4 The Einstein Equations

So far, we have managed to write the variation of the action (4.148) as

$\delta S={\displaystyle \int d^{4}x\sqrt{-g}\left(R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }\right)\delta g^{\mu \nu }}+\sqrt{-g}{g}^{\mu \nu }\delta R_{\mu \nu }$

We now need only worry about the final term. For this, we use:

This is a useful observation. At any point $p\in M$ we can choose to work in normal coordinates such that ${\partial }_{\rho }{g}_{\mu \nu }=0$ and, correspondingly, ${\mathrm{\Gamma }}_{\mu \nu }^{\rho }=0$. Then, to linear order in the variation, the change in the Christoffel symbol evaluated at $p$ is

	$\delta {\mathrm{\Gamma }}_{\mu \nu }^{\rho }$	$=$	${\displaystyle \frac{1}{2}}{g}^{\rho \sigma }\left({\partial }_{\mu }\delta g_{\sigma \nu }+{\partial }_{\nu }\delta g_{\sigma \mu }-{\partial }_{\sigma }\delta g_{\mu \nu }\right)$
		$=$	${\displaystyle \frac{1}{2}}{g}^{\rho \sigma }\left({\nabla }_{\mu }\delta g_{\sigma \nu }+{\nabla }_{\nu }\delta g_{\sigma \mu }-{\nabla }_{\sigma }\delta g_{\mu \nu }\right)$

where we’re at liberty to replace the partial derivatives with covariant derivatives because they differ only by the Christoffel symbols ${\mathrm{\Gamma }}_{\mu \nu }^{\rho }$ which, in normal coordinates, vanish at $p$. However, both the left and right-hand sides of this equation are tensors which means that although we derived this expression using normal coordinates, it must hold in any coordinate system. Moreover, the point $p$ was arbitrary so the final expression holds generally.

Next we look at the variation of the Riemann tensor. In normal coordinates, the expression (3.131) becomes

$R^{\sigma }{}_{\rho \mu \nu }={\partial }_{\mu }{\mathrm{\Gamma }}_{\nu \rho }^{\sigma }-{\partial }_{\nu }{\mathrm{\Gamma }}_{\mu \rho }^{\sigma }$

and the variation is

$\delta R^{\sigma }{}_{\rho \mu \nu }={\partial }_{\mu }\delta {\mathrm{\Gamma }}_{\nu \rho }^{\sigma }-{\partial }_{\nu }\delta {\mathrm{\Gamma }}_{\mu \rho }^{\sigma }={\nabla }_{\mu }\delta {\mathrm{\Gamma }}_{\nu \rho }^{\sigma }-{\nabla }_{\nu }\delta {\mathrm{\Gamma }}_{\mu \rho }^{\sigma }$

where, as before, we replace partial derivatives with covariant derivatives as we are working in normal coordinates where the Christoffel symbols vanish. Once again, our final expression relates two tensors and must, therefore, hold in any coordinate system. Contracting indices (and working to leading order), we have

$\delta R_{\rho \nu }={\nabla }_{\mu }\delta {\mathrm{\Gamma }}_{\nu \rho }^{\mu }-{\nabla }_{\nu }\delta {\mathrm{\Gamma }}_{\rho \mu }^{\mu }$

as claimed. $\mathrm{\square }$

The upshot of these calculations is that

$g^{\mu \nu }\delta R_{\mu \nu }={\nabla }_{\mu }{X}^{\mu }\mathit{\hspace{1em}\hspace{1em} }\mathrm{with}\mathit{\hspace{1em}\hspace{1em} }{X}^{\mu }=g^{\rho \nu }\delta {\mathrm{\Gamma }}_{\rho \nu }^{\mu }-g^{\mu \nu }\delta {\mathrm{\Gamma }}_{\nu \rho }^{\rho }$

The variation of the action (4.148) can then be written as

$\delta S={\displaystyle \int d^{4}x\sqrt{-g}\left[\left(R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }\right)\delta g^{\mu \nu }+{\nabla }_{\mu }{X}^{\mu }\right]}$

