Self-force and radiation reaction in general relativity

# Self-force and radiation reaction in general relativity

Leor Barack and Adam Pound Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
September 21, 2019
###### Abstract

The detection of gravitational waves from binary black-hole mergers by the LIGO-Virgo Collaboration marks the dawn of an era when general-relativistic dynamics in its most extreme manifestation is directly accessible to observation. In the future, planned (space-based) observatories operating in the millihertz band will detect the intricate gravitational-wave signals from the inspiral of compact objects into massive black holes residing in galactic centers. Such inspiral events are extremely effective probes of black-hole geometries, offering unparalleled precision tests of General Relativity in its most extreme regime. This prospect has in the past two decades motivated a programme to obtain an accurate theoretical model of the strong-field radiative dynamics in a two-body system with a small mass ratio. The problem naturally lends itself to a perturbative treatment based on a systematic expansion of the field equations in the small mass ratio. At leading order one has a pointlike particle moving in a geodesic orbit around the large black hole. At subsequent orders, interaction of the particle with its own gravitational perturbation gives rise to an effective “self-force”, which drives the radiative evolution of the orbit, and whose effects can be accounted for order by order in the mass ratio.

This review surveys the theory of gravitational self-force in curved spacetime and its application to the astrophysical inspiral problem. We first lay the relevant formal foundation, describing the rigorous derivation of the equation of self-forced motion using matched asymptotic expansions and other ideas. We then review the progress that has been achieved in numerically calculating the self-force and its physical effects in astrophysically realistic inspiral scenarios. We highlight the way in which, nowadays, self-force calculations make a fruitful contact with other approaches to the two-body problem and help inform an accurate universal model of binary black hole inspirals, valid across all mass ratios. We conclude with a summary of the state of the art, open problems and prospects.

Our review is aimed at non-specialist readers and is for the most part self-contained and non-technical; only elementary-level acquaintance with General Relativity is assumed. Where useful, we draw on analogies with familiar concepts from Newtonian gravity or classical electrodynamics.

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## 1 Introduction

Black holes are “the simplest macroscopic objects in the universe”, goes the famous quote from S. Chandrasekhar [1]. Indeed, we expect astrophysical black holes, when in isolation, to be described in exact form in terms of the 2-parameter family of Kerr solutions to the field equations of General Relativity (GR). But put two such black holes in orbit around each other, and they form a strikingly complex dynamical system. No closed-form solutions are known, and even numerical solutions have been forbiddingly hard to obtain until well into the 21st century. The complexity of the gravitational two-body problem in GR stands in stark contrast to its elementary nature in the context of point-particle Newtonian gravity, where all possible orbital configurations are simple conical sections. Point-particle idealizations are problematic in GR, but even in scenarios where they can make sense in some effective way, the orbital dynamics remains very complicated. In classical GR, a gravitationally bound system of two masses (subject only to gravitational forces) admits no stationary configurations: gravitational waves constantly carry orbital energy away from the system, and back-reaction from that radiation gradually drives the two objects closer together. Given enough time, the two bodies eventually merge. If the two bodies are initially Kerr black holes, a single, larger Kerr black hole eventually forms. The detailed description of this inspiral and merger process has been a major computational challenge and a key theme in gravitational research for almost 50 years. A central motivation has been the desire to predict the exact pattern of gravitational waves through which such systems can manifest themselves observationally.

The landmark observation of merging black holes by the LIGO-Virgo Collaboration in 2015 [2] has conclusively established the existence in nature of black holes, inspiralling black-hole binaries, and gravitational waves, all consistent with GR. Once mathematical curiosities, black holes and gravitational waves are now firmly in the realm of observational astronomy. For the first time, we are given direct observational access to a natural process where general-relativistic dynamics plays out at its most extreme. The continued advance of gravitational-wave astronomy will bring unprecedented opportunities to probe relativistic theory in its most dynamical regime. The theoretical modelling of gravitational-wave sources is an integral part of that programme, as the realization of the exciting science prospects relies crucially on the availability of accurate source models.

The LIGO-Virgo discoveries can serve as a case in point: without an accurate model of the inspiral and merger it would have not been possible to extract the physical parameters of GW150914 (the first event detected and the brightest so far) at the precision with which they were reported [2], and some of the other events would have likely been missed altogether [3]. Analysis by the LIGO-Virgo Collaboration [4] concluded that the quality of science extractable from future observations may well be limited not by experimental precision but by the accuracy of available theoretical models. (This would be the case if detected mergers were to involve more rapidly spinning black holes than in the mergers already observed, or larger mass disparities between the two merging black holes.) The desire to maximize the science return from gravitational-wave experiments thus continues to drive the theory programme to improve waveform models across the full parameter space relevant to observation.

Of particular interest is the inspiral scenario where one of the black holes is much lighter than the other: the so-called “extreme-mass-ratio inspiral”, or EMRI. Nature abounds with EMRIs. They come in the form of stellar-mass black holes (or neutron stars) that are captured into inspiral orbits around massive black holes—the kind of behemoth black holes, of masses of order to solar masses, that reside in the cores of many galaxies, including our own Milky Way. Astrophysical EMRIs emit gravitational waves in millihertz frequencies, which cannot be detected by existing detectors (seismic gravity-gradient perturbations restrict the operation of ground-based detectors to frequencies well above 1 Hz). But they will be prime targets for the planned space-based detector LISA (the Laser Interferometer Space Antenna [5, 6]), whose peak sensitivity will be in the millihertz band. EMRIs are extraordinary natural laboratories for strong-gravity physics. In EMRIs, back-reaction from emitted gravitational waves modifies the orbit on a timescale much larger than the orbital period, so the inspiral is slow and gradual—“adiabatic”. In a typical LISA EMRI, the captured object spends the last few years of insiral in a tight orbit around the massive hole (with an orbital revolution period of order an hour), moving at a significant fraction of the speed of light and emitting some detectable gravitational-wave cycles. The intricate gravitational-wave signal cleanly encodes within it an extremely detailed map of the spacetime geometry around the black hole: nature’s “ideal experiment” in strong gravity! Studies have shown how information from EMRI signals could be used to measure the central object’s mass and spin with exquisite accuracies, confirm whether it is a Kerr black hole as GR predicts, and eliminate or tightly constrain a host of proposed alternatives to GR. LISA could observe up to hundreds of such EMRIs per year [7].

