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 [122, 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
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 [81, 92, 82, 85, 101, 102, 89], and it is the results of [56, 55, 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 makes this general argument explicit for the specific case of higher-derivative gravity linearized about flat space.
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 [56, 55, 21], corresponding to the classical result for the metric in harmonic gauge, for which the Schwarzschild metric takes the form. It is natural that different values for are obtained when different definitions for are used, since these different definitions contribute differently to physical observables (on which all calculations must agree).
There is another ambiguity in the definition of the potential , 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 ): [56, 55]. If, instead, the full amplitude is used to define the interaction potential, Bjerrum–Bohr et al.  find  that these one-loop results for do not suffer from ambiguity due to the freedom to perform redefinitions of the form .
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  and Donoghue  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, .
Of course, the undetectability of these quantum corrections does not make them unimportant. Rather, the above calculations underline the following three conclusions:
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