(4.149)

This final term is a total derivative and, by the divergence theorem of Section 3.2.4, we ignore it. Requiring that the action is extremised, so $\delta S=0$, we have the equations of motion

$G_{\mu \nu }:=R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }=0$

(4.150)

where $G_{\mu \nu }$ is the Einstein tensor defined in Section 3.4.1. These are the Einstein field equations in the absence of any matter. In fact they simplify somewhat: if we contract (4.150) with $g^{\mu \nu }$, we find that we must have $R=0$. Substituting this back in, the vacuum Einstein equations are simply the requirement that the metric is Ricci flat,

$R_{\mu \nu }=0$

(4.151)

These deceptively simple equations hold a myriad of surprises. We will meet some of the solutions as we go along, notably gravitational waves in Section 5.2 and black holes in Section 6.

Before we proceed, a small comment. We happily discarded the boundary term in (4.149), a standard practice whenever we invoke the variational principle. It turns out that there are some situations in general relativity where we should not be quite so cavalier. In such circumstances, one can be more careful by invoking the so-called Gibbons-Hawking boundary term.

If we take the coordinates $x^{\mu }$ to have dimension of length, then the metric $g_{\mu \nu }$ is necessarily dimensionless. The Ricci scalar involves two spatial derivatives so has dimension $[R]=L^{-2}$. Including the integration measure, the action (4.147) then has dimensions $[S]=L^2$. However, actions should have dimensions of $\mathrm{energy}\times \mathrm{time}$ (it’s the same dimensions as $\mathrm{\hslash }$), or $[S]=M{L}^2{T}^{-1}$. This means that the Einstein-Hilbert action should be multiplied by a constant with the appropriate dimensions. We take

$S={\displaystyle \frac{c^3}{16\pi G}}{\displaystyle \int d^{4}x\sqrt{-g}R}$

where $c$ is the speed of light and $G$ is Newton’s constant,

$G\approx 6.67\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{s}^{-2}$

This factor doesn’t change the equation of motion in vacuum, but we will see in Section 4.5 that it determines the strength of the coupling between the gravitational field and matter, as we might expect.

It’s no fun carrying around a morass of fundamental constants in all our equations. For this reason, we often work in “natural units” in which various constants are set equal to 1. From now on, we will set $c=1$. (Any other choice of $c$, including $3\times {10}^8$, is simply dumb.) This means that units of length and time are equated.

However, different communities have different conventions when it comes to $G$. Relativists will typically set $G=1$. Since we have already set $c=1$, we have $[G]=L{M}^{-1}$. Setting $G=1$ then equates mass with length. This is useful when discussing gravitational phenomenon where the mass is often directly related to the length. For example, the Schwarzschild radius of black hole is $R_s=2GM/c^2$ which becomes simply $R_s=2M$ once we set $G=c=1$.

However, if you’re interested in phenomena other than gravity, then it’s no more sensible to set $G=1$ than to set, say, the Fermi coupling for the weak force $G_F=1$. Instead, it is more useful to choose the convention where $\mathrm{\hslash }=1$, a choice which equates energy with inverse time (also known as frequency). With this convention, Newton’s constant has dimension $[G]=M^{-2}$. The corresponding energy scale is known as the (reduced) Planck mass; it is given by

$M_{\mathrm{pl}}^2={\displaystyle \frac{\mathrm{\hslash }c}{8\pi G}}$

It is around ${10}^{18}\mathrm{GeV}$. This is a very high energy scale, way beyond anything we have probed in experiment. This can be traced to the weakness of the gravitational force. With $c=\mathrm{\hslash }=1$, we can equally well write the Einstein-Hilbert action as

$S={\displaystyle \frac{1}{2}}{M}_{\mathrm{pl}}^2{\displaystyle \int d^{4}x\sqrt{-g}R}$

You might be tempted to set $c=\mathrm{\hslash }=G=1$. This leaves us with no remaining dimensional quantities. It is typically a bad idea, not least because dimensional analysis is a powerful tool and one we do not want to lose. In these lectures, we will focus only on gravitational physics. Nonetheless, we will retain $G$ in all equations.

