Our interest is in the quantum and relativistic
corrections to this Newtonian limit, as described by the
gravitational action, Equation (28), plus the appropriate
source action (like, for instance, Equation (39
)). For point sources
which are separated by a large distance
we expect these
corrections to be weak, and so they should be calculable in
perturbation theory about flat space. The strength of the
gravitational interaction at large separation is controlled by two
small dimensionless quantities, which suggest themselves on
dimensional grounds. Temporarily re-instating factors of
and
, these small parameters are
and
. Both tend to zero for large
, and as we shall see, the first controls the size of
quantum corrections and the second controls the size of
relativistic corrections8
.
Because there is some freedom of choice in the definition of an interaction potential in a relativistic field theory, we first pause to consider some of the definitions which have been considered. Although more sophisticated possibilities are possible [123, 43], for systems near the flat-space limit a natural definition of the interaction potential between slowly-moving point masses can be made in terms of their scattering amplitudes.
Consider, then, two particles which scatter
non-relativistically, with each undergoing a momentum transfer,
, in the center-of-mass frame. The most
direct definition of the interaction potential
of these two particles is to define its matrix
elements within single-particle states to reproduce the full
field-theoretical amplitude for this scattering. For instance, if
the field-theoretic scattering matrix takes the form
, the potential
would be defined by
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Several other definitions for the interaction potential have also been considered by various workers, some of which we now briefly list.
We now describe the results of recent explicit
calculations of the gravitational potential just defined. A number
of these calculations have now been performed [82, 93, 83, 86, 102, 103, 90], and it is the
results of [57
, 56
, 21
, 20
] which are
summarized here.
For any of these potentials, scattering at large
distances () - i.e., large impact parameters -
corresponds to small momentum transfers,
. Because
corrections to the Newtonian limit involve the interchange of
massless gravitons, in general scattering amplitudes are not
analytic in this limit. In particular, in the present instance the
small-
limit to the scattering amplitude turns out to
behaves as
In position space the first three terms of
Equation (47) correspond to terms
which fall off with
like
,
, and
, respectively. By contrast,
the powers of
in
only contribute terms to
which are local, inasmuch as they are proportional
to
or its derivatives. Since our interest is only in
the long-distance interaction, the analytic contributions of
may be completely ignored in what follows.
The power-counting analysis described in earlier
sections suggest that the leading corrections to the Newtonian
result come either from (i)
relativistic contributions coming from tree-level calculations
within general relativity, (ii)
one-loop corrections to the classical potential, again using only
general relativity, or (iii) from
tree-level contributions containing precisely one vertex from the
curvature-squared terms of the effective theory, Equation (28). The interaction
potential therefore has the form
It is instructive to think of this -function contribution due to curvature-squared terms
in another way. To this end, consider the toy model of a massless
scalar field coupled to a classical
-function source,
whose Lagrangian is
This way of thinking of things is useful because it illustrates an important conceptual issue for effective field theories. Normally one considers higher-derivative theories to be anathema since higher-derivative field equations generically have unstable runaway solutions, and the above calculation shows why these do not pose problems for the effective field theory. To see why this is so, it is useful to pause to review how the runaway solutions arise.
At the classical level, runaway modes are
possible because of the additional initial data which
higher-derivative equations require. The reason for their origin in
the quantum theory is also easily seen using the toy theory defined
by Equation (50), for which at face
value the momentum-space scalar propagator would be
The reason these do not pose a problem for
effective field theories is that all of the higher-derivative terms
are required to be treated perturbatively, since these interactions are
defined by reproducing the results of the underlying physics
order-by-order in powers of inverse heavy masses . In the effective theory of Equation (50
) the
propagator (52
) must be read as
The square brackets, , in this
expression represent the relativistic corrections to the Newtonian
potential which already arise within classical general relativity,
and
is a known constant whose value depends on the
precise coordinate conditions used in the calculation. For example,
using the potential defined by the 1-particle-reducible scattering
amplitude gives
[57
, 56
, 21
], corresponding to
the classical result for the metric in harmonic gauge, for which
the Schwarzschild metric takes the form
There is another ambiguity in the definition of
the potential [90], which is related to
the freedom to redefine the coordinate , according to
. Of course, such a
coordinate change should drop out of physical observables, but how
this happens in this case involves a subtlety. The main point is
that the low-energy effective Lagrangian for the non-relativistic
particles contains two terms of the
same size at subleading order in the relativistic expansion, having
the schematic form
It follows from this observation that to the
extent that we focus on the long-distance interactions in , to the order we are working these must be
ultraviolet finite since they receive no contribution from the
amplitude’s analytic part. This means that the leading quantum
implications for
are unambiguous predictions which are
not complicated by the renormalization procedure.
Explicit calculation shows that the non-analytic
part of the quantum corrections to scattering are proportional to
, and so the leading one-loop quantum contribution to
the interaction potential is (again re-instating powers of
and
)
It is remarkable that the quantum corrections to the interaction potential can be so cleanly identified. In this section we summarize a few general inferences which follow from their size and dependence on physical parameters like mass and separation.
Conceptually, the main point is that the quantum
effects are calculable, and in principle can be distinguished from
purely classical corrections. For instance, the quantum
contribution (57) can be distinguished
from the classical relativistic corrections (54
) because the quantum
and the relativistic terms depend differently on
and the masses
and
. In particular, relativistic corrections are
controlled by the dimensionless quantity
, which is a
measure of typical orbital velocities
. The leading
quantum corrections, on the other hand, are
-independent and are controlled by the ratio
, where
is the Planck length.
Although the one-particle-reducible contributions need not be separately gauge-independent, Bjerrum-Borh [21] and Donoghue [60] argue that they may be usefully interpreted as defining long-distance quantum corrections to the metric external to various types of point sources. Besides obtaining corrections to the Schwarzschild metric in this way, they do the same for the Kerr-Newman and Reissner-Nordström metrics by incorporating spin and electric charge into the non-relativistic quantum source. Because the quantum corrections they find are source-independent, these authors suggest they be interpreted in terms of a running Newton’s constant, according to
Numerically, the quantum corrections are so
miniscule as to be unobservable within the solar system for the
forseeable future. Table 1 evaluates their
size using for definiteness a solar mass , and with
chosen equal to the solar radius
, or the
solar Schwarzschild radius
. Clearly the quantum-gravitational correction is
numerically extremely small when evaluated for garden-variety
gravitational fields in the solar system, and would remain so right
down to the event horizon even if the sun were a black hole. At
face value it is only for separations comparable to the Planck
length that quantum gravity effects become important. To the extent
that these estimates carry over to quantum effects right down to
the event horizon on curved black hole geometries (more about this
below) this makes quantum corrections irrelevant for physics
outside of the event horizon, unless the black hole mass is as
small as the Planck mass,
.
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Of course, the undetectability of these quantum corrections does not make them unimportant. Rather, the above calculations underline the following three conclusions: