Non-relativistic point masses in three spatial dimensions

4.1 Non-relativistic point masses in three spatial dimensions

The first example to be considered consists of quantum corrections to the potential energy $V (r )$ of gravitational interaction for two large, slowly-moving point masses separated by a distance $r$ . Working to leading order in source velocities $v$ , we expect the leading behavior for large source masses to be the Newtonian gravitational interaction of two classical, static point sources of energy:

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 $r$ 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 $ℏ$ and $c$ , these small parameters are $2 3 Gℏ ∕r c$ and $2 GMi ∕rc$ . Both tend to zero for large $r$ , and as we shall see, the first controls the size of quantum corrections and the second controls the size of relativistic corrections⁸ .

4.1.1 Definition of the potential

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, $Δp1 = − Δp2 ≡ q$ , in the center-of-mass frame. The most direct definition of the interaction potential $V(r)$ 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 $⟨f |T |i⟩ = (2π )4δ4(pf − pi)𝒜 (q)$ , the potential $V$ would be defined by

The position-space potential is then given by $−3∫ 3 &tidle; V (r) = N (2 π) d q V(q)$ . The overall normalization $N$ depends on the conventions used for the normalization of the initial and final states, and is chosen to ensure the proper form for the Newtonian interaction.

Figure 2:

The 1-particle-reducible Feynman graphs relevant to the definition of the interaction potential. The blobs represent self-energy and vertex corrections.

Several other definitions for the interaction potential have also been considered by various workers, some of which we now briefly list.

4.1.1.1 One-particle-reducible amplitude

An alternative definition, followed in [56

, 55

, 5 ], is to define the interaction potential in terms of the one-particle-reducible part of the amplitude $𝒜1PR$ – see Figure 2

– as is commonly done for quantum electrodynamics and quantum chromodynamics. The logic of this choice is that because the graviton propagator varies as $2 1∕q$ , for small $2 q$ graviton exchange dominates very-long-distance interactions. It has the disadvantage that the one-particle-reducible graphs are not observable in themselves, and so need not form a gauge-invariant subset. Nevertheless, the results obtained from this definition can be interpreted [59

, 21

] as giving the leading quantum corrections to the Schwarzschild, Kerr–Newman, and Reisner–Nordström metrics.

4.1.1.2 Vacuum polarization

Some early workers defined the interaction potential in terms of the purely vacuum polarization subset of the 1-particle-reducible graphs [62 , 33 , 79 ]. The motivation for such a choice is that these are the only graphs which would arise for a purely classical source, which macroscopic objects like planets or stars were expected to be. It is important to recognize that the power-counting arguments given earlier necessarily require the inclusion of vertex corrections at the same order in small quantities as the vacuum polarization graphs. The necessity for so doing shows that there is no limit in which a source for the gravitational field can be considered to be precisely classical. This non-classicality arises because the gravitational field itself carries energy, and its quantum fluctuations do not decouple in the large-mass limit due to the growth which the gravitational coupling experiences in this limit.

4.1.2 Calculation of the interaction potential

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 ( $r → ∞$ ) – i.e., large impact parameters – corresponds to small momentum transfers, $q2 → 0$ . 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- $2 q$ limit to the scattering amplitude turns out to behaves as

where $𝒜 = A + A q2 + ... an 0 2$ is an analytic function of $q2$ near $q2 = 0$ .

In position space the first three terms of Equation (47 ) correspond to terms which fall off with $r$ like $k2∕r$ , $2 k1∕r$ , and $3 k0∕r$ , respectively. By contrast, the powers of $2 q$ in $𝒜an$ only contribute terms to $V (r)$ which are local, inasmuch as they are proportional to $δ3(r)$ or its derivatives. Since our interest is only in the long-distance interaction, the analytic contributions of $𝒜an$ 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

respectively corresponding to the contributions of classical general relativity, the one-loop corrections within general relativity, and the classical curvature-squared contributions. The dependence of this latter term on the quantities $Mp$ , $a$ , and $b$ are written explicitly to emphasize which contributions depend on which parameters. We now describe the result which is obtained for each of these three types of contribution.

4.1.2.1 Curvature-squared terms

The simplest contribution to dispose of is that due to the curvature-squared terms⁹ . Because these terms are polynomials in momenta, they contribute only to the analytic part $𝒜an$ of the scattering amplitudes, and so give only local contributions to the interaction potential which involve $3 δ (r)$ or its derivatives. Their precise form is computed in [56

, 55

], who find

with $B$ given in terms of the constants $a$ and $b$ of Equation (28

) by $2 B = 128 π G (a + b)$ . Since they contribute only to $𝒜an$ , we see that these contributions are necessarily irrelevant to the large-distance interaction potential.

It is instructive to think of this $δ$ -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 $δ$ -function source, whose Lagrangian is

The higher-derivative term proportional to $κ$ in this model is the analogue of the curvature-squared gravitational interactions. The propagator for this theory satisfies the equation $2 4 (□ − κ □ )G κ(x,y) = δ (x − y)$ , which becomes (to linear order in $κ$ )

where $G (x, y) 0$ is the usual propagator when $κ = 0$ . We see the expected result that the leading contribution to $V (r )$ is purely local in position space (as might be expected for the low-energy implications of very-high-energy/very-short-range physics).

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

This shows how the higher-derivative term introduces a new pole into the propagator at $2 −1 p = − κ$ , but with a residue whose sign is unphysical (corresponding to a ghost mode with negative kinetic energy).