Varying the action as before now yields the Einstein equations,

$R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }=-\mathrm{\Lambda }{g}_{\mu \nu }$

This time, if we contract with $g^{\mu \nu }$, we get $R=4\mathrm{\Lambda }$. Substituting this back in, the vacuum Einstein equations in the presence of a cosmological constant become

$R_{\mu \nu }=\mathrm{\Lambda }{g}_{\mu \nu }$

We will solve these shortly in Section 4.2.

As in any field theory, higher derivative terms in the action only become relevant for fast varying fields. In General Relativity, they are unimportant for all observed physical phenomena and we will not discuss them further in this course.

However, not all the components of the metric $g_{\mu \nu }$ are physical. Two metrics which are related by a change of coordinates, $x^{\mu }\to {\tilde{x}}^{\mu }(x)$ describe the same physical spacetime. This means that there is a redundancy in any given representation of the metric, which removes precisely 4 of the 10 degrees of freedom, leaving just 6 behind.

Mathematically, this redundancy is implemented by diffeomorphisms. (We defined diffeomorphisms in Section 2.1.3.) Given a diffeomorphism, $\varphi :M\to M$, we can use this to map all fields, including the metric, on $M$ to a new set of fields on $M$. The end result is physically indistinguishable from where we started: it describes the same system, but in different coordinates. Such diffeomorphisms are analogous to the gauge symmetries that are familiar in Maxwell and Yang-Mills theory.

Let’s look more closely at the implication of these diffeomorphisms for the path integral. We’ll consider a diffeomorphism that takes a point with coordinates $x^{\mu }$ to a nearby point with coordinates

$x^{\mu }\to {\tilde{x}}^{\mu }(x)=x^{\mu }+\delta x^{\mu }$

We could view this either as an “active change”, in which one point with coordinates $x^{\mu }$ is mapped to another point with coordinates $x^{\mu }+\delta x^{\mu }$, or as a “passive” change, in which we use two different coordinate charts to label the same point. Ultimately, the two views lead to the same place. We’ll adopt the passive perspective here, simply because we have a lot of experience of changing coordinates. Later we’ll revert to the active picture.

We can think of the change of coordinates as being generated by an infinitesimal vector field $X$,

$\delta x^{\mu }=-X^{\mu }(x)$

The metric transforms as

$g_{\mu \nu }(x)\to {\tilde{g}}_{\mu \nu }(\tilde{x})$

$=$

${\displaystyle \frac{\partial x^{\rho }}{\partial {\tilde{x}}^{\mu }}}{\displaystyle \frac{\partial x^{\sigma }}{\partial {\tilde{x}}^{\nu }}}{g}_{\rho \sigma }(x)$

With our change of coordinate ${\tilde{x}}^{\mu }=x^{\mu }-X^{\mu }(x)$, with infinitesimal $X^{\mu }$, we can invert the Jacobian matrix to get

${\displaystyle \frac{\partial {\tilde{x}}^{\mu }}{\partial x^{\rho }}}={\delta }_{\rho }^{\mu }-{\partial }_{\rho }{X}^{\mu }\mathit{\hspace{1em}\hspace{1em} }\Rightarrow \mathit{\hspace{1em}\hspace{1em} }{\displaystyle \frac{\partial x^{\rho }}{\partial {\tilde{x}}^{\mu }}}={\delta }_{\mu }^{\rho }+{\partial }_{\mu }{X}^{\rho }$

where the inverse holds to leading order in the variation $X$. Continuing to work infinitesimally, we then have

	${\tilde{g}}_{\mu \nu }(\tilde{x})$	$=$	$\left({\delta }_{\mu }^{\rho }+{\partial }_{\mu }{X}^{\rho }\right)\left({\delta }_{\nu }^{\sigma }+{\partial }_{\nu }{X}^{\sigma }\right)g_{\rho \sigma }(x)$
		$=$	$g_{\mu \nu }(x)+g_{\mu \rho }(x){\partial }_{\nu }{X}^{\rho }+g_{\nu \rho }(x){\partial }_{\mu }{X}^{\rho }$

