Gravity and Strings

104 A perturbative introduction to general relativity
this is the theory we are going to obtain here. Its construction is useful for many purposes.
We will use it in constructing conserved quantities in spacetimes with arbitrary asymptotics
and we can use it to work with the Minkowskian Fierz–Pauli theory in arbitrary
coordinates. However, apart from these prosaic applications it will also teach us interesting
things, e.g. how to define masslessness in curved backgrounds.
To be as general as possible we will include a cosmological-constant term from the
beginning as in Eq. (3.275).
3.4.1 Linearized gravity
Let us first describe the setup: we consider a spacetime metric gµν that solves the d-
dimensional cosmological Einstein equations for some matter energy–momentum tensor
T µν
matter (here we set c = 1, as usual),
where Gc µν is the cosmological Einstein tensor,
Gc µν = 8πG (d) µν
N T matter, (3.277)
Gc µν ≡ G µν +
d − 2
2 gµν . (3.278)
The metric gµν must be such that we can consider it as produced by a small perturbation of
the background metric ¯gµν, i.e. we can write
gµν =¯gµν + hµν, (3.279)
where the perturbation hµν goes to zero at infinity fast enough that the metric gµν is asymptotically
¯gµν. Furthermore, hµν and its derivatives are assumed to be small enough that we
can ignore higher-order terms. 52
Usually, the background metric ¯gµν will be the vacuum metric, i.e. a maximally symmetric
solution of the vacuum Einstein equations
¯Gc µν = 0. (3.280)
Therefore, the metrics gµν that we consider describe in the gravitational language isolated
systems. There are no matter sources of the gravitational field at infinity. In the absence of a
cosmological constant, the vacuum metric ¯gµν = ηµν, the Minkowski metric, and the metrics
gµν will be asymptotically flat. With positive (negative) cosmological constant, the
(maximally symmetric) vacuum solution is the (anti-)de Sitter ((A)dSd) spacetime and the
metrics gµν will be asymptotically (anti-)de Sitter. However, we will keep the background
metric completely general in order to cover other interesting cases in which a solution gµν
goes asymptotically to a ¯gµν that is not the vacuum solution or even a solution of the vacuum
Einstein equations. Thus, we will use only Eqs. (3.277) and (3.279) to find the equation satisfied
by the perturbation hµν. Later on, we will impose the condition that the background
metric solves the Einstein equation (3.280).
52 Here we have absorbed the coupling constant χ = 16πG (d)
N into hµν.