frac{\partial g_{\mu \nu }}{\partial y}=0$$

However, we have met coordinates like y before: they become the ignorable coordinates in the Lagrangian for a particle moving in the metric $g_{\mu \nu }.$ resulting in conserved quantities. We once again see the familiar link between symmetries and conserved quantities. We’ll explore this more in Section 4.3.2 and again later in the lectures.

There is a group structure underlying these symmetries. Or, more precisely, a Lie algebra structure. This follows from the result (2.15)

$${\mathcal{L}}_X{\mathcal{L}}_Y-{\mathcal{L}}_Y{\mathcal{L}}_X={\mathcal{L}}_{[X,Y]}$$

(Strictly speaking, we showed this in (2.15) only for Lie derivatives acting on vector fields, but it can be checked that it holds on arbitrary tensor fields.) This means that Killing vectors too form a Lie algebra. This is the Lie algebra of the isometry group of the manifold.

An Example: Minkowski Space

As a particularly simple example, consider Minkowski spacetime with $g_{\mu \nu }={\eta }_{\mu \nu }$ . The Killing equation is

$${\partial }_{\mu }{K}_{\nu }+{\partial }_{\nu }{K}_{\mu }=0$$

There are two forms of solutions. We can take

$$K_{\mu }=c_{\mu }$$

for any constant vector $c_{\mu }$ . These generate translations in Minkowski space. Alternatively, we can take

$$K_{\mu }={\omega }_{\mu \nu }{x}^{\nu }$$

with ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$ . These generate rotations and Lorentz boosts in Minkowski space.

We can see the emergence of the algebra structure more clearly. We define Killing vectors

$$P_{\mu }={\partial }_{\mu }\mathrm{and}{M}_{\mu \nu }={\eta }_{\mu \rho }{x}^{\rho }{\partial }_{\nu }-{\eta }_{\nu \rho }{x}^{\rho }{\partial }_{\mu }\text{\hspace{1em}(4.30)}$$

There are 10 such Killing vectors in total; 4 from translations and six from rotations and boosts. A short calculation shows that they obey

$$\begin{array}{rl}& [P_{\mu },P_{\nu }]=0,[M_{\mu \nu },P_{\sigma }]=-{\eta }_{\mu \sigma }{P}_{\nu }+{\eta }_{\sigma \nu }{P}_{\mu }\\ & [M_{\mu \nu },M_{\rho \sigma }]={\eta }_{\mu \sigma }{M}_{\nu \rho }+{\eta }_{\nu \rho }{M}_{\mu \sigma }-{\eta }_{\mu \rho }{M}_{\nu \sigma }-{\eta }_{\nu \sigma }{M}_{\mu \rho }\end{array}$$

which we recognise as the Lie algebra of the Poincar´e group ${\mathbf{R}}^4\times SO(1,3)$

More Examples: de Sitter and anti-de Sitter

The isometries of de Sitter and anti-de Sitter are simplest to see from their embeddings. The constraint (4.17) that defines de Sitter space is invariant under the rotations of ${\mathbf{R}}^{1,4}$ and so de Sitter inherits the $SO(1,4)$ isometry group. Similarly, the constraint (4.26) that defines anti-de Sitter is invariant under the rotations of ${\mathbf{R}}^{2,3}$ . Correspondingly, AdS has the isometry group $SO(2,3)$ . Note that both of these groups are 10 dimensional: in terms of counting, these spaces are just as symmetric as Minkowski space.

It is simple to write down the 10 Killing vectors in the parent 5d spacetime: they are

$$M_{AB}={\eta }_{AC}{X}^C{\partial }_B-{\eta }_{BC}{X}^C{\partial }_A$$

where $X^A,A=0,1,2,3.$ 4 are coordinates in 5d and ${\eta }_{AB}$ is the appropriate Minkowski metric, with signature $(-++++)$ for de Sitter and $(---+++)$ for anti-de Sitter. In either case, the Lie algebra is that of the appropriate Lorentz group,

$$[M_{AB},M_{CD}]={\eta }_{AD}{M}_{BC}+{\eta }_{BC}{M}_{AD}-{\eta }_{AC}{M}_{BD}-{\eta }_{BD}{M}_{AC}$$

Importantly, the embedding hyperbolae (4.17) and (4.26) are both invariant under these Killing vectors, in the sense that the flows generated by $M_{AB}$ take us from one point on the hyperbolae to another. This means that the Killing vectors are inherited by de Sitter and anti-de Sitter spaces respectively.