Meanwhile, we can Taylor expand the left-hand side

${\tilde{g}}_{\mu \nu }(\tilde{x})={\tilde{g}}_{\mu \nu }(x+\delta x)={\tilde{g}}_{\mu \nu }(x)-X^{\lambda }{\partial }_{\lambda }{\tilde{g}}_{\mu \nu }(x)$

Comparing the the different metrics at the same point $x$, we find that the metric undergoes the infinitesimal change

$\delta g_{\mu \nu }(x)={\tilde{g}}_{\mu \nu }(x)-g_{\mu \nu }(x)=X^{\lambda }{\partial }_{\lambda }{g}_{\mu \nu }+g_{\mu \rho }{\partial }_{\nu }{X}^{\rho }+g_{\nu \rho }{\partial }_{\mu }{X}^{\rho }$

(4.152)

But this is something we’ve seen before: it is the Lie derivative of the metric. In other words, if we act with an infinitesimal diffeomorphism along $X$, the metric changes as

$\delta g_{\mu \nu }={({ℒ}_{X}g)}_{\mu \nu }$

This makes sense: it’s like the leading term in a Taylor expansion along $X$.

In fact, we can also massage (4.152) into a slightly different form. We lower the index on $X^{\rho }$ in the last two $\partial X^{\rho }$ terms by taking the metric inside the derivative. This results in two further terms in which the derivative hits the metric, and these must be cancelled off. We’re left with

$\delta g_{\mu \nu }={\partial }_{\mu }{X}_{\nu }+{\partial }_{\nu }{X}_{\mu }+X^{\rho }\left({\partial }_{\rho }{g}_{\mu \nu }-{\partial }_{\mu }{g}_{\rho \nu }-{\partial }_{\nu }{g}_{\mu \rho }\right)$

But the terms in the brackets are the Christoffel symbols, $2g_{\rho \sigma }{\mathrm{\Gamma }}_{\mu \nu }^{\sigma }$. We learn that the infinitesimal change in the metric can be written as

$\delta g_{\mu \nu }={\nabla }_{\mu }{X}_{\nu }+{\nabla }_{\nu }{X}_{\mu }$

(4.153)

Let’s now see what this means for the path integral. Under a general change of the metric, the Einstein-Hilbert action changes as (4.149)

$\delta S={\displaystyle \int d^{4}x\sqrt{-g}{G}^{\mu \nu }\delta g_{\mu \nu }}$

where we have discarded the boundary term. Insisting that $\delta S=0$ for any variation $\delta g_{\mu \nu }$ gives the equation of motion $G^{\mu \nu }=0$. In contrast, symmetries of the action are those variations $\delta g_{\mu \nu }$ for which $\delta S=0$ for any choice of metric. Since diffeomorphisms are (gauge) symmetries, we know that the action is invariant under changes of the form (4.153). Using the fact that $G_{\mu \nu }$ is symmetric, we must have

$\delta S=2{\displaystyle \int d^{4}x\sqrt{-g}{G}^{\mu \nu }{\nabla }_{\mu }{X}_{\nu }}=0\mathit{\hspace{1em}\hspace{1em} }\text{for all}{X}_{\mu }(x)$

After integrating by parts, we find that this results in something familiar: the Bianchi identity

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

We already know that the Bianchi identity holds from our work in Section 3.4, but the derivation there was a little fiddly. Here we learn that, from the path integral perspective, the Bianchi identity is a result of diffeomorphism invariance.

In fact it makes sense that the two are connected. Naively, the Einstein equation $G_{\mu \nu }=0$ comprises ten independent equations. But, as we’ve seen, diffeomorphism invariance means that there aren’t ten independent components of the metric, so one might worry that the Einstein equations are overdetermined. Happily, diffeomorphisms also ensure that not all the Einstein equations are independent either; they are related by the four Bianchi constraints. We see that, in fact, the Einstein equations give only six independent conditions on the six independent degrees of freedom in the metric.

