Gravity and Strings

3.1 Scalar SRFTs of gravity 47
Afree scalar propagating in Minkowski spacetime is described by the action
S = d d x 1
2 (∂φ)2 , (∂φ) 2 ≡ η µν ∂µφ∂νφ, (3.1)
and has as equation of motion
∂ 2 φ = 0, ∂ 2 ≡ η µν ∂µ∂ν. (3.2)
The source for the Newtonian gravitational field is the gravitational mass of matter which
is experimentally found to be proportional (equal in appropriate units) to the inertial mass
for all material bodies. In special relativity the inertial mass, the energy, and the momentum
of a physical system are combined into the energy–momentum tensor T µν and, therefore,
the source for the gravitational field will be the matter energy–momentum tensor. This is
an object of utmost importance and was studied in some detail in Chapter 2.
3.1.1 Scalar gravity coupled to matter
From our previous discussion, the source of the scalar gravitational field (the r.h.s. of
Eq. (3.2)) must be a scalar built out of the energy–momentum tensor of the matter fields.
The simplest scalar is the trace Tmatter ≡ Tmatter µ µ, and using it, and taking into account all
factors of c, wearrive at the action for matter coupled to scalar gravity
S = 1
d
c
d
1
x
2Cc2 (∂φ)2 + φ
c2 Tmatter
+ Lmatter , (3.3)
where C is a proportionality constant to be determined. From this action we can derive the
equation of motion for the scalar gravitational field,
∂ 2 φ = CTmatter, (3.4)
and the equation of motion for matter in the gravitational field.
Observe that the conservation of the matter energy–momentum tensor plays no role whatsoever
in the construction of this theory. In fact, if it was required in some sense for consistency,
we would be in trouble because, after the coupling to the gravitational field, the
matter energy–momentum tensor is no longer conserved: only the total energy–momentum
tensor of the above Lagrangian (the matter energy–momentum tensor, plus the gravitational
energy–momentum tensor, plus an interaction term) is conserved. However, the equation of
motion that we have obtained is perfectly consistent as it stands.
Observe also that nowhere is it required that the energy–momentum tensor is symmetric
(although only its symmetric part contributes to the trace). In fact, there are no conditions
that we can impose on the energy–momentum tensor to select only one out of the infinitely
many possible energy–momentum tensors that we can obtain by adding terms proportional
to the equations of motion or superpotential terms. We can view this as a weakness of
scalar SRFTs of gravity. In the cases that we are going to consider, we will simply take the
canonical energy–momentum tensor obtained from the matter action in its simplest form.
Now, to determine the constant C, wecan require φ to be identical to the Newtonian
gravitational potential in the static, non-relativistic limit in which only the Tmatter00 =−ρc 2