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The Einstein Equations

It is now time to do some physics. The force of gravity is mediated by a gravitational field. The glory of general relativity is that this field is identified with a metric $g_{\mu \nu}(x)$ on a 4d Lorentzian manifold that we call spacetime.

This metric is not something fixed; it is, like all other fields in Nature, a dynamical object. This means that there are rules which govern how this field evolves in time. The purpose of this section is to explore these rules and some of their consequences.

All our fundamental theories of physics are described by action principles. Gravity is no different. Furthermore, the straight-jacket of differential geometry places enormous restrictions on the kind of actions that we can write down. These restrictions ensure that the action is something intrinsic to the metric itself, rather than depending on our choice of coordinates.

Spacetime is a manifold $M$, equipped with a metric of Lorentzian signature. An action is an integral over $M$. We know from Section 2.4.4 that we need a volume-form to integrate over a manifold. Happily, as we have seen, the metric provides a canonical volume form, which we can then multiply by any scalar function. Given that we only have the metric to play with, the simplest such (non-trivial) function is the Ricci scalar $R$. This motivates us to consider the wonderfully concise action
$$S=\int d^{4} x \sqrt{-g} R$$
This is the Einstein-Hilbert action. Note that the minus sign under the square-root arises because we are in a Lorentzian spacetime: the metric has a single negative eigenvalue and so its determinant, $g=\operatorname{det} g_{\mu \nu}$, is negative.

As a quick sanity check, recall that the Ricci tensor takes the schematic form (3.39) $R \sim \partial \Gamma+\Gamma \Gamma$ while the Levi-Civita connection itself is $\Gamma \sim \partial g .$ This means that the Einstein-Hilbert action is second order in derivatives, just like most other actions we consider in physics.

When Gravity is Weak

The elegance of the Einstein field equations ensures that they hold a special place in the hearts of many physicists. However, any fondness you may feel for these equations will be severely tested if you ever try to solve them. The Einstein equations comprise ten, coupled partial differential equations. While a number of important solutions which exhibit large amounts symmetry are known, the general solution remains a formidable challenge.

We can make progress by considering situations in which the metric is almost flat. We work with $\Lambda=0$ and consider metrics which, in so-called almost-inertial coordinates $x^{\mu}$, takes the form
$$g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}$$
Here $\eta_{\mu \nu}=\operatorname{diag}(-1,+1,+1,+1)$ is the Minkowski metric. The components $h_{\mu \nu}$ are assumed to be small perturbation of this metric: $h_{\mu \nu} \ll 1$.

Our strategy is to expand the Einstein equations to linear order in the small perturbation $h_{\mu \nu}$. At this order, we can think of gravity as a symmetric “spin 2 ” field $h_{\mu \nu}$ propagating in flat Minkowski space $\eta_{\mu \nu}$. To this end, all indices will now be raised and lowered with $\eta_{\mu \nu}$ rather than $g_{\mu \nu}$. For example, we have
$$h^{\mu \nu}=\eta^{\mu \rho} \eta^{\nu \sigma} h_{\rho \sigma}$$
Our theory will exhibit a Lorentz invariance, under which $x^{\mu} \rightarrow \Lambda_{\nu}^{\mu} x^{\nu}$ and the gravitational field transforms as
$$h^{\mu \nu}(x) \rightarrow \Lambda^{\mu}{ }_{\rho} \Lambda_{\sigma}^{\nu} h^{\rho \sigma}\left(\Lambda^{-1} x\right)$$
In this way, we construct a theory around flat space that starts to look very much like the other field theories that we meet in physics.

Black Holes

The Schwarzschild Solution

We have already met the simplest black hole solution back in Section 1.3: this is the Schwarzschild solution, with metric
$$d s^{2}=-\left(1-\frac{2 G M}{r}\right) d t^{2}+\left(1-\frac{2 G M}{r}\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$
It is not hard to show that this solves the vacuum Einstein equations $R_{\mu \nu}=0$. Indeed, the calculations can be found in Section $4.2$ where we first met de Sitter space. The Schwarzschild solution is a special case of the more general metric $(4.9)$ with $f(r)^{2}=$ $1-2 G M / r$ and it’s simple to check that this obeys the Einstein equation which, as we’ve seen, reduces to the simple differential equations $(4.10)$ and $(4.11)$.
$\mathrm{M}$ is for Mass
The Schwarzschild solution depends on a single parameter, $M$, which should be thought of as the mass of the black hole. This interpretation already follows from the relation to Newtonian gravity that we first discussed way back in Section $1.2$ where we anticipated that the $g_{00}$ component of the metric should be $(1.26)$
$$g_{00}=-(1+2 \Phi)$$
with $\Phi$ the Newtonian potential. We made this intuition more precise in Section 5.1.2 where we discussed the Newtonian limit. For the Schwarzschild metric, we clearly have
$$\Phi=-\frac{G M}{r}$$
which is indeed the Newtonian potential for a point mass $M$ at the origin.
The black hole also provides an opportunity to roadtest the technology of Komar integrals developed in Section 4.3.3. The Schwarzschild spacetime admits a timelike Killing vector $K=\partial_{t}$. The dual one-form is then
$$K=g_{00} d t=-\left(1-\frac{2 G M}{r}\right) d t$$

Following the steps described in Section 4.3.3, we can then construct the 2 -form
$$F=d K=-\frac{2 G M}{r^{2}} d r \wedge d t$$
which takes a form similar to that of an electric field, with the characteristic $1 / r^{2}$ fall-off. The Komar integral instructs us to compute the mass by integrating
$$M_{\text {Komar }}=-\frac{1}{8 \pi G} \int_{\mathbf{S}^{2}} \star F$$
where $\mathbf{S}^{2}$ is any sphere with radius larger than the horizon $r=2 G M$. It doesn’t matter which radius we choose; they all give the same answer, just like all Gaussian surfaces outside a charge distribution give the same answer in electromagnetism. Since the area of a sphere at radius $r$ is $4 \pi r^{2}$, the integral gives
$$M_{\text {Komar }}=M$$
for the Schwarzschild black hole.
There’s something a little strange about the Komar mass integral. As we saw in Section 4.3.3, the 2 -form $F=d K$ obeys something very similar to the Maxwell equations, $d \star F=0$. But these are the vacuum Maxwell equations in the absence of any current, so we would expect any “electric charge” to vanish. Yet this “electric charge” is precisely the mass $M_{\text {Komar }}$ which, as we have seen, is distinctly not zero. What’s happening is that, for the black hole, the mass is all localised at the origin $r=0$, where the field strength $F$ diverges.

We might expect that the Schwarzschild solution only describes something physically sensible when $M \geq 0$. (The $M=0$ Schwarzschild solution is simply Minkowski spacetime.) However, the metric (6.1) is a solution of the Einstein equations for all values of $M$. As we proceed, we’ll see that the $M<0$ solution does indeed have some rather screwy features that make it unphysical.

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