While this restriction is natural geometrically, it is rather unusual from the perspective of a physical theory. It is not a holonomic constraint on the physical degrees of freedom: instead it is an inequality (together with the requirement that $g_{\mu \nu }$ has one, rather than three, negative eigenvalues). Other fields in the Standard Model don’t come with such restrictions. Instead, it is reminiscent of fluid mechanics where one has to insist that matter density obeys $\rho (𝐱,t)>0$. Ultimately, it seems likely that this restriction is telling us that the gravitational field is not fundamental and should be replaced by something else in regimes where $det{g}_{\mu \nu }$ is getting small.

The restriction that $det{g}_{\mu \nu }\ne 0$ means that the simplest Ricci flat metric is Minkowski space, with

$d{s}^2=-d{t}^2+d{𝐱}^2$

Of course, this is far from the only metric obeying $R_{\mu \nu }=0$. Another example is provided by the Schwarzschild metric,

$d{s}^2=-\left(1-{\displaystyle \frac{2GM}{r}}\right)d{t}^2+{\left(1-{\displaystyle \frac{2GM}{r}}\right)}^{-1}d{r}^2+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$

(4.154)

which we will discuss further in Section 6. We will meet more solutions as the course progresses.

We can put some more flesh on this. Because the metric (4.158) has a similar form to the Schwarzschild metric, we simply need to follow the steps that we already took in Section 1.3. First we write down the action for a particle in de Sitter space. We denote the proper time of the particle as $\sigma$. (In Section 1.3, we used $\tau$ to denote proper time, but we’ll need this for a different time coordinate defined below.) Working with the more general metric (4.155), the action is

$S_{dS}={\displaystyle \int 𝑑\sigma \left[-f{(r)}^2{\dot{t}}^2+f{(r)}^{-2}{\dot{r}}^2+r^2({\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2)\right]}$

(4.159)

where ${\dot{x}}^{\mu }=d{x}^{\mu }/d\sigma$.

Any degree of freedom which appears only with time derivatives in the Lagrangian is called ignorable. They lead to conserved quantities. The Lagrangian above has two ignorable degrees of freedom: $\varphi (\sigma )$ and $t(\sigma )$. The first leads to the conserved quantity that we call angular momentum,

$l={\displaystyle \frac{1}{2}}{\displaystyle \frac{dL}{d\dot{\varphi }}}=r^2{\sin }^2\theta \dot{\varphi }$

where the factor of $1/2$ in front of $dL/d\dot{\varphi }$ arises because the kinetic terms in (4.159) don’t come with the usual factor of $1/2$. Meanwhile, the conserved quantity associated to $t(\sigma )$ is usually referred to as the energy

$E=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{dL}{d\dot{t}}}=f{(r)}^2\dot{t}$

(4.160)

The equations of motion arising from the action (4.159) should be supplemented with the constraint that tells us whether we’re dealing with a massive or massless particle. For a massive particle, the constraint ensures that the trajectory is timelike,

$-f{(r)}^2{\dot{t}}^2+f{(r)}^{-2}{\dot{r}}^2+r^2({\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2)=-1$

Without loss of generality, we can restrict to geodesics that lie in the $\theta =\pi /2$ plane, so $\dot{\theta }=0$ and ${\sin }^2\theta =1$. Replacing $\dot{t}$ and $\dot{\varphi }$ with $E$ and $l$ respectively, the constraint becomes

${\dot{r}}^2+V_{\mathrm{eff}}(r)=E^2$

We focus on geodesics with vanishing angular momentum, $l=0$. In this case, the potential is an inverted harmonic oscillator. A particle sitting stationary at $r=0$ is a geodesic, but it is unstable: if it has some initial velocity then it will move away from the origin, following the trajectory

$r(\sigma )=R\sqrt{E^2-1}\sinh \left({\displaystyle \frac{\sigma }{R}}\right)$

(4.161)

The metric (4.158) is singular at $r=R$, which might make us suspect that something fishy is going on there. But whatever this fishiness is, it’s not visible in the solution (4.161) which shows that any observer reaches $r=R$ in finite proper time $\sigma$.