However, even the most luminous EMRI signals are expected to be weaker than the instrumental noise of LISA in its current design. They could be extracted only by filtering the signal against accurate theoretical waveform templates. A detailed model of the radiative evolution and emitted radiation in EMRI systems is thus a prerequisite to being able to tap into the rich science encapsulated in their signals. Gravitational-wave detectors measure phase modulations, so the model has to capture correctly the coherent phase evolution of the emitted EMRI signal over the entire inspiral; and it must ideally do so over the entire relevant parameter space of astrophysical EMRIs, allowing for arbitrary orbital configurations, masses, and any other relevant physical attributes. The exciting prospects of observing EMRIs with LISA have, over the past 20 years or so, driven a concerted effort by theorists to develop a faithful model of EMRIs within GR.

The EMRI domain of the relativistic two-body problem presents unique challenges. The disparate lengthscales and long inspiral time make it hard to tackle using direct Numerical Relativity (NR) methods of the kind that inform the LIGO-Virgo searches (i.e., ones based on numerically solving the full nonlinear set of Einstein’s equations). These methods are extremely computationally expensive already in the comparable-masses regime relevant to LIGO-Virgo searches (indeed, these searches rely on phenomenological approximants that interpolate the very sparse database of NR waveforms available), and they become completely intractable for EMRIs. Also inadequate are methods based on weak-gravity or slow-motion approximations, such as the post-Newtonian (PN) approach, in which GR is treated perturbatively as an expansion about Newtonian gravity. PN methods have been greatly successful in predicting and explaining a variety of observed GR phenomena, from Mercury’s perihelion advance to the orbital decay in the famous Hulse–Taylor binary pulsar [8]. They also play an important part in modelling the early stage of inspiral for LIGO-Virgo merger searches. But PN methods are wholly inappropriate in the EMRI case, where the entire interesting part of the inspiral occurs low inside the deep gravitational well of the central black hole, where gravity is extreme. One must not rely on any weak-field approximation when modelling EMRIs.

Fortunately, the EMRI problem is naturally amenable to a different type of perturbative treatment: one based on an expansion in the small mass ratio (the small body’s mass divided by the massive black hole’s mass), which, for EMRIs detectable by LISA, is as small as . At “zeroth order”, the small object is a pointlike test particle, whose motion is unaffected by its own gravitational field or internal structure. The particle traces a geodesic orbit—the curved-space equivalent of a straight line—in the Kerr spacetime associated with the large black hole. Geodesic orbits around a Kerr black hole can be highly complex (in particular, they are generically ergodic—i.e., space-filling; see Sec. 5.1 and Fig. 4 below), but they are well understood and can be described, essentially, in closed form. Such geodesic orbits approximate true EMRI orbits well only over periods much shorter than the timescale of gravitational radiation-reaction. Radiation-reaction effects come into play at the next order of the expansion: one now treats the gravitational field of the small object as a small perturbation of the background Kerr geometry, satisfying a linearized version of Einstein’s field equations, and one then considers the back-reaction from that perturbation on the particle’s orbit. Still viewed as a trajectory in the fixed Kerr background, the particle’s orbit now experiences a small acceleration with respect to the original geodesic. This acceleration is interpreted as being caused by an effective “gravitational self-force” (GSF) attributed to the particle’s interaction with its own gravitational field. In this picture, it is the GSF which drives the slow radiative inspiral, and whose work upon the particle converts orbital energy into gravitational-wave energy. Higher orders in the perturbative expansion account for nonlinear interactions of the gravitational field with the particle and with itself, as well as for dynamical effects coming from the small object’s internal structure. (The leading-order effect of the particle’s spin enters the equation of motion already at linear order.)

In this manner, one obtains successive approximations for the EMRI orbit and emitted radiation through a systematic expansion in the mass ratio. At each order, one derives an effective equation of motion that describes the EMRI orbit as a trajectory on the fixed Kerr geometry, coupled to a set of field equations that govern the gravitational field, including observable radiation. Analysis suggests [9] that to construct a sufficiently accurate EMRI model for future LISA searches, one must derive and solve the equations of motion through second order in the perturbative expansion (i.e., accounting for self-acceleration terms that scale with the square of the mass ratio). This is a formidable theoretical and computational challenge. To tackle it with sufficient rigour one must take a step back and revisit a fundamental question about the nature of motion in GR: how does a “small” object move in a curved spacetime, which it itself influences?

Historically, many attempts to address this question have been in the PN context of weak fields, going at least as far back as the seminal work of Einstein, Infeld, and Hoffmann [10]. In the fully relativistic arena, significant effort has gone into simply establishing that the equivalence principle (or geodesic principle) is actually a derivable result of the Einstein field equations, not an independent postulate: in the limit of zero mass and size, the Einstein equations alone dictate that all objects, no matter their internal composition, move on geodesics of the external spacetime. Ref. [11] provides an early review of work along these lines, and Ref. [12] a very recent one.

A major step beyond this was taken in the 1970s by Dixon [13], who showed precisely how the geodesic approximation is altered by the finite size of a body (building on earlier work by Mathisson [14] and Papapetrou [15], among others). He derived an exact equation of motion for an arbitrary material body, in which the body’s multipole moments couple to the curvature of the external spacetime to induce corrections to geodesic motion. However, his method fails if the body’s own gravity is strong, and he hence restricted his result to test bodies, which are of finite size but do not affect the spacetime geometry. He reasoned that extending his method to gravitating bodies would require identifying and somehow subtracting the body’s own (highly nonlinear) “self-field”, which dominates the metric in the body’s neighbourhood but, in analogy with Newtonian gravity, might be expected to have no direct effect on its motion.

