Gravity and Strings

3
A perturbative introduction to general relativity
The standard approach to general relativity (GR) is purely geometrical: spacetime is
curved by its energy content according to Einstein’s equation and test particles move along
geodesics. This point of view is what makes GR a theory completely different from the theories
that describe all the other known interactions that are special-relativistic field theories
(SRFTs) that, after quantization, explain the interaction between two charged bodies as the
interchange of quanta of the field.
The enormous success of relativistic quantum field theories with a gauge principle made
it unavoidable to try to find a theory of that kind to describe gravitational interactions at a
classical and quantum level. This path was followed by many people and it was found that
such a theory, whose starting point is the linear perturbation theory of GR (the Fierz–Pauli
theory for a free, massless spin-2 particle), would be self-consistent only after the introduction
of an infinite number of non-linear terms whose summation should be equivalent to the
full non-linear GR theory. 1 Thus, this approach may lead to a different justification of Einstein’s
theory and provides an alternative interpretation of it that is worth studying. 2 Some
of the predictions of GR can be obtained at leading or next to leading order in this approach.
Since this is not the standard approach, there are only a few complete treatments in the literature:
the book [386], based on Feynman’s lectures on gravitation, that also contains many
references, some of which we will follow in Section 3.2; and also Deser’s lectures on the
gravitational field [300]. Reference [30] is also an excellent review with many references.
In this chapter, as a warm-up exercise, we are first going to study the construction of
SRFTs of gravity based on a scalar field. This is the simplest possibility in the search for
aSRFT of the gravitational interaction and it will offer us the possibility of studying, in a
simple setting, problems that we will find later on.
As is well known, scalar theories of gravity predict no global bending of light rays (in
contrast to observation) and a value for the precession of the perihelion of Mercury which
1 There are other alternative special-relativistic field theories for spin-2 particles. See, for example, [659] in
which gravity is based on a massive (with extremely small mass) spin-2 field.
2 Some string theories have a massless spin-2 particle in their spectra. If these string theories are consistent, the
argument we will develop will imply that they contain gravity, which, to the lowest order, will be described
by Einstein’s theory.
45