The fishiness reveals itself if we look at the coordinate time $t$. This also has the interpretation of the time experienced by someone sitting at the point $r=0$. Using (4.160), the trajectory (4.161) evolves as

${\displaystyle \frac{dt}{d\sigma }}=E{\left(1-{\displaystyle \frac{r^2}{R^2}}\right)}^{-1}$

It is simple to check that $t(\sigma )\to \mathrm{\infty }$ as $r(\sigma )\to R$. (For example, suppose that we have $r(\sigma )=R$ at some value $\sigma ={\sigma }_{\star }$ of proper time. Then look at what happens just before this time by expanding $\sigma ={\sigma }_{\star }-ϵ$ with $ϵ$ small. The equation above becomes $dt/dϵ=-\alpha /ϵ$ for some constant $\alpha$, telling us that $t(ϵ)\sim -\log (ϵ/R)$ and we indeed find that $t\to \mathrm{\infty }$ as $ϵ\to 0$.) This means that while a guy on the trajectory (4.161) sails right through the point $r=R$ in finite proper time, according to his companion waiting at $r=0$ this will take infinite time.

This strange behaviour is, it turns out, similar to what happens at the horizon of a black hole, which is the surface $r=2GM$ in the metric (4.154). (We will look at more closely at this in Section 6.) However, the Schwarzschild metic also has a singularity at $r=0$, whereas the de Sitter metric looks just like flat space at $r=0$. (To see this, simply Taylor expand the coefficients of the metric around $r=0$.) Instead, de Sitter space seems like an inverted black hole in which particles are pushed outwards to $r=R$. But how should we interpret this radius? We will get more intuition for this as we proceed.

The choice of coordinates (4.164) and (4.165) are not the most intuitive. First, they single out $X^4$ as special, when the constraint (4.163) does no such thing. This hides some of the symmetry of de Sitter space. Moreover, the coordinates do not cover the whole of the hyperboloid, since they restrict only to $X^4\ge 0$.

We can do better. Consider instead the solution to the constraint (4.163)

$X^0=R\sinh (\tau /R)\mathit{\hspace{1em}\hspace{1em} }\mathrm{and}\mathit{\hspace{1em}\hspace{1em} }{X}^i=R\cosh (\tau /R)y^i$

(4.166)

where the $y^i$, with $i=1,2,3,4$, obey ${\sum }_i{(y^i)}^2=1$ and so parameterise a unit 3-sphere. These coordinates have the advantage that they retain (more of) the symmetry of de Sitter space, and cover the whole space. Substituting this into the 5d Minkowski metric (4.162) gives a rather different metric on de Sitter space,

$d{s}^2=-d{\tau }^2+R^2{\cosh }^2(\tau /R)d{\mathrm{\Omega }}_3^2$

(4.167)

where $d{\mathrm{\Omega }}_3^2$ denotes the metric on the unit 3-sphere. These are known as global coordinates, since they cover the whole space. (Admittedly, any choice of coordinates on ${𝐒}^3$ will suffer from the familiar problem of coordinate singularities at the poles.) Since this metric is related to (4.158) by a change of coordinates, it too must obey the Einstein equation. (We’ll check this explicitly in Section 4.6 where we discuss a class of metrics of this form.)

These coordinates provide a much clearer intuition for the physics of de Sitter space: it is a time-dependent solution in which a spatial ${𝐒}^3$ first shrinks to a minimal radius $R$, and subsequently expands. This is shown in the figure. The expansionary phase is a fairly good approximation to our current universe on large scales; you can learn more about this in the lectures on Cosmology.

The global coordinates clearly show that there is nothing fishy happening when $X^4=0$, the surface which corresponds to $r=R$ in (4.158). This is telling us that this is nothing but a coordinate singularity. (As, indeed, is the $r=2GM$ singularity in the Schwarzschild metric.) Nonetheless, there is still some physics lurking in this coordinate singularity, which we will extract over the next few sections.