For example, we can consider de Sitter space in the static patch with $r^2=(X^1{)}^2+$ $(X^2{)}^2+(X^3{)}^2$ and (4.19)

$$X^0=\sqrt{R^2-r^2}\sinh (t/R)\mathrm{and}{X}^4=\sqrt{R^2-r^2}\cosh (t/R)$$

We know that the metric in the static patch (4.12) is independent of time. This means that $K={\partial }_t$ is a Killing vector. Pushed forwards to the 5d space, this becomes

$$\frac{\partial }{\partial t}=\frac{\partial X^A}{\partial t}\frac{\partial }{\partial X^A}=\frac{1}{R}\left(X^4\frac{\partial }{\partial X^0}+X^0\frac{\partial }{\partial X^4}\right)\text{\hspace{1em}(4.31)}$$

In fact, this Killing vector highlights a rather important subtlety with de Sitter space. As we go on, we will see that timelike Killing vectors – those obeying $g_{\mu \nu }{K}^{\mu }{K}^{\nu }<0$ everywhere – play a rather special role because we can use them to define energy. (We’ll describe this for particles in the next section.)

In anti-de Sitter space, there is no problem in finding a timelike Killing vector. Indeed, we can see it by eye in the global coordinates (4.22), where it is simply $K={\partial }_t$ But de Sitter is another story.

If we work in the static patch (4.12), then the Killing vector (4.31) is a timelike Killing vector. Indeed, we used this to derive the conserved energy E when discussing geodesics in de Sitter in Section 4.2.1. But we know that the static patch does not cover all of de Sitter spacetime.

Indeed, if we extend the Killing vector (4.31) over the entire space, it is not timelike everywhere. To see this, note that when $X^4>0$ and $X^0=0$ , the Killing vector pushes us forwards in the $X^0$ direction, but when $X_4<0$ and $X^0=0$ it pushes us backwards in the $X^0$ direction. This means that the Killing vector field points to the future in some parts of space and to the past in others! Correspondingly, if we try to define an energy using this Killing vector it will be positive in some parts of the space and negative in others. Relatedly, in parts of the space where $X^4=0$ and $X^0\ne 0$ , the Killing vector pushes us in the $X^4$ direction, and so is spacelike.

The upshot of this discussion is an important feature of de Sitter space: there is no global, positive conserved energy. This tallies with our metric in global coordinates (4.21) which is time dependent and so does not obviously have a timelike Killing vector. The lack of a globally defined energy is one of several puzzling aspects of de Sitter space: we’ll meet more as we proceed.

4.3.2 A First Look at Conserved Quantities

Emmy Noether taught us that symmetries are closely related to conserved quantities. In the present context, this means that any dynamics taking place in a spacetime with an isometry will have a conserved quantity.

There are a number of different scenarios in which we can ask about conserved quantities. We could look at particles moving in a fixed background; these are the kinds of calculations that we did in Section 1. Alternatively, we could ask about fields moving in a fixed background; we will address this in Section 4.5.5. Finally, we could ask about the energy stored in the spacetime itself. We will provide a formula for this in Section 4.3.3, and also make some further comments in Section 4.5.5.

