4. 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 gµ⌫(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.
We will start by understanding the dynamics of the gravitational field in the absence of any matter. We will then turn to understand how the gravitational field responds to matter – or, more precisely, to energy and momentum – in Section 4.5.
4.1 The Einstein-Hilbert Action All our fundamental theories of physics are described by action principles. Gravity is no di↵erent. Furthermore, the straight-jacket of di↵erential 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.WeknowfromSection2.4.4 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 = d4x p gR (4.1) Z 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 =detgµ⌫,isnegative.
As a quick sanity check, recall that the Ricci tensor takes the schematic form (3.39) R @ + whiletheLevi-Civitaconnectionitselfis @g.Thismeansthatthe ⇠ ⇠ Einstein-Hilbert action is second order in derivatives, just like most other actions we consider in physics.
–140– Varying the Einstein-Hilbert Action We would like to determine the Euler-Lagrange equations arising from the action (4.1).
We do this in the usual way, by starting with some fixed metric gµ⌫(x)andseeinghow the action changes when we shift
g (x) g (x)+ g (x) µ⌫ ! µ⌫ µ⌫ µ⌫ Writing the Ricci scalar as R = g Rµ⌫, the Einstein-Hilbert action clearly changes as