In the 1980s, another major step was taken by Thorne and Hartle [16], this time building on work by D’Eath [17] and Kates [18], among others. Using a perturbative expansion in the limit of small mass and size, they established a sort of generalized equivalence principle: at least at low orders in perturbation theory, a gravitating object, be it a material body or a black hole, moves as a test body, governed by Dixon’s laws, in what locally appears to be the external spacetime. If finite-size effects are neglected, the object moves on a geodesic of that external metric. However, unlike the metric in the usual equivalence principle, this metric is only effectively external. It is influenced by the object itself, in a way that Thorne and Hartle did not determine. Hence, like Dixon, they left open the difficult task of finding an appropriate division of the physical metric into a self-field and an “external” remainder.

A decade later, motivated by the emerging need for EMRI models, Mino, Sasaki, and Tanaka [19] and Quinn and Wald [20] overcame that hurdle, deriving an equation of motion, now known as the MiSaTaQuWa equation, that included the effect of the object’s own field at first order in perturbation theory. Detweiler and Whiting [21, 22] then showed that the MiSaTaQuWa equation is equivalent to the geodesic equation in a certain linearized vacuum metric, thereby identifying (through first perturbative order) the effective external metric required to complete Thorne and Hartle’s results.

This article reviews the significant progress that has been made on the EMRI problem over the two decades since the derivation of the MiSaTaQuWa equation. Our intention is to provide readers with a non-specialist introduction to the subject (the first of this kind, we believe), covering both foundational and computational aspects. We begin with an elementary-level introduction to self-force theory in curved spacetime: Section 2 reviews the foundations of electromagnetic self-force theory in flat and curved spacetimes, and Sec. 3 then covers the essentials of gravitational self-force theory. In Sec. 4 we survey the computational techniques that have been developed in order to enable the application of GSF theory to the astrophysical EMRI problem. Then, in Sec. 5, we describe a perturbative approach to the problem of the long-term radiative evolution of the orbit (given the GSF), based on a systematic two-timescale expansion. At this point we turn to discuss actual calculations of the GSF and its effects in EMRI systems. Section 6 covers calculations of dissipative effects and the long-term orbital evolution in EMRIs, while in Sec. 7 we focus on non-dissipative physical effects, such as the GSF-induced modification of the rates of periastron advance and spin precession. Then, in Sec. 8, we review work comparing the predictions of GSF calculations with those of full NR simulations and of PN calculations. Such synergistic studies can inform the development of a universal model of binary inspirals, valid across all mass ratios; we discuss this idea, and the “effective one body” (EOB) framework that enables it. We conclude, in Sec. 9, with a summery of progress and a discussion of open problems and prospects.

There already exist several, more expert-oriented review texts on EMRI physics and the self-force. The most comprehensive review of self-force foundations, including self-contained derivations, is the Living Review article by Poisson, Pound and Vega [23] (last updated in 2011). There is a more recent, pedagogical review of foundations by A. Pound [24], which covers also more recent work of importance. A. Harte’s [25] reviews the non-perturbative approach to the problem of motion in GR, carrying on Dixon’s programme. Computational methods for EMRIs have been reviewed by Barack in [26] and, more recently, by Wardell in [27]. For reviews of EMRI science with LISA, see, for example, [28, 29, 7].

## 2 Electromagnetic self-force in flat and curved spacetimes

In order to understand GSF physics, it is worthwhile to examine the ways in which it differs from Newtonian physics—and the perhaps surprising ways in which it remains the same. Consider the Newtonian analog of an EMRI: the Kepler problem, specifically the case in which a smaller body, such as a planet, orbits a much larger one, such as the Sun. We learn early in our physics education that if the smaller body is perfectly spherical and of uniform density, then it can be treated as a point mass. It creates a gravitational potential111 To keep expressions simple, throughout this review we use geometric units in which . , which diverges at its location, . But we also learn that it does not “feel” the field created by this potential; instead, it only responds to the external field of the larger body, obeying the equation of motion

 md2→zdt2=−m→∇Φext(→z), (1)

where and is the trajectory of the larger body. Finally, when we solve the equation of motion (1), we learn that the (bound) solutions are eternal, periodic ellipses.

To a large extent, GSF physics describes the breakdown of each of these results, and in this review we will detail how that breakdown occurs. However, we will also frequently emphasise the countervailing fact: that so long as its various elements are appropriately generalized, much of the Newtonian picture remains remarkably valid.

### 2.1 Nonrelativistic self-force

Perhaps some years after learning about the Kepler problem, we first encounter a self-force, through which an object does “feel its own field”, in the case of a nonrelativistic accelerating charge in flat spacetime. At first, the facts in electromagnetism appear very much the same as in Newtonian gravity: if a point charge is static, it produces a Coulomb potential , just like the gravitational potential; and if we can treat it as a test charge, then it feels only the external fields, just like the point mass does, obeying the Lorentz-force law

 md2→zdt2=→Fext:=q(→Eext+→v×→Bext), (2)

where . However, this is no longer the case if we stop treating it as a test charge. If we actually take into account the change in the field due to the particle’s acceleration, we find the motion obeys the Abraham-Lorentz equation:222 The force is often written as , making the equation of motion third order in time and leading to physically pathological solutions. We instead write it in the “order-reduced” form [30] (sometimes called the Landau-Lifshitz equation), in which is replaced by its leading approximant . Solutions to the order-reduced equation approximate solutions to the third-order equation in a meaningful sense, but they are physically well behaved [31, 32, 33]. Moreover, the order-reduced form is in fact the correct one from a more fundamental perspective. The third-order form is derived from treating the charge as an exact point particle. The order-reduced form, on the other hand, automatically follows from considering an asymptotically small but extended charge distribution [34]. As we discuss below, here we take the position that in a classical theory, point particles are only ever approximations to extended objects, and so we favour the order-reduced form as a matter of principle.

 md2→zdt2=→Fext+23q2md→Fextdt. (3)

The second term is a self-force. More specifically, it is a radiation-reaction force. Unlike the Coulomb potential of the static charge, the Liénard-Wiechert potentials of the accelerating charge contain an unbound piece, which carries energy-momentum out to infinity in the form of radiation. That emission causes a recoil, pushing the particle in the opposite direction. Because of this effect, Eq. (3) differs from Newtonian gravity not only in the presence of a self-force, but in the fact that the self-force is dissipative. For that reason, the classical hydrogen atom, consisting of an electron in orbit around a proton, is famously unstable: due to its emission of radiation, the orbiting electron would lose energy, causing it to spiral inward until the atom collapsed. Hence, it is impossible to construct an electromagnetic analog of the Kepler problem.