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 $1∕m$ . In the effective theory of Equation (50 ) the propagator (52 ) must be read as

since the Lagrangian itself is only accurate to leading order in $κ$ . The ghost pole does not arise perturbatively in $κp2$ , since its location is up at high energies, $p2 = − κ−1$ . Simon [136 ] makes this general argument explicit for the specific case of higher-derivative gravity linearized about flat space.

4.1.2.2 Classical general relativity

The leading contributions for large $r$ due to the relativistic corrections of general relativity have the large- $r$ form (with factors of $c$ restored)

where $G = 1 ∕(8πM 2p)$ is Newton’s constant, $M1$ and $M2$ are the masses whose potential energy is of interest, and which are separated by the distance $r$ .

The square brackets, $[ ] 1 + ...$ , in this expression represent the relativistic corrections to the Newtonian potential which already arise within classical general relativity, and $λ$ 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 $λ1PR = − 1$ [56 , 55 , 21 ], corresponding to the classical result for the metric in harmonic gauge, for which the Schwarzschild metric takes the form

Alternatively, using the potential defined by the full scattering amplitude $𝒜tot$ instead gives $λtot = +3$ [20

]. It is natural that different values for $λ$ are obtained when different definitions for $V$ 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 [89 ], which is related to the freedom to redefine the coordinate $r$ , according to $′ r → r = r(1 + aGM ∕r + ...)$ . 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

where $M$ and $v$ are the mass and velocity of the non-relativistic particle of interest. The main point is that the constants $λ$ and $′ λ$ are redundant interactions in the sense defined earlier, inasmuch as all physical observables only depend on a single combination of these two constants. Observables only depend on one combination because the other combination can be removed by performing the coordinate transformation $r → r(1 + aGM ∕r + ...)$ as above. From this we see that the coefficient $λ$ of $2 GM ∕r$ obtained for $V (r)$ can also differ from one another, provided that the coefficient $λ′$ also differs in such a way as to give the same results for physical observables.

4.1.2.3 One-loop general relativity

The final term in $V (r)$ arises from the one-loop contribution as computed within general relativity, which is extracted by calculating the one-loop corrections to the scattering amplitude $𝒜q$ . Although these corrections typically diverge in the ultraviolet, on general grounds such divergences contribute only polynomials in momenta, and so can contribute only to the non-relativistic amplitude’s analytic part $𝒜 (q2) an$ . Indeed, this is required for the one-loop divergences to be absorbed by renormalizing the effective couplings $a$ and $b$ of the higher-curvature terms of the gravitational action (28

¹⁰ .

It follows from this observation that to the extent that we focus on the long-distance interactions in $V (r)$ , 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 $V (r)$ 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 $log q2$ , and so the leading one-loop quantum contribution to the interaction potential is (again re-instating powers of $ℏ$ and $c$ )

where $ξ$ is a calculable number. If the potential is computed using only the one-particle-reducible scattering amplitude, the result for pure gravity is [21

Notice that this corrects an error in the earlier result for the same quantity, given in [56 , 55 ]. If, instead, the full amplitude $𝒜tot$ is used to define the interaction potential, Bjerrum–Bohr et al. [20

] find

It is argued in [20 ] that these one-loop results for $ξ$ do not suffer from ambiguity due to the freedom to perform redefinitions of the form $r → r(1 + aG2∕r2 + ...)$ .

4.1.3 Implications

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 $G$ and the masses $M1$ and $M2$ . In particular, relativistic corrections are controlled by the dimensionless quantity $GMtot ∕rc2$ , which is a measure of typical orbital velocities $v2∕c2$ . The leading quantum corrections, on the other hand, are $M$ -independent and are controlled by the ratio $2 2 ℓp∕r$ , where $3 1∕2 − 35 ℓp = (Gℏ ∕c ) ∼ 10 m$ is the Planck length.

Although the one-particle-reducible contributions need not be separately gauge-independent, Bjerrum–Borh [21 ] and Donoghue [59 ] 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 $M ⊙$ , and with $r$ chosen equal to the solar radius $R⊙ ∼ 109 m$ , or the solar Schwarzschild radius $rs = 2GM ⊙∕c2 ∼ 103 m$ . 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, $Mhole ∼ Mp ∼ 10−5 g$ .

Table 1:

The generic size of relativistic and quantum corrections to the Sun’s gravitational field.

	$GM--⊙- rc2$	$-G-ℏ r2c3$

$r = R⊙$	$− 6 10$	$−88 10$
$2 r = 2GM ⊙∕c$	$0.5$	$−76 10$

Of course, the undetectability of these quantum corrections does not make them unimportant. Rather, the above calculations underline the following three conclusions:

One need not throw up one’s hands when contemplating quantum gravity effects, because quantum corrections in gravity are often unambiguous and calculable.
Although the small size of the above quantum corrections in the solar system mean that they are unlikely to be measured, they also show that the great experimental success of classical general relativity in the solar system should also be regarded as a triumph of quantum gravity! Classical calculations are not a poor substitute for some poorly-understood quantum theory, they are rather an extremely good approximation for which quantum corrections are exceedingly small.
Despite the above two points, the mysteries of quantum gravity remain real and profound. But the above calculations show that these are high-energy (or short-distance) mysteries, and so point to cosmological singularities or primordial black holes as being the places to look for quantum gravitational effects.