The first term in Equation (28) is the cosmological
constant, which is dropped in what follows since the observed size
of the universe implies
is extremely small. There is,
of course, no real theoretical understanding why the cosmological
constant should be this small (a comprehensive review of the
cosmological constant problem is given in [152]; for a recent suggestion in the spirit
of effective field theories see [3, 27]). Once the
cosmological term is dropped, the leading term in the derivative
expansion is the one linear in
, which is the usual
Einstein-Hilbert action of general relativity. Its coefficient
defines Newton’s constant (and so also the Planck mass,
).
The explicit mass scales,
and
, are introduced to ensure that the remaining
constants
,
,
,
and
appearing in Equation (28
) are dimensionless.
Since it appears in the denominator, the mass scale
can be considered as the smallest microscopic scale to have been
integrated out to obtain Equation (28
). For definiteness we
might take the electron mass,
,
for
when considering applications at energies below the
masses of all elementary particles. (Notice that contributions like
or
could also exist, but these
are completely negligible compared to the terms displayed in
Equation (28
).)
As discussed in the previous section, some of
the interactions in the Lagrangian (28) may be redundant, in
the sense that they do not contribute independently to physical
observables (like graviton scattering amplitudes about some fixed
geometry, say). To eliminate these we are free to drop any terms
which are either total derivatives or which vanish when evaluated
at solutions to the lower-order equations of motion.
The freedom to drop total derivatives allows us
to set the couplings and
to zero. We can drop
because
, and we can drop
because the quantity
The freedom to perform field redefinitions allows
further simplification (just as was found for the toy model in
earlier sections). To see how this works, consider the
infinitesimal field redefinition , under
which the leading term in
undergoes the variation
Since the lowest-order equations of motion for
pure gravity (without a cosmological constant) imply , we see that all of the
interactions beyond the Einstein-Hilbert term which are explicitly
written in Equation (28
) can be removed in one
of these two ways. The first interaction which can have physical
effects (for pure gravity with no cosmological constant) in this
low-energy expansion is therefore proportional to the cube of the
Riemann tensor.
This last conclusion changes if matter or a
cosmological constant are present, however, since then the
lowest-order field equations become for some
nonzero tensor
. Then terms like
or
no longer vanish when
evaluated at the solutions to the equations of motion, but are
instead equivalent to interactions of the form
,
, or
. Since some of
our later applications of
are to the gravitational
potential energy of various localized energy sources, we shall find
that these terms can generate contact interactions amongst these
sources.