This connection between local and global effects is an important theme in self-force theory. Locally, the particle’s energy can only be changed by the self-force. But the self-force removes an amount of energy from the particle precisely equal to that carried off to infinity in the form of radiation. In fact, the most straightforward way of arriving at Eq. (3) is by finding the simplest force that ensures this energy balance.

A more rigorous way of deriving Eq. (3), and one which introduces a second central theme, is by considering a small charge distribution and then taking the limit as it shrinks to zero size in a self-similar way, such that its charge, mass, and size all go to zero [34]. The notion of a point charge, and in particular a test charge, arises as the leading nontrivial approximation in this limit; the Abraham-Lorentz force (along with, in general, some finite-size effects) appears at the first subleading order. This type of limiting procedure plays a crucial role in self-force theory.

### 2.2 Relativistic self-force in flat spacetime

Despite our description of it, the Abraham-Lorentz equation in the form (3) does not evince a particularly direct relationship between the self-force and the particle’s field. To obtain a more direct, physically compelling picture, and to begin to recover some of the Newtonian description, it is convenient to consider the relativistic generalization of Eq. (3), known as the Abraham-Lorentz-Dirac equation:

 mD2zμdτ2=Fμext+23q2m(δμν+uμuν)DFνextdτ. (4)

Here is the covariant form of the Lorentz force, is the particle’s proper time, its four-velocity, and the covariant derivative along its worldline.

Like the Abraham-Lorentz force, the self-force in Eq. (4) is dissipative. And like Eq. (3), Eq. (4) can be surmised from a simple conservation law, as Dirac did [35]—specifically, from the conservation of stress-energy inside a small tube around the particle’s worldline. Alternatively, it can be rigorously derived from conservation of stress-energy of a small, extended charge distribution, using the limiting procedure mentioned above [34]; a second-order extension of the equation has also been recently derived using the same method [36]. This, besides being more physically compelling, bypasses the need for an infinite mass renormalization, which is essential in Dirac’s derivation. However, we are primarily interested in the form the equation takes, not in how it is derived. Specifically, we wish to establish how the radiation-reaction force in it relates to the particle’s own field.

We first examine the form of the field. If the Lorenz gauge condition is satisfied, then the potential sourced by the particle satisfies the wave equation

 □Aμ=−4πjμ, (5)

where is the flat-space d’Alembertian, is the metric of flat spacetime, and

 jμ=quμδ3(→x−→z)ut (6)

is the particle’s charge-current density. The standard retarded and advanced solutions to this equation are

 Aμ±(x)=∫Gμ±μ′(x,x′)jμ′(x′)d4x′, (7)

where, in Cartesian coordinates ,

 Gμ±μ′=δμμ′δ(t−t′∓|→x−→x′|)|→x−→x′| (8)

are the retarded (upper sign) and advanced (lower sign) Green’s functions for , and is the spacetime volume element. Primed indices correspond to tensors at . Due to the delta function in Eq. (8), the retarded and advanced solutions are entirely determined by the state of the particle at the retarded and advanced time, respectively; see Fig. 1.

Of course we are primarily interested in the physical, retarded solution. It contains both time-symmetric and time-antisymmetric pieces, which we can obtain by splitting the retarded Green’s function into corresponding pieces:

 Gμ+μ′=GμSμ′+GμRμ′, (9)

where

 GμSμ′=12(Gμ+μ′+Gμ−μ′) (10)

and

 GμRμ′=12(Gμ+μ′−Gμ−μ′). (11)

is a symmetric Green’s function, satisfying and . , on the other hand, is an antisymmetric homogeneous solution, satisfying and . (Note that we use the symbols and interchangeably to label a point.) Substituting this split into Eq. (7) gives us the corresponding split of the retarded field,

 Aμ+=AμS+AμR. (12)

The singular field is a relativistic generalization of the Coulomb field. It satisfies Eq. (5), it is time-symmetric, and locally, near the particle, it behaves as , becoming singular at the particle’s location. The regular field , on the other hand, is an unbound field. It satisfies the homogeneous equation

 □AμR=0, (13)

it is time-antisymmetric, and it is regular (actually, smooth) at the particle’s location.

Now return to the equation of motion (4). By evaluating the Faraday tensor associated with the regular field, , on the particle, and comparing the result to the right-hand side of Eq. (4) (before the order reduction described in footnote 3), one finds that the equation of motion can also be written as

 mD2zμdτ2=Fμext+qFμRνuν. (14)

In line with its interpretation as a generalization of the Coulomb field, does not appear in the equation of motion. But exerts an ordinary Lorentz force on the particle. Combined with the fact that is a homogeneous field, this suggests that from the particle’s perspective, is indistinguishable from an external field. If we define the effective external field (and associated Faraday tensor ), then the equation of motion is simply the Lorentz-force law

 mD2zμdτ2=q~Fμextνuν. (15)

Equation (15) provides an alternative description of the self-force, one that is not tied to dissipation or radiation-reaction, and one much closer to the Newtonian picture: whatever its field does, a particle is always governed by the Lorentz force exerted by what it perceives to be the “external” field. However, we stress that this is only an effective external field. Away from the particle, is not physical. It depends not only on the retarded point on the particle’s worldline, but also on the advanced point (see Fig. 1). Hence, it is not causal. Only in the limit to the particle, where the retarded and advanced points merge, does it become physically meaningful.

After the inception of the EMRI modelling programme in 1996, this idea of a particle (or small object) behaving as a test particle in an effective external field was most famously advocated by Detweiler. It occupies a central place in self-force theory, and we will return to it at every stage of this review.