Sometimes this metric is written by introducing the coordinate $r=R\sinh \rho$, after which it takes the form

$d{s}^2=-{\cosh }^2\rho d{t}^2+R^{2}d{\rho }^2+R^2{\sinh }^2\rho \left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right)$

(4.169)

Now there’s no mysterious coordinate singularity in the $r$ direction and, indeed, we will see shortly that these coordinates now cover the entire space.

In contrast, if the particle also has angular momentum then the potential has a minimum at $r_{\star }^2=Rl$. This geodesic is like a motorcycle wall-of-death trick, with the angular momentum keeping the particle pinned up the potential. Other geodsics spin in the same fashion, while oscillating about $r_{\star }$. Importantly, particles with finite energy $E$ cannot escape to $r\to \mathrm{\infty }$: they are trapped by the spacetime to live within some finite distance of the origin.

The picture that emerges from this analysis is that AdS is like a harmonic trap, pushing particles to the origin. This comes with something of a puzzle however because, as we will see below (and more in Section 4.3), AdS is a homogeneous space which, roughly speaking, means that all points are the same. How is it possible that AdS acts acts like a harmonic trap, pushing particles to $r=0$, yet is also a homogeneous space?!

To answer this question, consider a guy sitting stationary at the origin $r=0$. This is a geodesic. From his perspective, intrepid AdS explorers on other geodesics (with, say, $l=0$) will oscillate backwards and forwards about the origin $r=0$, just like a particle in a harmonic trap. However, since these explorers will themselves be travelling on a geodesic, they are perfectly entitled to view themselves as sedentary, stay-at-home types, sitting perfectly still at their ‘origin’, watching the other folk fly around them. In this way, just as everyone in de Sitter can view themselves at the centre of the universe, with other observers moving away from them, everyone in anti-de Sitter can view themselves in the centre of the universe, with other observers flying around them.

We can also look at the fate of massless particles. This time the action is supplemented by the constraint

$-f{(r)}^2{\dot{t}}^2+f{(r)}^{-2}{\dot{r}}^2+r^2({\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2)=0$

This tells us that the particle follows a null geodesic. The equation (4.170) gets replaced by

${\dot{r}}^2+V_{\mathrm{null}}(r)=E^2$

with the effective potential now given by

$V_{\mathrm{null}}(r)={\displaystyle \frac{l^2}{r^2}}\left(1+{\displaystyle \frac{r^2}{R^2}}\right)$

This potential is again shown in Figure 32. This time the potential is finite as $r\to \mathrm{\infty }$, which tells us that there is no obstacle to light travelling as far as it likes: it suffers only the usual gravitational redshift. We learn that AdS spacetime confines massive particles, but not massless ones.

To solve the equations for a massless particle, it’s simplest to work in the coordinates $r=R\sinh \rho$ that we introduced in (4.169). If we restrict to vanishing angular momentum, $l=0$, the equation above becomes

$R\dot{\rho }=\pm {\displaystyle \frac{E}{\cosh \rho }}\mathit{\hspace{1em}\hspace{1em} }\Rightarrow \mathit{\hspace{1em}\hspace{1em} }R\sinh \rho =E(\sigma -{\sigma }_0)$

where $\sigma$ is the affine geodesic parameter. We see that $\rho \to \mathrm{\infty }$ only in infinite affine time, $\sigma \to \mathrm{\infty }$. However, it’s more interesting to see what happens in coordinate time. This follows by recalling the definition of $E$ in (4.160),

$E={\cosh }^2\rho \dot{t}$

(Equivalently, you can see this by dint of the fact that we have a null geodesic, with $\cosh \rho \dot{t}=\pm R\dot{\rho }$.) We then find

$R\tan (t/R)=E(\sigma -{\sigma }_0)$

So as $\sigma \to \mathrm{\infty }$, the coordinate time tends to $t\to \pi R/2$. We learn that not only do light rays escape to $\rho =\mathrm{\infty }$, but they do so in a finite coordinate time $t$. This means that to make sense of dynamics in AdS, we must specify some boundary conditions at infinity to dictate what happens to massless particles or fields when they reach it.

Anti-de Sitter space does not appear to have any cosmological applications. However, it turns out to be the place where we best understand quantum gravity, and so has been the object of a great deal of study.

In fact there is one small subtlety: the embedding hyperboloid has topology ${𝐒}^1\times {𝐑}^3$, with ${𝐒}^1$ corresponding to a compact time direction. This can be seen in the parameterisation (4.173), where the time coordinate takes values $t\in [0,2\pi R)$. However, the AdS metrics (4.168) or (4.169) have no such restriction, with $t\in (-\mathrm{\infty },+\mathrm{\infty })$. They are the universal covering of the hyperboloid (4.172).

There is another parameterisation of the hyperboloid that is also useful. It takes the rather convoluted form

$X^i={\displaystyle \frac{r}{R}}{x}^i\mathrm{for}i=0,1,2\mathit{\hspace{1em}},X^4-X^3=r\mathit{\hspace{1em}},X^4+X^3={\displaystyle \frac{R^2}{r}}+{\displaystyle \frac{r}{R^2}}{\eta }_{ij}{x}^i{x}^j$

with $r\in [0,\mathrm{\infty })$. Although the change of coordinates is tricky, the metric is very straightforward, taking the form

$d{s}^2=R^2{\displaystyle \frac{d{r}^2}{r^2}}+{\displaystyle \frac{r^2}{R^2}}{\eta }_{ij}d{x}^{i}d{x}^j$

(4.174)

These coordinates don’t cover the whole of AdS; instead they cover only one-half of the hyperboloid, restricted to $X^4-X^3>0$. This is known as the Poincaré patch of AdS. Moreover, the time coordinate, which already extends over the full range $x^0\in (-\mathrm{\infty },+\mathrm{\infty })$, cannot be further extended. This means that as $x^0$ goes from $-\mathrm{\infty }$ to $+\mathrm{\infty }$ in (4.174), in global coordinates (4.168), the time coordinate $t$ goes only from $0$ to $2\pi R$.

Two other choices of coordinates are also commonly used to describe the Poincaré patch. If we set $z=R^2/r$, then we have

$d{s}^2={\displaystyle \frac{R^2}{z^2}}\left(d{z}^2+{\eta }_{ij}d{x}^{i}d{x}^j\right)$

Alternatively, if we set $r=R{e}^{\rho }$, we have

$d{s}^2=R^{2}d{\rho }^2+e^{2\rho }{\eta }_{ij}d{x}^{i}d{x}^j$

In each case, massive particles fall towards $r=0$, or $z=\mathrm{\infty }$, or $\rho =-\mathrm{\infty }$.

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 $K\in 𝔛(M)$ which, at each point in $M$, points along tangent vectors to the flow

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

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

${ℒ}_{K}g=0\mathit{\hspace{1em}\hspace{1em} }\iff \mathit{\hspace{1em}\hspace{1em} }{\nabla }_{\mu }{K}_{\nu }+{\nabla }_{\nu }{K}_{\mu }=0$

(4.175)

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

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

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 (2.70)

${ℒ}_X{ℒ}_Y-{ℒ}_Y{ℒ}_X={ℒ}_{[X,Y]}$

(Strictly speaking, we showed this in (2.70) 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.

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

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

(4.176)

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

	$[P_{\mu },P_{\nu }]=0\mathit{\hspace{1em}\hspace{1em} },[M_{\mu \nu },P_{\sigma }]=-{\eta }_{\mu \sigma }{P}_{\nu }+{\eta }_{\sigma \nu }{P}_{\mu }$
	$[M_{\mu \nu },M_{\rho \sigma }]={\eta }_{\mu \sigma }{M}_{\nu \rho }+{\eta }_{\nu \rho }{M}_{\mu \sigma }-{\eta }_{\mu \rho }{M}_{\nu \sigma }-{\eta }_{\nu \sigma }{M}_{\mu \rho }$

which we recognise as the Lie algebra of the Poincaré group ${𝐑}^4\times SO(1,3)$.