Here, we consider a massive particle moving in a spacetime with metric $g.$ The particle will follow some trajectory $x^{\mu }(\tau )$ , with τ the proper time. If the spacetime admits a Killing vector $K.$ , then we can construct the quantity that is conserved along the geodesic,

$$Q=K_{\mu }\frac{d{x}^{\mu }}{d\tau }\text{\hspace{1em}(4.32)}$$

To see that Q is indeed unchanging, we compute

$$\begin{array}{rl}& \frac{dQ}{d\tau }={\partial }_{\nu }{K}_{\mu }\frac{d{x}^{\nu }}{d\tau }\frac{d{x}^{\mu }}{d\tau }+K_{\mu }\frac{d^2{x}^{\mu }}{d{\tau }^2}\\ & ={\partial }_{\nu }{K}_{\mu }\frac{d{x}^{\nu }}{d\tau }\frac{d{x}^{\mu }}{d\tau }-K_{\mu }{\Gamma }_{\rho \sigma }^{\mu }\frac{d{x}^{\rho }}{d\tau }\frac{d{x}^{\sigma }}{d\tau }\\ & ={\nabla }_{\nu }{K}_{\mu }\frac{d{x}^{\nu }}{d\tau }\frac{d{x}^{\mu }}{d\tau }=0\end{array}$$

where in the second line we’ve used the geodesic equation and, in the final equality, we’ve used the symmetry of the Killing equation.

The derivation above looks rather different from our usual formulation of Noether’s theorem. For this reason, it’s useful to re-derive the Killing equation and corresponding conserved charge by playing the usual Noether games. We can do this by looking at the action for a massive particle (in the form (1.32))

$$S=\int d\tau g_{\mu \nu }(x)\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }$$

Now consider the infinitesimal transformation

$$\delta x^{\mu }=K^{\mu }(x)$$

The action transforms as

$$\begin{array}{rl}& \delta S=\int d\tau {\partial }_{\rho }{g}_{\mu \nu }\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }{K}^{\rho }+2g_{\mu \nu }\frac{d{x}^{\mu }}{d\tau }\frac{d{K}^{\nu }}{d\tau }\\ & =\int d\tau {\partial }_{\rho }{g}_{\mu \nu }\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }{K}^{\rho }+2\frac{d{x}^{\mu }}{d\tau }\left(\frac{d{K}_{\mu }}{d\tau }-K^{\nu }\frac{d{g}_{\mu \nu }}{d\tau }\right)\\ & =\int d\tau ({\partial }_{\rho }{g}_{\mu \nu }{K}^{\rho }-2K^{\rho }{\partial }_{\nu }{g}_{\mu \rho }+2{\partial }_{\nu }{K}_{\mu })\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }\\ & =\int d\tau 2{\nabla }_{\nu }{K}_{\mu }\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }\end{array}$$

The transformation is a symmetry of the action if $\delta S=0$ . From the symmetry of the $\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }$ terms, this is true provided that $K_{\mu }$ obeys the Killing equation

$${\nabla }_{(\nu }{K}_{\mu )}=0$$

Noether’s theorem then identifies the charge $Q$ defined in (4.32) as the conserved quantity arising from this symmetry.

We met examples of these conserved quantities in Section 4.2 when discussing geodesics in de Sitter and anti-de Sitter spacetimes. (And, indeed, in Section 1.3 when discussing the geodesic orbits around a black hole). Both the energy E and the angular momentum l are Noether charges of this form.

Killing vectors have further roles to play in identifying conserved quantities. In Section 4.5.5, we’ll describe how we can use Killing vectors to define energy and momentum of fields in a background spacetime.

4.3.3 Komar Integrals

If we have a Killing vector, there is a rather pretty way of associating a conserved quantity to the spacetime itself.

Given a Killing vector $K=K^{\mu }{\partial }_{\mu }$ , we can construct the 1-form $K=K_{\mu }d{x}^{\mu }$ . From this 1-form, we can then construct a 2-form

$$F=dK$$

Alternatively, in components, we have $F=\frac{1}{2}{F}_{\mu \nu }d{x}^{\mu }\wedge d{x}^{\nu }$ with

$$F_{\mu \nu }={\nabla }_{\mu }{K}_{\nu }-{\nabla }_{\nu }{K}_{\mu }$$

We’ve called this 2-form F, in analogy with the electromagnetic 2-form. Indeed, the key idea of the Komar integral is that we can think of F very much like the electromagnetic field strength. Indeed, we claim the following is true:

Claim

If the vacuum Einstein equations are obeyed, so $R_{\mu \nu }=0$ , then $F$ obeys the vacuum Maxwell equations

$$d\star F=0$$

Alternatively, as shown in (3.24), we can write this as

$${\nabla }^{\mu }{F}_{\mu \nu }=0$$

Proof

To see this, we start with the Ricci identity (3.16) which, applied to the Killing vector $K^{\sigma }$ , reads

$$({\nabla }_{\mu }{\nabla }_{\nu }-{\nabla }_{\nu }{\nabla }_{\mu })K^{\sigma }=R_{\rho \mu \nu }^{\sigma }{K}^{\rho }$$

Contracting the $\mu$ and $\sigma$ indices then gives

$$({\nabla }_{\mu }{\nabla }_{\nu }-{\nabla }_{\nu }{\nabla }_{\mu })K^{\mu }=R_{\rho \nu }{K}^{\rho }$$

But $K^{\mu }$ is a Killing vector and so obeys the Killing equation ${\nabla }_{(\mu }{K}_{\nu )}=0$ and so ${\nabla }_{\mu }{K}^{\mu }=0$ . This means that the Ricci identity simplifies to

$${\nabla }_{\mu }{\nabla }_{\nu }{K}^{\mu }=R_{\rho \nu }{K}^{\rho }\text{\hspace{1em}(4.33)}$$

With this in hand, we now look at ${\nabla }^{\mu }{F}_{\mu \nu }$ . We have

$${\nabla }^{\mu }{F}_{\mu \nu }={\nabla }^{\mu }{\nabla }_{\mu }{K}_{\nu }-{\nabla }^{\mu }{\nabla }_{\nu }{K}_{\mu }=-2{\nabla }^{\mu }{\nabla }_{\nu }{K}_{\mu }=-2R_{\rho \nu }{K}^{\rho }$$

where $\mathrm{w}{\mathrm{e}}^{\prime }$ ve used the Killing equation in the second equality and the Ricci identity (4.33) in the third. This then gives the promised result: $d\star F=0$ provided that the Einstein equations $R_{\rho \nu }=0$ hold. □

Since the 2-form F obeys the vacuum Maxwell equations, we can use it to construct the Komar charge, or Komar integral. We integrate over some three-dimensional spatial submanifold Σ,

$$Q_{\mathrm{Komar}}=-\frac{1}{8\pi G}{\int }_{\Sigma }d\star F=-\frac{1}{8\pi G}{\int }_{\partial \Sigma }\star F=-\frac{1}{8\pi G}{\int }_{\partial \Sigma }\star dK$$

Here the factor of $1/8\pi G$ is for later convenience. Because $d\star F=0$ , the same kind of argument that we met in Section 3.2.5 then tells us that $Q_{\mathrm{Komar}}$ is conserved.

Just as for the point particle discussed previously, the interpretation of the Komar integrals depends on the Killing vector at hand. For example, if $K^{\mu }$ is everywhere timelike, meaning $g_{\mu \nu }{K}^{\mu }{K}^{\nu }<0$ at all points, then the Komar integral can be identified with the energy, or equivalently, the mass of the spacetime

$$M_{\mathrm{Komar}}=-\frac{1}{8\pi G}{\int }_{\partial \Sigma }\star dK$$

In particular, we’ll later see how the Komar charge can be non-vanishing, even though d $\star F=0$ when the vacuum Einstein equations are obeyed. Relatedly, if the Killing vector is related to rotations, the conserved charge is identified with the angular momentum of the spacetime.

At this point, it would obviously be nice to give some examples of Komar integrals. Sadly, we don’t yet have any useful examples at our disposal! However, we will use this technology throughout Section 6 to identify the mass and angular momentum of black holes.

As an aside: later in Section 4.5, we will look at what happens if we couple matter to gravity. There we will learn that the Einstein equations are no longer $R_{\mu \nu }=0$ , but instead the right-hand side gets altered by the energy and momentum of the matter. In this case, we can again form the field strength $F,$ but now it obeys the Maxwell equation with a source, d ⋆ $F=\star J$ , where the current J can be related to the energymomentum tensor. However, it turns out that for our applications in Section 6 the case of the vacuum Einstein equations $R_{\mu \nu }=0$ is all we’ll need.

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