### 2.3 Electromagnetic self-force in curved spacetime

The move to curved spacetime brings a major change to the physics of the problem. In flat spacetime, waves propagate at the speed of light, along null rays: but in curved spacetime, waves scatter off the spacetime curvature, causing solutions to propagate not just on lightcones, but also within them. Because of this, the retarded potential depends not only on the state of the particle at the retarded point , but on its state at all prior points , as illustrated in Fig. 2. This causes an important change to the equation of motion (4), which becomes

 mD2zμdτ2=Fμext+q2(δμν+uμuν)(23mDFνextdτ+13Rνρuρ)+2q2uν∫τ−−∞∇[μGν]+μ′uμ′dτ, (16)

where is the Ricci tensor of the spacetime, and is the retarded Green’s function for the curved-space wave equation [Eq. (17), below]. The final term in this equation is a “tail”. It is an integral over the entire past history of the particle, up to , accounting for all the waves that have scattered back to the particle after having been created by it in its past.

Equation (16) was first derived by DeWitt and Brehme [37] (as corrected by Hobbs [38]) using the same approach as Dirac, considering conservation of stress-energy within a small tube around the particle. Like in the case of flat spacetime, the most rigorous derivation follows from considering the point-particle limit of an extended charge distribution; this has been done by Harte [39, 25], who derived the exact equation of motion of an arbitrary charge distribution and then took the point-particle limit. But also like in the flat-space case, for the moment we are more interested in the form of the equation than its derivation.

Despite the changes in the physics of the solution, the fundamental picture from the preceding section remains valid: the particle feels a Lorentz force due to an effective external field , and the equation of motion (16) can be rewritten in the form (15).

To motivate the form of the regular field , we begin with the field equation that the particle’s potential satisfies. In the Lorenz gauge, it reads

 □Aμ−RμνAν=−4πjμ, (17)

where , and is the metric of the spacetime. The retarded solution is given by , where is a covariant volume element, with being the determinant of at the integration point . We wish to split this solution into appropriate singular and regular pieces in analogy with Eq. (12). We first note that in curved spacetime, the self-force can plainly not be described as the Lorentz force exerted by the potential : just as the retarded solution depends on the entire past history of the particle, the advanced solution depends on its entire future history, as shown in Fig. 2. This acausality persists even in the limit to the particle, unlike that of the regular field in flat space, and so it cannot give rise to the correct, physical self-force. An appropriate alternative was found by Detweiler and Whiting [22], who defined the modified two-point functions

 GμSμ′=12(Gμ+μ′+Gμ−μ′−Hμμ′) (18)

and

 GμRμ′=12(Gμ+μ′−Gμ−μ′+Hμμ′). (19)

Here is a symmetric homogeneous solution, satisfying and . It is chosen such that has support at all points except those in the chronological past of . This ensures that the corresponding field has no dependence on points in the chronological future; once again, see Fig. 2.

With this choice of , the singular and regular fields and possess all of the same key properties as in flat spacetime. satisfies the inhomogeneous Eq. (17), and near the particle it behaves as a Coulomb field. satisfies the homogeneous wave equation and is smooth at the particle. Off the particle, it is acausal, depending on all points on the worldline prior to the advanced point . But like in flat spacetime, when evaluated on the particle, it becomes causal, depending only on points in the past. Most importantly, evaluating and its derivatives on the particle reveals that the DeWitt-Brehme equation, (16), is equivalent to Eq. (14) and therefore to Eq. (15).

However, there is one significant change from flat spacetime. While the Green’s function remains symmetric in its arguments and indices, does not remain antisymmetric, due to the presence of in Eq. (19). Because of this, unlike the purely dissipative self-force in flat spacetime, the self-force in curved spacetime has a conservative piece; it is no longer simply a radiation-reaction force. But despite this change, and despite the more complicated physics of wave propagation in curved spacetime, the essential picture remains unchanged: the particle obeys the Lorentz-force law (15) in what it perceives to be the external field.

## 3 Gravitational self-force and the generalized equivalence principle

### 3.1 Perturbation theory in GR and the failure of the point particle description

In our discussion of the electromagnetic self-force, we said that the results are rigorously justified by considering the limit of an asymptotically small charge distribution, with the point particle and its field emerging from that limit. However, in the case of a gravitating source in GR, this is no longer true: the point-particle approximation fails; the field of the small object cannot, in general, be expressed as that of a point particle.

This failure stems from the nonlinearity of the Einstein field equations. From a physical perspective, we know the Einstein equations imply that a sufficiently dense mass distribution will collapse to form a black hole, not a point particle. From a mathematical perspective, we know that the Einstein equations with a point-particle source do not have a well-defined solution within any suitable class of functions [40, 41].

Let us examine how this failure manifests itself in our problem. We consider an object of mass moving in a spacetime with a much larger external length scale ; in an EMRI, can be the mass of the large black hole, for example (in geometrical units where mass has dimensions of length). Now we wish to take advantage of the separation of scales by expanding the exact metric of our system, , in the limit . The metric reads

 gμν=gμν+ϵh(1)μν+ϵ2h(2)μν+O(ϵ3), (20)

where we have introduced as a formal expansion parameter to count powers of ; it will be set to unity at the end of a calculation. The zeroth-order term in Eq. (20), , is referred to as the background metric. In the case of an EMRI, it is the metric of the large black hole. The corrections describe the gravitational perturbations created by the small object.

The metric (20) must satisfy the Einstein equation , where is the Einstein tensor of the spacetime and is the stress-energy tensor of the system’s matter content. For simplicity, suppose that the small object represents the only matter, such that is the stress-energy tensor of the small object itself. If we substitute (20) into the Einstein equation, then the left-hand side becomes

 Gμν[g]=Gμν[g]+ϵδGμν[h(1)]+ϵ2(δGμν[h(2)]+δ2Gμν[h(1)])+O(ϵ3), (21)

where is linear in and has the schematic form . Let us also suppose that in this limit, is approximately that of a point particle, such that , where is the stress-energy of a point mass moving in the background . (We will momentarily delay the question of whether this makes sense in the case of a black hole, for which identically vanishes at all points in the spacetime manifold.)

Through first order in , no fundamental problem arises. Keeping only the first-order terms in the Einstein equation, we arrive at the linearized Einstein equation with a point-particle source:

 δGμν[h(1)]=8πT(1)μν. (22)

This equation is analogous to Eq. (17) for the electromagnetic potential. Its solutions can be expressed in terms of Green’s functions, just as in the preceding sections. Like in the electromagnetic case, the retarded field splits into Detweiler-Whiting singular and regular fields. The singular field behaves as near the particle, where is a measure of distance to the particle’s worldline. The regular field is again a smooth vacuum solution that contains the backscattered waves that arise from propagation within, not just on, the light cones of the background spacetime.

But now suppose we read off the second-order term in the Einstein equation. It is

 δGμν[h(2)]=8πT(2)μν−δ2Gμν[h(1)]. (23)

The second-order perturbation is sourced by the quadratic combinations of in , which generically behave like near the particle.333 We will, however, mention a fine-tuned way of skirting this generic behavior in Sec. 3.6. This singularity is too strong to be integrated, and because it is constructed from a quadratic operation on an integrable function (as opposed to a linear one), it is not even well defined as a distribution. The other source in Eq. (23), , if it is well defined at all, must be a distribution solely supported on the particle’s worldline. Hence, it cannot cure ’s nondistributional divergence, and the field equation (23) is itself ill defined.

Because of this failure of the point-particle treatment, in gravity we must face the small object’s extended size head on. However, since the object is small, we still wish to avoid directly including its potentially complicated internal dynamics in the Einstein equations. A principal goal of self-force theory is therefore to generalize the point-particle approximation: to reduce the object to a few “bulk” properties (such as mass and spin) supported on a worldline, without representing it as a delta function stress-energy tensor. Like the point-particle approximation in electromagnetism, this reduction is achieved by considering an extended object in the limit of zero mass and size. Of course, a critical ingredient in the reduction is an equation of motion for the representative worldline, and it is there that the GSF appears.

Before proceeding to describe the limiting procedure and its results, we first note one of its crucial outcomes: at linear order, it establishes that the point-particle approximation is valid. That is, Eq. (22) is correct even though Eq. (23) is not. It is also correct even if is not the first-order approximation to ; it remains valid even if the small object is a black hole and identically vanishes, for example. Given these facts, much of the GSF literature takes Eq. (22) as its starting point, and from that point it describes the GSF in a manner precisely analogous to the electromagnetic case. However, rather than presenting that description immediately, we will instead examine how it emerges from the more fundamental picture of an asymptotically small object.

### 3.2 Point particle limits and multipole moments

The key idea in generalizing the notion of a point particle is to focus not on the small object itself, but on the gravitational field in its immediate neighbourhood. Rather than thinking of the point-particle approximation as a statement about the object’s stress-energy tensor, we can take it to be a statement about the object’s field.

To illustrate this idea, we return to the simpler context of Newtonian gravity. Consider an isolated material body described by a mass distribution . It sources a Newtonian potential satisfying

 ∂i∂iΦ=4πρ, (24)

where we have adopted Cartesian coordinates . Written in terms of the Green’s function , the potential reads

 Φ(→x)=∫G(→x,→x′)ρ(→x′)d3x′. (25)

Now, suppose we are only interested in the potential outside the body. Further suppose that the body is small, with a characteristic size , and choose any representative worldline in its interior. We can then expand in the integrand of Eq. (25) as

 G(→x,→x′)=−1r−δx′inir2−(δx′iδx′j−r′2δij)ninjr3+O(ℓ3/r4), (26)

where , , and . Equation (25) then becomes an expansion in terms of the body’s multipole moments:

 Φ=−mr−minir2−mijninjr3+O(mℓ3/r4), (27)

where is the body’s mass, its dipole moment, and its quadrupole moment. The dipole moment measures the distance between and the body’s center of mass. Hence, we can set by choosing the representative worldline to be the body’s center of mass.

Equation (27) is the potential outside a small but extended body. But suppose we took it to be the potential at all points off the body’s worldline [i.e., ]. From Eq. (25), one can easily see that it then corresponds precisely to the potential sourced by a “structured” point particle with a mass distribution

 ρ=mδ3(xi−zi)+mi∂iδ3(xi−zi)+mij∂i∂jδ3(xi−zi)+… (28)

If we include only the first term, this is a standard, structureless point mass. The potential it sources in that case is identical to one sourced by a perfectly spherical body, with at the body’s center.

The above expansion procedure gives us a precise way of thinking of point particles as approximations to extended objects. But it also demonstrates that, in a meaningful sense, the delta function source (28) is not fundamental to the approximation. Rather than thinking of the mass distribution as the defining characteristic of the approximation, we can instead think of the singularity in the potential, , as primitive; the delta distribution is simply an intermediary that encodes this behavior of the field. This way of thinking, far from being novel, is how point particles were thought of prior to Dirac’s invention of his delta function, and it played a significant role in early derivations of equations of motion in GR [42, 10, 43, 11]. Unlike a delta function, it survives in the nonlinear arena of Einstein’s theory.

It is worth stressing that here we take the view that a point particle is purely an approximation to an extended object, with no fundamental status on its own. It emerges from an asymptotic expansion in the limit of small size, —or equivalently, large relative distance, . As we get closer to the object, we “see” more of its structure, in the form of sensitivity to its higher multipole moments. If we are very close, at distances comparable to the object’s size, , then the approximation breaks down entirely.

### 3.3 Matched asymptotic expansions and the local form of the metric

Even in GR, any sufficiently well-behaved stress-energy tensor can be reduced to an infinite set of multipole moments, as Dixon showed [13]. But unlike in Newtonian gravity, there is no simple way of translating these moments into an expression for the gravitational field outside the object. Fortunately, there is actually no need to do so: we can obtain the field in a small region outside the object, expressed in terms of a discrete set of multipole moments, directly from the field equations. Instead of specifying a stress-energy tensor, we need only specify the object’s moments.444 Of course, if one wishes to model a particular type of body with a specific , then one must infer its moments from that . However, that becomes, in some sense, an independent problem. We achieve this using the method of matched asymptotic expansions [44, 45]. This method, which derives from singular perturbation theory, has become a standard means of obtaining equations of motion of small objects; see Refs. [17, 46, 18, 47, 16, 48, 49, 19, 21, 50, 51, 52, 53, 54, 55, 56, 57] for a sample. In particular, this method was used in the first derivation of the MiSaTaQuWa equation [19], and it has been the basis of most later foundational work. Our presentation of the method follows Refs. [55, 58].

Matched expansions are used when an ordinary expansion breaks down in a small region. The key idea of the method is to introduce a second limit, one which magnifies this problematic region, and to perform a second expansion in that new limit. In our case, this region is a neighbourhood of the small object itself. As we mentioned in the previous section, the point-particle approximation breaks down at a distance from the object. The expansion (20) of the metric fails at that same distance. This is intuitively sensible: Eq. (20) treats the small object’s field as a small perturbation of the external universe, but sufficiently near the object, its gravity will dominate over that of external sources, and it cannot be treated as a small perturbation.

Let us assume the object is compact, such that , and now counts powers of or . If we think of as being of order 1, then the region , which we call the “body zone”, is equivalent to . To zoom in on this region, we introduce a scaled distance . In terms of this scaled distance, the body zone corresponds to . Hence, we can zoom in on the body zone by taking the limit at fixed , and we perform our new expansion in this limit. This contrasts with the original, ordinary expansion (20), which is performed in the limit at fixed . That limit shrinks the object to zero size while holding external lengths fixed. The limit at fixed instead keeps lengths of size fixed—in particular, the size of the small object—and blows up external lengths out to infinity.

We shall illustrate the method in the Newtonian case. Suppose we have a small body of mass density in an external gravitational field. Again assume the object is compact (but otherwise arbitrarily structured), and take to be the typical scale over which the external field varies. In a region outside the external sources, the total field satisfies . Outside all sources, including the body, it satisfies the homogeneous equation . If we expand in powers of at fixed , we have , analogous to Eq. (20). If we expand in powers of at fixed , we have instead . The background field in the first expansion, , is the field due to external sources in the absence of the body; the background in the second expansion, , is the field of the small body itself in the absence of external sources. Let us call the first expansion an “outer expansion” and denote it , and the second expansion an “inner expansion” and denote it , where represent the expansion operations.

Since and are both expansions of the same function, they must agree with (or “match”) each other when suitably compared. To make the comparison, we perform an inner expansion of , giving us , and we perform an outer expansion of , giving us . When written as functions of and , each of these operations yields a double expansion in the limit and , with the forms

 EinΦout =∞∑n=0∞∑p=−∞rpϵnΦn,p, (29) EoutΦin =∞∑n=0∞∑q=−∞(ϵr)qϵn^Φn,q, (30)

where the coefficients are independent of and . For a sufficiently well-behaved , we will have ,555 For discussions of the criteria that guarantee this condition is satisfied, see Ref. [44]. Reference [53] uses a particularly clear, though unnecessarily strong set of criteria. implying the matching condition

 Φn,p=^Φn+p,−p. (31)

Equation (29) represents the behaviour of the outer expansion in the limit , very near the worldline relative to the external length scale , while Eq. (30) represents the behaviour of the inner expansion in the limit , very far from the body relative to the internal length scale . We can expect the resulting double expansions to be accurate when and are both small. As illustrated in Fig. 3, this range, , describes a “buffer region” between the body zone and the external universe. It can be thought of as a “local far-field” region, being simultaneously in the small body’s local neighbourhood () and in its far field ().

Without performing any calculations, we can use the matching condition (31) to constrain the forms of and . Since there are no negative powers of in the outer expansion (i.e., no for ), Eq. (31) implies that vanishes for ; likewise, since there are no negative powers of in (i.e., no for ), Eq. (31) dictates that vanishes for . Therefore we have

 ^Φ0 =^Φ0,0 +ϵr^Φ0,1 +ϵ2r2^Φ0,2 +O(ϵ3/r3), (32a) ^Φ1 =r^Φ1,0 +ϵ^Φ1,1 +ϵ2r^Φ1,2 +O(ϵ3/r2), (32b) ^Φ2 =r2^Φ2,0 +ϵr^Φ2,1 +ϵ2^Φ2,2 +O(ϵ3/r), (32c)

and

 Φ0 =Φ0,0 +rΦ0,1 +r2Φ0,2 +O(r3), (33a) Φ1 =1rΦ1,−1 +r0Φ1,0 +rΦ1,1 +O(r2), (33b) Φ2 =1r2Φ2,−2 +1rΦ2,−1 +r0Φ2,0 +O(r). (33c)

The matching condition further dictates that the coefficients in the th row of Eq. (32) match, term by term, those in the th column of Eq. (33). However, we often only care about the particular case of the first row and first column.

In our context, we are primarily interested in the outer expansion; we only use the inner expansion to inform the outer. The form (33) of the outer expansion near the worldline is valid regardless of what field equation satisfies. We can now further constrain it using the field equation. Substituting Eq. (33) into , one finds the familiar result that the terms of the form (with ) must be linear combinations of spherical harmonics , and those of the form (with ) must be linear combinations of harmonics . We can also write this as and for some symmetric and trace-free tensors and .

Now let us interpret the terms in the expansion. With suggestive renamings of the tensors , we can conveniently sort the terms into two groups,

 ΦS1 =−m(t)r, (34a) ΦS2 =−mi(t)nir2−δm(t)r, (34b) ⋮

and

 ΦRn=ϕ(n)(t)+ϕ(n)i(t)xi+ϕ(n)ij(t)xixj+…, (35)

with the total field given by their sum, . Although derived in a very different way, these two groups represent (Newtonian versions of) Detweiler-Whiting singular and regular fields. We can think of as the body’s self-field. It is characterized by a set of multipole moments and corrections to them, with containing moments up to . We can identify these moments as those of the body itself, without having to integrate over the body’s interior. To understand this, note that the most singular term at a given order , , corresponds to a term in the expansion (32a). That expansion is identical to Eq. (27), the far-field expansion of the field of an isolated body, and instead of defining the body’s moments as integrals over its interior, we can define them directly as the coefficients in that far-field expansion. On the other hand, the R field defined from Eq. (35) has the form of a Taylor expansion of a smooth external field, with no direct dependence on the body’s moments. Hence, we can think of as an effective external field. In Sec. 3.6 we will present an example of how the fields actually arise in a Newtonian context.

From the above analysis, we see that solely from the vacuum field equation and the matching condition, we can tightly constrain the local form of the field outside the body: it is given by Eqs. (34) and (35), expressed in terms of the body’s multipole moments and some smooth fields . We can freely specify the multipole moments; this is equivalent to specifying the body’s material composition. We can also freely specify the fields ; this is equivalent to specifying the external environment. However, it is more practical to leave the free at this stage. They will be determined by solving the global problem with whatever external sources are present.

With the local form of the field known, we can now forget about the matching procedure and extend our solution down to as we did when discussing the point-particle approximation. This effectively replaces the physical field inside the body with a fictitious one. But crucially, it does not alter the physical field outside the body. At , the S field becomes singular, while the R field remains a smooth solution to the vacuum equation , in precise analogy with the Detweiler-Whiting fields.

Although the calculations become far more involved in GR, all of the essential ideas remain the same. We assume an outer expansion of the form (20) along with a complementary inner expansion at fixed . The matching condition dictates that the fields have forms precisely analogous to those in Eq. (33). After further constraining the form of the perturbations using the vacuum Einstein equations, one finds that they can again be conveniently written in the form , where in analogy with Eq. (34), the S field has the schematic form [55, 58]

 hS(1)μν ∼mr+O(r0), (36)
 hS(2)tt∼hS(2)ij ∼m2+minir2+O(1/r), (37a) hS(2)ti ∼ϵijksjnkr2+O(1/r), (37b)

and so on at higher orders. [Here, in the curved-space case, are specialized to be comoving coordinates centred on .] In analogy with Eq. (35), the R field has the form

 hR(n)μν=ϕ(n)μν(t)+ϕ(n)μνi(t)xi+ϕ(n)μνij(t)xixj+… (38)

As is clear from the presence of the object’s spin in , the forms of these fields do differ somewhat from the Newtonian ones. In the Newtonian case, the S field was written entirely in terms of “mass moments” ; in GR there are an additional set of moments , called “current moments”, of which is the first. In , both the mass and current quadrupole moments, and , would appear; in , the octupole moments; and so forth. Similarly, the R field is more complex than its Newtonian analogue. In Newtonian gravity each of the coefficients in the expansion (35) was symmetric and trace-free; in GR the tensors are not all symmetric and trace-free, though they are uniquely defined in terms of a certain set of symmetric-trace-free “seed” tensors, in a procedure best explained in Refs. [58, 59].

These differences aside, the S and R fields’ properties closely parallel those of the Newtonian fields. In particular, the singular field carries local information about the object; every term in its expansion, at all orders in and , is proportional to one of the object’s multipole moments (or to some nonlinear combination of them). The regular field is a smooth vacuum perturbation for all , locally independent of the object; prior to imposing boundary conditions, no terms in the series (38) depend on any of the object’s moments. One can also show that, like the Detweiler-Whiting regular field in electromagnetism, is causal on the worldline [59]. We can combine it with the external background to form a metric that we can interpret as the effective external geometry. This interpretation will be further bolstered when we consider the equation of motion.

Nevertheless, though the S and R fields have desirable properties, one should keep in mind that only their sum represents a truly physical field. We could have split that field into alternative choices of S and R fields, or foregone the split altogether. For concreteness we have adopted the definitions in Refs. [56, 58], but there are many (in fact, infinitely many) other possible choices. Unlike in the Newtonian case, the singular fields in GR, under most sensible definitions, will contain nondivergent terms proportional to with , and because of this, there is no obviously preferred singular-regular split. Ultimately, one can make any convenient choice and derive useful equations in terms of that choice. In Sec. 3.5 we will comment on some alternative choices that have been made in the literature.

Before proceeding, we recapitulate the essential point of this section: Eqs. (36) and (37), along with the Taylor series for , represent the form of the metric in the buffer region outside any compact object. This form is valid regardless of whether the small object is a material body or a black hole. If it is a material body, then is some measure of distance to a representative worldline in the body’s interior, as in the Newtonian case. If the object is instead a black hole, then clearly there is no representative worldline in its physical interior. However, we can associate a worldline with it based on the behaviour of the field outside of it, with serving to define that worldline. This associated worldline emerges from the limiting procedure, existing not in the physical spacetime but in the background spacetime with metric (or equivalently, in the effective spacetime with metric ).

### 3.4 Point particles, punctures, and effective sources

As we described in Sec. 3.1, a traditional point-particle description fails at nonlinear orders. However, after the local analysis of the last section, we are now equipped to adopt the more general viewpoint of Sec. 3.2: rather than approximating the object as a delta function stress-energy tensor, we replace it with a local singularity in the metric. We call this singularity a puncture in the spacetime.

To understand how we use this idea, first begin with the vacuum Einstein equations outside the object. At first and second order, they have the form of Eqs. (22) and (23) with the stress-energy terms set to zero,

 δGμν[h(1)] =0, (39) δGμν[h(2)] =−δ2Gμν[h(1)]. (40)

The physically correct solutions to these equations satisfy a free boundary-value problem: in a small region around , they must satisfy Eqs. (36)–(37), which we can think of as boundary conditions; and the location of that boundary region (or equivalently, of ) is free to move in response to the solution.

We solve this free boundary value problem by taking the following steps. We first extend the locally derived fields down to all , replacing the physical field in the object’s interior but leaving intact the physical field in its exterior. The fields then satisfy Eqs. (39)–(40) for all . But they do not satisfy these equations on the domain ; and for , they cannot (in general) be made to satisfy distributional equations on that domain. To obtain equations that can be solved on that domain, we next change our field variable from to, essentially, . More precisely, we introduce a “puncture field” by truncating the expansions (36) and (37) at some finite order, and we then solve for the “residual field”